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" ) )" /( # 5OLUTION "/ ") D/ (4.52 + 242 ) - 14 = 34.8 cm " : ") 8 /( " % " " " ") ( ( $ &1 96 %! % & " /" ( ( /( " %! % & . q % 4.5 q $% 24 ( ' ") " )" $ / " % * ( "( "( * )" )" ' )" $ " AB ' 69.62 + 69.62 + 2 ¥ 69.6 ¥ 69.6 ¥ cos (2 ¥ 10.62∞) = 136.8 N ' ¥ Bmufsobujwfmz 'XAMPLE!drci #% &°®¼¹¬ɏŶȍŶŻ¨ɏº¯¶¾ºɏ»¾¶ɏ½¬ª»¶¹ºɏ0ɏ¨µ«ɏ1ȍɏș¨Țɏ$¬»¬¹´°µ¬ɏ»¯¬ɏ´¨®µ°»¼«¬ɏ2ɏ¶ɏ»¯¬ɏ ½¬ª»¶¹ɏº¼´ɏ2ɏǛɏ0ɏǖɏ1ȍɏș©Țɏ$¬»¬¹´°µ¬ɏ»¯¬ɏ¨µ®³¬ɏaɏ©¬»¾¬¬µɏ2ɏ¨µ«ɏ»¯¬ɏ·¶º°»°½¬ɏ¿Ɋ¨¿°ºȍɏ șªȚɏ7¹°»¬ɏ2ɏ¨ºɏ¨ɏ½¬ª»¶¹ɏ°µɏ»¬¹´ºɏ¶ɏ»¯¬ɏ¼µ°»ɏ½¬ª»¶¹ºɏ°ɏ¨µ«ɏ±ɏ¨µ«ɏ»¯¬µɏ¾¹°»¬ɏ¨ɏ¼µ°»ɏ½¬ª»¶¹ɏ¨³¶µ®ɏ»¯¬ɏ½¬ª»¶¹ɏ º¼´ɏ2ȍɏș«Țɏ$¬»¬¹´°µ¬ɏ»¯¬ɏ´¨®µ°»¼«¬ɏ3ɏ¶ɏ»¯¬ɏ½¬ª»¶¹ɏ«°Ĺ¬¹¬µª¬ɏ3ɏǛɏ0ɏɌɏ1ȍ ()* $+ ɏ ŸŻ &¶¹ª¬ɏ3Àº»¬´ºɏ¨µ«ɏ2¬º¼³»¨µ»ºɏ y y y S P = 4 units P = 4 units P j i 75° a 45° x 30° R 105° –Q x x b Q Q = 3 units Q = 3 units (a) (b) (c) $ 5OLUTION ! " # % & $ q % + $& ' % *& , ' / " / b \ 7 " b P sin q 4 sin 75∞ = = 0.957 Q + P cos q 3 + 4 cos 75∞ a b:% $% * " a' ' 7 7 " " " " a :% fi q // : b &* $% * H ! $#' / " && ¥ %* ' R (5.43i + 1.328 j) = ' 5.59 : ' : 7 , ; : *& & 7 " %* H &$ ' // " / , P 2 + Q 2 + 2 PQ cos q = 42 + 32 + 2 ¥ 4 ¥ 3 ¥ cos 75∞ = 5.59 units ' 7 * " ! % ?A ; $# , * 7 P 2 + Q 2 + 2 PQ cos q = 42 + 32 + 2 ¥ 4 ¥ 3 ¥ cos 105∞ = 4.33 units , < Bmufsobujwfmz + " - , 7 7 7 A 7 a " / : / > $–$& %–:% " ' $–$& ' %–:% &– / / ' & ?A ; , " < a / " $–$& :%–:% / % ' $ %% " < < < ? @A $#' ?A ; 'j H # $#' ! % ?A ; , < $ %%–; ?A ; % ?A ; < , < Ÿżɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº 'XAMPLE!drcj 2¬º¶³½¬ɏ»¯¬ɏ¶¹ª¬ɏ&Ȏɏº¯¶¾µɏ°µɏ&°®ȍɏŶȍŶż¨Ȏɏ°µ»¶ɏ¨ɏª¶´·¶µ¬µ»ɏ·¬¹·¬µ«°ª¼³¨¹ɏ»¶ɏ!"ɏ ¨µ«ɏ¨ɏª¶´·¶µ¬µ»ɏ·¨¹¨³³¬³ɏ»¶ɏ"#ȍ ()* %+ D F F N Q 3 4 C B A q 25° 25° A g P M q C a P M B B (a) q b b a D Q F g (b) (c) % 5OLUTION B % & " ; + , % $ a : a E q " %H$ / % CD CFG? - - % ;* : & fi & @I " 3 Q sin 65∞ Q ¥ 0.906 = = 4 P + Q cos 65∞ P + Q ¥ 0.423 " % ' & ' %& *;& & ' & ' & % BmufsobujwfmzJ b g / b q *;&& & & ;& ## G ? 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' " % " % 5 wLx wx % 5 % % wL % wL2 8 fi % fi " % # wL % $ %- ' ' ! wL % ' ! ;#! fi wL ! ¥!6 !¥ % % ! 0 B C 7/!!!Tjnqmz!Tvqqpsufe!Cfbn!xjui!VEM!pwfs!Foujsf!Tqbo! " '* +!, " # $ %./. ! 0 +!, ' ' !3% " # $ %4 4 ' ' " 8 " Parabola + (d ) BMD Gjh/!6/34! % .5. 0 – –Wa/L (b) SFD S wL/2 B A 0 A x (b) 1 wL/2 C + wx M S1 Wb/L 0 wL/2 (a) b + 1 ' ɏ ŵŷŻ "¬¨´ºɏ¨µ«ɏ&¹¨´¬ºɏ 6/9! .1!grj �! EJGGFSFOUJBM!FRVBUJPOT!PG!FRVJMJCSJVNÚSFMBUJPO! CFUXFFO!w-!S!BOE!M! ! %º»¨©³°º¯ɏ»¯¬ɏ¹¬³¨»°¶µɏ ©¬»¾¬¬µɏº¯¬¨¹ɏ¶¹ª¬Ȏɏ ©¬µ«°µ®ɏ´¶´¬µ»ɏ¨µ«ɏ ³¶¨«ȍ ' " ' ' # $ %$ < " ! ' ! + " : # $ .= " # $ %$ 0 " > ! ' !2 ! ? ' %2 % % 2 w dx w M S M + dM B A x VA dx VB dx (a) Simply-supported beam S + dS (b) Free body diagram of small element Gjh/!6/36! % @ # $ %$ 4 A S %5 %6 % 5 " ¥ ! dS dx fi 5 $. 3 ' 0 4 ! ! " S ÚS fi dS % @ x - Ú w ◊ dx fi %5% x ' " x - Ú w ◊ dx x %6 % " 4 ! " ! ! % ! 0 ! B " ;#! 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" # $+ + .4 ¥,4,01 fi . 0 9!lO 4 , ¥ $ 4 , ¥ 0 1 fi . . 0 43!lOn *" &' &6 *" 6 . / . 7 &' 3 8 . 63 .3 %6 0 , 9: ! %. 0 !, ¥ % ; , 0 < 9: &6% 4 kN 1 kN/m B A 4m C 2m 2.5 kN 3 kN (a) 1 kN/m 4 kN 4 kN B 2m MA C D E A A 1.5 m 1m C 6m 2m 0.5 m (a) VA 7.5 (b) 4.5 8 4 + 0 + A B + 0 A B A B C C (b) SFD 2.5 D E (c) SFD 0 A B C – C – –8 –8.25 –15 (d ) BMD Gjh/!6/37 D E 0 – –32 # $$% *" *" ! 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S !>!1;! 10 N VA B 3m 10 Nm 20 Nm 10 N E C D 2.5 m 2.5 m VE 3m (b) FBD of beam Gjh/!6/51 HE ŵŸżɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº 'XAMPLE!grci &¶¹ɏ»¯¬ɏ¹¨´¬ɏ³¶¨«¬«ɏ¨ºɏº¯¶¾µɏ°µɏ&°®ȍɏŹȍŸŵ¨Ȏɏª¨³ª¼³¨»¬ɏ¨³³ɏ»¯¬ɏ¶¹ª¬ºɏ¨ª»°µ®ɏ¶µɏ»¯¬ɏ ´¬´©¬¹ɏ!"#ȍɏ \MP!6/6^ C B C C 0.2 m 0.2 m D 8 kNm0.2 m D D 8 kNm0.2 m D E A B 0.2 m HA F E A VA 0.3 m VC B FBE 0.2 m F VF A HA VA 0.3 m 0.3 m (a) Frame HC (b) FBD of Frame (c) FBD of member AC Gjh/!6/52 5OLUTION! Gjhvsf!6/52 !tipxt!uif!gsff!cpez!ejbhsbn!pg!b!gsbnf!xjui! B!boe! B!bt!sfbdujpo!dpnqpofout!bu! uif!qpjou!B!boe! G!bt!uif!sfbdujpo!bu!G/!Uif!gsff!cpez!ejbhsbn!pg!uif!nfncfs!BCD!xjui!bmm!uif!gpsdft!bdujoh! bu!qpjout!B-!C!boe!D!jt!tipxo!jo!Gjh/!6/52 /!! Bqqmzjoh!frvbujpot!pg!frvjmjcsjvn!up!gsff!cpez!ejbhsbn!pg!uif!gsbnf-!xf!hfu S }G! >!1;! ) B!¥!1/4*!Ï!9!>!1! fi! B!>!37/78!lO S !! >!1;! B!>!1 Up!dbmdvmbuf!gpsdft!bu!kpjout!C!boe!D-!xf!bqqmz!frvbujpot!pg!frvjmjcsjvn!up!uif!GCE!pg!uif!nfncfs!uif! BCD!)Gjh/!6/52 *S }D!>!1;! ) B!¥!1/4*!Ï!) CF!¥!1/5*!>!1 fi )37/78!¥!1/4*!Ï!) 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"/ 9 ! ! fi M 2 = - 0.42( x - 2) - 1.67 x + 10 # # # % < % ¥ ¥ 7 ¥ ! ! ! D D D ! 3 fi C8 ! ! ; B ! + % £!£ , $ % # ! & ' $* M , % , " ! . /0 6 0 # *9/ / * , !!/0 * 89 # 89 / - 9 ! &" ' * ¥ % + 1 S fi ! # + ! & % ! , - fi !# * 7 ¥ ! " fi % £!£ M , % $* HK! HK ¥ HK! 9 ! H K EF + EF HK ¥ ɏ ŵŹŻ "¬¨´ºɏ¨µ«ɏ&¹¨´¬ºɏ 'XAMPLE!grdh !ɏº°´·³Àɏº¼··¶¹»¬«ɏ©¬¨´ɏ¶ɏŻɊ´ɏº·¨µɏ¾°»¯ɏ¶µ¬ɏ¶½¬¹¯¨µ®ɏ¶ɏŶɊ´ɏ³¬µ®»¯ɏª¨¹¹°¬ºɏ¨ɏ ª¶µª¬µ»¹¨»¬«ɏ´¶´¬µ»ɏ¶ɏŷŴŴŴɏ.´ɏ¨»ɏ¨ɏ·¶°µ»ɏŶɏ´ɏ¨¾¨Àɏ¹¶´ɏ»¯¬ɏº¼··¶¹»¬«ɏ¬µ«Ȏɏ ¨µ«ɏ¨ɏ·¶°µ»ɏ³¶¨«ɏ¶ɏŻŹŴɊ.ɏ´¨®µ°»¼«¬ɏ¨»ɏ»¯¬ɏ¨¹ɏ¬µ«ɏ¶ɏ¶½¬¹¯¨µ®ɏ¨ºɏº¯¶¾µɏ°µɏ&°®ȍɏŹȍŹŴ¨ȍɏ#¨³ª¼³¨»¬ɏ º¼··¶¹»ɏ¹¬¨ª»°¶µºɏ¨µ«ɏ·³¶»ɏº¯¬¨¹ɏ¶¹ª¬ɏ¨µ«ɏ©¬µ«°µ®ɏ´¶´¬µ»ɏ«°¨®¹¨´ºɏ¶¹ɏ»¯¬ɏ©¬¨´ȍɏɏ \MP!6/6-!MP!6/8^ 5OLUTION J 9!! - S ¥ J # D$% I% ! / N " I% ' // / # F ' // 9 9 I% % ! 6 ! fi K 411!O 750 N A C B VA 2m ' ! "- // 6 /0 / ! # ! 300 + * C B A D (b) SFD * * . * J # P9 * P9 / # # + ¥ ", " F H F - A C D B – * –1500 –2400 # # ! "Q ! ", H F (c) BMD ¥ Gjh/!6/61 F !; $ R9 I% * # ' 'XAMPLE!grdi 5OLUTION! J # ! $* - "' * K "Q ¥ F !ɏ ©¬¨´ɏ °ºɏ ³¶¨«¬«ɏ ¨ºɏ º¯¶¾µɏ °µɏ &°®ȍɏ ŹȍŹŵ¨ȍɏ $¹¨¾ɏ °»ºɏ º¯¬¨¹ɏ ¶¹ª¬ɏ «°¨®¹¨´ɏ ¨µ«ɏ ©¬µ«°µ®ɏ´¶´¬µ»ɏ«°¨®¹¨´ȍ \MP!6/8^ 9!! / 9/ S -0 9 ¥ ¥ S \ 750 450 600 P9 / # I J D VC 2m 3000 Nm 3m (a) # * . K F ! ! . $* 89 / G. # 89 / - 9 " 561!O * . -0 ' D$% ' // - O * ! ! " ! * # * " 89 / F %" # ! D # . J D$% 1 -0 * fi K ! -0 !!/0 * 89 ¥K K S / 9/ + ¥ 32/98!lO * 89 fi ¥ ¥ # 89 / - 9 EF 7 fi 39/24!lO ŵŹżɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº % ! ! * P9 % % %C #C 1 +K EF ¥ + EF + EF S ! ¥ ¥ ¥ ¥ ¥ ¥ %", H H EF %"Q ¥ + EF C $* " ! . /0 * ! P9 / # # ! P9 * # ! ! C J D$% I% % % ' // // 7 EF 10 kN 10 kNm A 30 kNm C 1.5 m VA D E 1m 0.5 m 2m VA 1m VB VB (a) C D E 5 0 A B 5 C + – A D – –5 –8.13 E B 2m (a) 21.87 + 3 kNm D C 2m 2m B A C EF H 20 kN 20 kN/m ! ! 6 B + E –5 – –11 (b) SFD (b) SFD –28.13 20 36.25 10 28.13 Parabolic Linear 10.13 C A ! 6.24 D 0 A + E 10 0.593 m D F + C B E B – –6 (c) BMD (c) BMD Gjh/!6/62! Gjh/!6/63 'XAMPLE!grdj &°®¼¹¬ɏŹȍŹŶ¨ɏº¯¶¾ºɏ¨µɏ¶½¬¹¯¨µ®ɏ©¬¨´ɏª¨¹¹À°µ®ɏ5$,Ȏɏ·¶°µ»ɏ³¶¨«ɏ¨µ«ɏª¶µª¬µ»¹¨»¬«ɏ ´¶´¬µ»ȍɏ0³¶»ɏº¯¬¨¹ɏ¶¹ª¬ɏ¨µ«ɏ©¬µ«°µ®ɏ´¶´¬µ»ɏ«°¨®¹¨´ºɏ¶¹ɏ»¯¬ɏ©¬¨´ȍɏ,¶ª¨»¬ɏ »¯¬ɏ·¶°µ»ɏ¶ɏª¶µ»¹¨ĸ¬¿¼¹¬ɏ¨³º¶Ȏɏ°ɏ¨µÀȍɏɏ \MP!6/8^ 5OLUTION! S S ¥ ' ¥ !¥ " ¥ " ¥# " fi fi ' "$ %& fi ## ## "$ ( %& % ) , , ) ) ) % + %+ +. %+ / ( %& ( %& %+ - " * ( " ) 0 ( %& +! * ) 1! ɏ ŵŹŽ "¬¨´ºɏ¨µ«ɏ&¹¨´¬ºɏ * , , 1 ! ) ) ) 3 ) 4 ) ) * ! 1 ! ( ! %1 !! ( " # ! % ( ¥ #! "" %& 0"" ' "$ % / % -' %3 3 ) ) * 2 * 2 2 %& %& 5 1 23 , , ) ) ) 1 ) ) 6 * ) * 9 : 2 ) ;) 641 + + ! 3 ( ¥ # " %& " '" # %& (¥ " ¥#'" " %& ¥ #! ¥ " %& 8 1 8 2 15 - ) 4 2 ++1 7 *) 4! 4 ! ' #!# = # ! 5$ "$! ) ) 2 71 ) 5 ;) * 8 5 < !# * fi ##! ' "# 2) * ) ) ) 'XAMPLE!grdk ) * ! 9 : ) ! 1/6:4!n 25 4 5 (5(# 2 $¹¨¾ɏ »¯¬ɏ º¯¬¨¹ɏ ¶¹ª¬ɏ ¨µ«ɏ ©¬µ«°µ®ɏ ´¶´¬µ»ɏ «°¨®¹¨´ºɏ ¶ɏ »¯¬ɏ ©¬¨´ɏ º¯¶¾µɏ °µɏ &°®ȍɏŹȍŹŷ¨ȍɏ \MP!6/8^ ) 5OLUTION! S ¥#' S ( ) 5( ¥ " ¥ " ! ¥ 23 ¥ 1 5( ¥ " ¥ " ' fi fi ( "5 $ %& ( "5 $ 5 %& % ) ) ) ) + 2 + B/ ) C) 2 2 % %+ %1 % % "5 $ %& * ) "5 $ %& ' 5( ¥ 5( ¥ ( 5 ' "5#( #5 A %& 5( ¥ " ¥ " "5 $ ( 5 %& 5 %& 1 ) ) ) ) ;) 641 + 2 + + 1 1 2 71 ) 4 5 (5( ! ¥ " "5 $ %& ¥ 5( 5( ¥ 5( ¥ (! ¥ ! 2 * 8 5 1 3 ¥" 2 +! "5#( %& +! ŵźŴɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº 20 kN 15 kN/m C A 10 kN/m VD 2m B 1m VA 1m 1m (a) VB 22.5 22.5 2.5 + 1.67 0 A + B (b) SFD B C 1m 52.5 (a) 0 A D B VA C A 100 kNm C D C D 2.5 0 – 75 –2.08 –3.33 (b) SFD + 1.67 0 A 1.25 0 – B –2.5 + 0 A C ! –25 B (c) BMD (c) BMD Gjh/!6/64! Gjh/!6/65 'XAMPLE!grel $¹¨¾ɏº¯¬¨¹ɏ¶¹ª¬ɏ¨µ«ɏ©¬µ«°µ®ɏ´¶´¬µ»ɏ«°¨®¹¨´ºɏ¶¹ɏ»¯¬ɏº°´·³Àɏº¼··¶¹»¬«ɏ©¬¨´ɏ º¯¶¾µɏ°µɏ&°®ȍɏŹȍŹŸ¨ȍ \MP!6/8^ 5OLUTION! S "( ¥ #! ¥ #=#! ' " S "( ¥ # # ' # ¥ fi 1 ' 1 ¥ ( fi 1 1 ( #5( %& * 2 #5(! 2! (#5( %& % % % ) ) , ) ) + (#5( %& "( ¥ # (#5( "( ¥ # # (#5( %+ / ##5( %& * # #5( %& ) * ) B 1 ) ,) 2 2 %1 %1 1 "( ¥ # # #5( %& (#5( 1 ) ! # #5( %& +! ) 1! ɏ ŵźŵ "¬¨´ºɏ¨µ«ɏ&¹¨´¬ºɏ , ) - ¥# "( ¥ #! ¥ " (#5( ¥ # ¥# ¥ "( ¥ #! ¥ " " (#5( ¥ # "( ¥ #! ¥ " ' "! " (#5( ¥ ! 2 * 8 5 $( %& * , ) ) ) ;) 641 ) + 1 2 71 / + 1 ) ! 4 5 (5( " 2 ! #( %& " #5( %& 57//#4; H % difdl! zpvs!voefstuboejohI!6* ) 2J * K/ * 2 .1!grc! $°º»°µ®¼°º¯ɏ©¬»¾¬¬µɏ©¬¨´ºɏ¨µ«ɏª¶³¼´µºɏ°µɏ¨ɏ·¶¹»¨³ɏ ¹¨´¬ȍ ;) ) < ) ) 5 ;) * : * 8 * 8 2 % 7$$3 2 % 5 ;) 8 2 5+ ( $ ! $898 2 5 * 25 .1!grd! ,°º»ɏ»¯¬ɏµ¨´¬ºɏ¶ɏ«°Ĺ¬¹¬µ»ɏ»À·¬ºɏ¶ɏ³¶¨«°µ®ºɏ¶ɏ©¬¨´ºȍ "! D 2 #! E 2 2 2 E1-! !E 8 2 !4: 2 5 2 !+ * .1!gre! ,°º»ɏ»¯¬ɏµ¨´¬ºɏ¶ɏ«°Ĺ¬¹¬µ»ɏ»À·¬ºɏ¶ɏº¼··¶¹»ºȍ "! / #! F 2 2 ! 2 .1!grf! #¨»¬®¶¹°º¬ɏ«°Ĺ¬¹¬µ»ɏ»À·¬ºɏ¶ɏ©¬¨´ºȍ "! 6 !+ 2 5 #! + 8 ! ) 8 ) ! 4: 2 (! 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EfÝof!tifbs!gpsdf/!Xifo!epft!ju!efwfmpq!jo!b!cfbn@ ! ! 6) * * 2 8 2 2 8 ) 8 ) ) 5 5/! EfÝof!cfoejoh!npnfou/!Xifo!epft!ju!efwfmpq!jo!b!cfbn@! ! ! 2 2 ) 4 1 2/! Xibu!jt!uif!ejggfsfodf!cfuxffo!b!cfbn!boe!b!gsbnf@! 2 ! 2 5*146 #059'4!37'56+105 ) ! 2 \MP!6/9^ 8 ! dS dx ) E1- * ! ) 71 ! ! ) *) 2 +/2146#06!(14/7.#' L / ! : ) 5 ! ! * 2 2 * ) M 25 \MP!6/7^ 2 ) *) 2 ) * * * 5 H) 2 2 2 8 2 ) 5 6/! Xibu!jt!uif!tjhojÝdbodf!pg!tifbs!gpsdf!boe!cfoejoh!npnfou!ejbhsbnt@! ! ;) 2 82 2 2 ) 2 54 : J * 2 : 8 ) * 2 2 ) 5 2 2 \MP!6/7^ * 8 * \MP!6/8^ * ) ɏ ŵźŷ "¬¨´ºɏ¨µ«ɏ&¹¨´¬ºɏ ! 7/! Xiz!jt!b!qpjou!pg!dpousbÞfyvsf!dbmmfe!tp@! ) ! ) * 8 ! ! \MP!6/8^ 2 2 ! ) *) !5 2 5 5 8 25 ;) ) 8/! Xibu!jt!uif!sfmbujpotijq!cfuxffo!cfoejoh!npnfou-!tifbs!gpsdf!boe!mpbejoh!bu!b!qpjou!po!b!cfbn@! \MP!6/9^ dS d 2M dM =- 2 . ! % ! dx dx dx ! ! 9/! Xibu!jt!uif!ejggfsfodf!cfuxffo!nfdibojdbm!boe!tusvduvsbm!qpsubm!gsbnft@! ! ;) J ! *) * 2 ) \MP!6/21^ ) J 25 -';!6'4/5 .1!grc .1!grd D + .1!gre 2 .1!grf 6) 6 / E 8 + F * + 2 + 2 * : 8 641 71 9 : * ) 4: 2 2 .1!gri 2 2 E1- D .1!grh 8 ) 4: 2 D 2 * 8 + ! 4'8+'9!37'56+105 ! 2/! 6 ) ! 3/! 6 2 2 2 2 : ! 4/! H) 2 2 2 ) ) 2 8 ! 7/! 3: ! 8/! F ! 9/! 1 O ! :/! H) * 9 : 5 ) I ) * \MP!6/5^ 2 5 \MP!6/21^ 9 : 2 9 : \MP!6/8^ I ! 5/! H ! 6/! F ) ) ) ) 5 2 ) N ) * 2 : !D 5 2 I 3: \MP!6/21^ 2 ) ) \MP!6/4^ * ) ! 7 *) * M5 H) ) 2 * ) O: 2 2 2 * 2 2 I * * * 2 ) I H) \MP!6/4^ ) * 8 * ) 2 8 ! I \MP!6/4^ ) 5 \MP!6/7^ I \MP!6/8^ ŵźŸɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº ! 1$,'%6+8'!37'56+105 #! /ULTIPLE %HOICE!3UESTIONS! + 8 ! 2/! ! ) ! : 2 2 8 * * ! 3/! ;) ! ! 2 2 # ! \MP!6/3^ 2 2 *) * ) 9 : ! ) * * 2 \MP!6/6^ =# ! 8 ) \MP!6/9^ ! 2 9 * I \MP!6/8^ ! # 2 2 2 ! ) ! O: 2 ! 8 ) ! 8/! 4 ) ! ! " ! 7/! ) ! ! ) ) ! ! 6/! F 2 8 \MP!6/6^ ) 2 2 ) 8 ! ! = ! 2 ) ! 5/! ;) 2 ! 4/! 4 2 ) ! ) ! 2 2 ) ) 2 \MP!6/5^ 2 2 " E1- \MP!6/8^ " " " " " ! ! 9/! \MP!6/8^ " " " " ! :/! " \MP!6/9^ $! (ILL $LANKS!3UESTIONS ( # $$$$$$$$$ ! 2/! % \MP!6/3^ ! 3/! " & $$$$$$$ % \MP!6/7^ ! 4/! " & ! % ! 5/! ' ! 6/! " ( ! ) ! $$$$$$$$$ % $$$$$$$ % $$$$$$$$$ \MP!6/9^ \MP!6/5^ \MP!6/8^ ɏ ŵźŹ "¬¨´ºɏ¨µ«ɏ&¹¨´¬ºɏ ! 7/! * $$$$$$ ! 8/! $$$$$$$$$$$$ $$$$$$$$$% ! 9/! , & $$$$$$$$$$ %! % \MP!6/9^ ! + \MP!6/8^ \MP!6/9^ %! /ATCH %OLUMNS!3UESTIONS # ! 2/! Dpmvno!J! , . 013 Dpmvno!JJ! - ( / " 4 5 " ( 5 ! 3/! Dpmvno!J! 0 0 * ! 4% * ! \MP!6/8^ ,.1 Dpmvno!JJ! - 5 / 5 4 5 5 , 66 \MP!6/8^ ! ! Dpmvno!J! * * ! ! % Dpmvno!JJ! " % 5 ! 5 ! % / " 4 " 5 " \MP!6/8^ #NSWERS!TO!1BJECTIVE!3UESTIONS !)B*! Nvmujqmf.Dipjdf!Rvftujpot ! 2/ ! 3/! ! 4/! ! 5/! ! 6/ ! 7/ ! 8/! 9/! !)C*! Gjmm.Cmbolt!Rvftujpot 2/ 5/! 8/ 3/! 6/! 9/! 7 ! 4/! 7/ !)D*! 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Gjh/!7/8 /! 1 2 B C B FCE FBE A FAE 1 A 1 C FBC 1 2 E D FDE D 2 RA 2 RD (a) Truss being cut by two sections (b) Isolated parts of the truss Gjh/!7/8! & Tpnfujnft-!cpui!uif!bcpwf!nfuipet!bsf!vtfe!up!bobmztf!b!mbshf!usvtt/!Jo!tvdi!dbtft-!cfgpsf!dvuujoh!b! tfdujpo-!ju!nbz!cf!vtfgvm!up!Ýoe!uif!gpsdft!jo!tpnf!nfncfst!cz!uif!nfuipe!pg!kpjout/!Tfmepn-!{fsp!gpsdf! nfncfst!bsf!uifsf!jo!b!usvtt/!Ju!jt!vtfgvm!up!jefoujgz!tvdi!nfncfst!xijmf!bobmztjoh!usvttft/!Uif!gpmmpxjoh! hvjefmjoft!nbz!cf!dpotjefsfe!up!jefoujgz!{fsp.gpsdf!nfncfst; ! ) *! Jg!b!kpjou!pomz!ibt!uxp!nfncfst!boe!op!fyufsobm!mpbe!boe0ps!op!tvqqpsu-!uifo!uiptf!uxp!nfncfst!bsf! {fsp.gpsdf!nfncfst!)Gjh/!7/9 */ ! ) *! Jg!b!kpjou!pomz!ibt!uxp!nfncfst!boe!jt!mpbefe-!boe!jg!uif!mjof!pg!bdujpo!pg!sftvmubou!gpsdf!gspn!bqqmjfe! mpbet!bu!uif!kpjou!jt!dpmmjofbs!xjui!pof!pg!uif!nfncfst!uifo!uif!puifs!nfncfs!jt!b!{fsp.gpsdf!nfncfs/! Jg!uif!sftvmubou!gpsdf!bu!uif!kpjou!jt!opu!dpmmjofbs!xjui!fjuifs!nfncfs!uifo!cpui!nfncfst!bsf!opu!{fsp. gpsdf!nfncfst!)Gjh/!7/9 */ ! ) *! Jg!b!kpjou!ibt!uisff!nfncfst!boe!op!fyufsobm!mpbe!boe0ps!tvqqpsu-!boe!jg!uxp!pg!nfncfst!bsf!dpmmjofbs! uifo!uif!opo.dpmmjofbs!nfncfs!jt!b!{fsp.gpsdf!nfncfs!)Gjh/!7/9 */ P P P (a) (b) Gjh/!7/9! ' 'XAMPLE!hrc (c) ( ) &°µ«ɏ»¯¬ɏ¶¹ª¬ºɏ°µɏ»¯¬ɏ´¬´©¬¹ºɏ!"Ȏɏ!#ɏ¨µ«ɏ"#ɏ¶ɏ»¯¬ɏ»¹¼ººɏº¯¶¾µɏ°µɏ&°®ȍɏźȍŽ¨ɏ©Àɏ»¯¬ɏɏ ´¬»¯¶«ɏ¶ɏ±¶°µ»ºȍ \MP!7/4^ ŵŻżɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº 200 kN FAB B VB 200 kN 200 kN A A 60° 30° C VC 5m (a) Triangular truss B 60° A FBC 100 173.21 FAC 30° VB C VC (b) FBD of joints A, B and C B 60° 30° 86.6 VB C VC 5m (c) Analysed truss (forces in kN) Gjh/!7/: 5OLUTION! Jo!uif!usjbohvmbs!usvtt-!–CBD!>!:1¡/!Uifsfgpsf-!BD!>!CD!dpt!41¡!>!5/44!n/ Frvbujoh!uif!tvn!pg!npnfout!pg!bmm!gpsdft!bcpvu!uif!tvqqpsu!D!up!{fsp-!xf!hfu S&}D!>!1;! Ï!*C!¥!6!,!311!¥!)5/44!dpt!41¡*!>!1! fi! *C!>!261!lO Frvbujoh!uif!tvn!pg!wfsujdbm!gpsdft!up!{fsp-!xf!hfu S# !>!1;! *C!Ï!311!,!*D!>!1! fi! *D!>!311!Ï!261!>!61!lO Mfu!vt!bqqmz!uif!nfuipe!pg!kpjout/!Gsff!cpez!ejbhsbnt!pg!bmm!uif!kpjout!bsf!tipxo!jo!Gjh/!7/: /!Uif!gpsdft! jo!bmm!uif!nfncfst!bsf!bttvnfe!ufotjmf/ Kpjou!C;!Bqqmzjoh!frvbujpot!pg!frvjmjcsjvn!jo!wfsujdbm!boe!ipsj{poubm!ejsfdujpot-!xf!hfu -150 !>!Ï284/23!lO!>!284/32!lO!)D*+ sin 60∞ S# !>!1;! #BC!dpt!71¡!,!#CD! fi! #CD!>!Ï )Ï284/32!dpt!71¡*!>!97/7!lO!)U* S# !>!1;! *C!,!#BC!tjo!71¡!>!1! fi! #BC!>! Kpjou!D;!Bqqmzjoh!frvbujpot!pg!frvjmjcsjvn!jo!wfsujdbm!boe!ipsj{poubm!ejsfdujpot-!xf!hfu 50 *C !>!Ï!211!lO!>!211!lO!)D* =sin 30∞ sin 30∞ Uif!bobmztfe!usvtt!jt!tipxo!jo!Gjh/!7/: /!Opuf!uibu!uif!gpsdft!xjui!ofhbujwf!tjho!bsf!tipxo!bt!dpnqsfttjwf/ S# !>!1;! #BD!tjo!41¡!,!*D!>!1! fi! #BD!>!Ï! 'XAMPLE!hrd &°µ«ɏ»¯¬ɏ¨¿°¨³ɏ¶¹ª¬ºɏ°µɏ¨³³ɏ´¬´©¬¹ºɏ¶ɏ»¯¬ɏ»¹¼ººɏº¯¶¾µɏ°µɏ&°®ȍɏźȍŵŴ¨ȍ \MP!7/4^ 5OLUTION! Frvbujoh!uif!tvn!pg!wfsujdbm!gpsdft!up!{fsp-!xf!hfu! S# !>!1;! +B!,!+E!Ï!3!Ï!5!>!1! fi! +B!,!+E!>!7! )* Opx-!frvbujoh!uif!tvn!pg!npnfout!pg!bmm!gpsdft!bcpvu!uif!tvqqpsu!B!up!{fsp-!xf!hfu S&}B!>!1;! +E!¥!7!Ï!3!¥!2/6!Ï!5!¥!5/6!>!1! fi! +E!>!4/6!lO! Gspn!Fr/!) *-! +B! >!7!Ï!+E!>!7!Ï!4/6!>!3/6!lO Mfu!vt!bqqmz!uif!nfuipe!pg!kpjout/!Gsff!cpez!ejbhsbnt!pg!bmm!uif!kpjout!bsf!tipxo!jo!Gjh/!7/21 <!gpsdft!jo! bmm!nfncfst!bsf!bttvnfe!ufotjmf/ ! +!B!nfncfs!gpsdf!gpmmpxfe!cz!)D*!jt!dpnqsfttjwf!gpsdf-!boe!uibu!gpmmpxfe!cz!)U*!jt!ufotjmf!gpsdf/! ɏ ŵŻŽ 4¹¼ºº¬ºɏ¨µ«ɏ#¨©³¬ºɏ 4 kN 2 kN B B C FAB A 60° 60° 60° 60° D A E RA 3m 4 kN 2 kN 3m C FBC FCE FBE FAE E FCD FDE 2.5 kN RD (a) 2D truss carrying loads D 3.5 kN (b) Free body diagrams of all joints Gjh/!7/21 Kpjou!B;!Bqqmzjoh!frvbujpot!pg!frvjmjcsjvn-!xf!hfu 2.5 !>!Ï!3/9:!lO!>!3/9:!lO!)D* sin 60∞ S# ! >!1;! #BF!,!#BC!dpt!71¡!>!1! fi! #BF!>!Ï!1/6!#BC!>!Ï!1/6!¥!)Ï!3/9:*!>!2/55!lO!)U* S# ! >!1;! #BC!tjo!71¡!,!+B!>!1! fi! #BC!>!!- Kpjou!C;!Bqqmzjoh!frvbujpot!pg!frvjmjcsjvn-!xf!hfu S# ! >!1;! Ï!#BC!tjo!71¡!Ï!#CF!tjo!71¡!Ï!3!>!1 - 2 - #AB sin 60∞ - 2 - (- 2.89) sin 60∞ !>!1/69!lO!)U* !>! sin 60∞ sin 60∞ S# ! >!1;! Ï!#BC!dpt!71¡!,!#CF!dpt!71¡!,!#CD!>!1 fi! #CD!>!)Ï!3/9:*!¥!1/6!Ï!1/69!¥!1/6!>!Ï2/84!lO!>!2/84!lO!)D* fi! #CF!>! Kpjou!F;!Bqqmzjoh!frvbujpot!pg!frvjmjcsjvn-!xf!hfu S# ! >!1;! #CF!tjo!71¡!,!#DF!tjo!71¡!>!1 - #BE sin 60∞ !>!Ï!1/69!lO!>!1/69!lO!)D* sin 60∞ S# ! >!1;! #EF!Ï!#BF!Ï!#CF!dpt!71¡!,!#DF!dpt!71¡!>!1 fi! #EF!>!2/55!,!1/69!¥!1/6!Ï!)Ï!1/69*!¥!1/6!>!3/13!lO!)U* fi! #DF!>! Kpjou!E;!Bqqmzjoh!uif!frvbujpo!pg!frvjmjcsjvn!jo!uif!ipsj{poubm!ejsfdujpo-!xf!hfu S# ! >!1;! Ï!#EF!Ï!#DE!dpt!71¡!>!1 2.02 fi! #DE!>! !>!Ï!5/15!lO!>!5/15!lO!)D* cos 60∞ Uif!bobmztfe!usvtt!jt!tipxo!jo!Gjh/!7/22/!Opuf!uibu!uif!gpsdft!xjui!ofhbujwf!tjho!bsf!tipxo!bt!dpnqsfttjwf/ ŵżŴɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº 4 kN 2 kN B C 1.73 2.89 0.58 4.04 0.58 A D 1.44 2.02 E 3m 3m 2.5 kN 3.5 kN ) Gjh/!7/22! 'XAMPLE!hre ( lO* !µ¨³Àº¬ɏ»¯¬ɏ»¹¼ººɏº¯¶¾µɏ°µɏ&°®ȍɏźȍŵŴ¨ɏș%¿¨´·³¬ɏźȍŶȚɏ¼º°µ®ɏ»¯¬ɏ´¬»¯¶«ɏ¶ɏº¬ª»°¶µºȍ \MP!7/4^ 5OLUTION! Ibwjoh!efufsnjofe!uif!tvqqpsu!sfbdujpot!jo!Fybnqmf!7/3-!mfu!vt!dvu!uif!usvtt!xjui!b!tfdujpo!2.2! qbttjoh!uispvhi!nfncfst!CD-!CF!boe!BF-!boe!bopuifs!tfdujpo!3.3!qbttjoh!uispvhi!CD-!DF!boe!EF!bt!tipxo! jo!Gjh/!7/23 /!Xf!dpotjefs!gsff!cpez!ejbhsbnt!pg!uif!uxp!qbsut!pg!uif!usvtt/! 4 kN 2 kN 4 kN 2 kN 1 2 1 B 2 B C C FBC FBE A 60° 60° 60° E 1 RA 60° D A FCE 60° 60° FAE 2 3m FBC FDE 1 33m m RD (a) Truss being cut by two sections D 2 2.5 kN 3.5 kN (b) Part of truss left of 1-1 (c) Part of truss right of 2-2 Gjh/!7/23 # )j*!!Qbsu!pg!uif!usvtt!tipxo!jo!Gjh/!7/23c; S& (() * !" ¥ #!" $ # ¥%¥ fi & ' # 2.5 ¥ 1.5 3 ¥ sin 60∞ 2/55!lO!)U* * ) S# $# S# # & '$ +# fi # , fi ++ & '+# $#!4% 7 , # fi 2/84!lO!)D* - 2 + 2.5 1/69!lO!)U* sin 60∞ # , $ !" ¥ !"- $ #!// ɏ ŵżŵ 4¹¼ºº¬ºɏ¨µ«ɏ#¨©³¬ºɏ )jj*!Qbsu!pg!uif!usvtt!tipxo!jo!Gjh/!7/23d;! (() S# +8 $ #, # $ !"- 7 #,8 2.5 sin 60∞ $ #8 $ #,8 fi & ' (() ) $ !-; 7 3.5 ¥ 1.5 3 ¥ sin 60∞ #8 3/13!lO!)U* 8 fi & ' #,8 - 2.02 cos 60∞ $ /! / 7 5/15!lO!)D* !ɏ¹¶¶ɏ»¹¼ººɏ°ºɏº¯¶¾µɏ°µɏ&°®ȍɏźȍŵŷȍɏ$¬»¬¹´°µ¬ɏ»¯¬ɏµ¨»¼¹¬ɏ¨µ«ɏ´¨®µ°»¼«¬ɏ¶ɏ»¯¬ɏ¶¹ª¬ºɏ °µɏ´¬´©¬¹ºɏ#$Ȏɏ#&ɏ¨µ«ɏ&'Ȏɏ¼º°µ®ɏ»¯¬ɏ´¬»¯¶«ɏ¶ɏº¬ª»°¶µºȍ \MP!7/4^ 5OLUTION! ! < ( ) # = 7 ) C 10 kN # >$ > & ' >$? > & '@ & ' > ?# $ & '@ ?# $ * & '@ #!" )) S& ( C RA Gjh/!7/24 C )) 1 60° 30° F 2m 14 kN (a) Truss being cut by Section 1-1 E RE Gjh/!7/25 FFG ( #/ 7 FCD FCF 1 60° 60° G 2m 6 kN 14 kN # D 30° 60° 2m 60° 30° F 2m 6 kN = + B A 60° 60° G 2m #=# ! &!#/ ! : ( 10 kN 30° 60° 2m A RA + ¥ & $ #/ ¥ / $ & ¥ $ # ¥ fi + #%!-% 7 >@ ! &!#/ ! : # >! D B A #,8 #, 3/9:!lO!)D* ! &!##! 'XAMPLE!hrf B ? ! ! ,8 , 1/69!lO!)D* , +8 ¥ #!" $ #8 ¥ % ¥ S# - # * ) S& , : fi & ' + !" #8 : 3.5 - 4 sin 60∞ #, (() S# : * ) fi & '$/ # : * 60° F ) C ) 1 D 1m 60° 30° E 2m RE 14 kN (b) FBD of right-hand portion 1 E RE ŵżŶɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº < ( #,8 8 + #,8 ¥ # & ' + #,8 fi #,8 $#%!-% ¥ % ' + + $ #/ ) = S& S# & ' # !, * ) = ( + ¥ #, 38/77!lO!)D* - (- 27.66) ¥ 0.5 - 13.83 + 14 27/27!lO!)U* sin 60∞ $ # > $ #,8 % ' $ #, & ' fi # > $ ?$ 4!&&@ ¥ !-&& $ #&!#& ¥ !" 26/98!lO!)U* fi #, S# 7/6! .1!hrf , ) ) ) D %½¨³¼¨»¬ɏº¯¨·¬Ȏɏ º¼··¶¹»ɏ¹¬¨ª»°¶µºȎɏ¨µ«ɏ ¶¹ª¬ºɏ¨»ɏ½¨¹°¶¼ºɏ·¶°µ»ºɏ °µɏ¨ɏª¨©³¬ɏª¨¹¹À°µ®ɏ»¯¬ɏ ·¶°µ»ɏ³¶¨«ºȎɏ¼µ°¶¹´³Àɏ «°º»¹°©¼»¬«ɏ³¶¨«ɏ¨µ«ɏ°»ºɏ º¬³Ɋ¾¬°®¯»ȍ ( �! DBCMFT! ) ( )) E C ) E ) ) ? !, ) ) ! &!#"@ ( (( ) D !: ) ) ( ( I D !: D ) ! ) D ! :F (( ) ) ) D C ) ) , ) D ( ) ) ! =D ) ) D D !, ) )) )) ) ) ( )) ) (( ( !G ) ( !, H ( !, ) ) �! 9ORTHY!OF!0OTE Cvdlmjoh ( D ) D C ! Tower Tower Steel cables Suspenders Deck Piers Piers Gjh/!7/26! !: D ) ) ! ɏ ŵżŷ 4¹¼ºº¬ºɏ¨µ«ɏ#¨©³¬ºɏ Bttvnqujpot!Nbef!jo!uif!Bobmztjt!pg!Dbcmft , #! ( ) ! ! ! " %!! Fwfsz!qpjou!jo!b!dbcmf!jt!ijohfe ! ! ! 7/6/2! ) ) ) ) D ) )) ) * ) ( ) !: * ! ! ))! Dbcmft!Tvckfdufe!up!Dpodfousbufe!Mpbet! J ) D ) &!#& ) )( ) ( ( ## )) ) ) (( @ * 8! : , * ) ( ( ) # !: # (( H ? (() * K , 8! : ! ! )) % * ) !L ( D ) ! L A HA RA HB D F2 C F1 TAC VC VD a1 RB D C H l3 l2 l1 h B yD yC a3 H TDB F2 F1 (b) Tensions in segments AC and DB (a) Cable subjected to point loads Gjh/!7/27! Gpsdft!jo!uif!Dbcmf : ) ) ( , uisvtu! ) !: A ) C ) ) *, M a# - D ) ) a#! : , ! &!#& ! : !: A D ! * ) !8 ) C )) ! &!#& ) ) D )D ipsj{poubm! - ( ) ( , ) ( D *D D H 2 + VC2 , *, ) ?&!#@ ( ) *! ?&!#@ ) ) , - 2 + ( - tan a1 ) 2 ( A - 1 + tan 2 a1 ,! ) C( D ) , ŵżŸɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº - 2 + ( - tan a 3 ) 2 - 1 + tan 2 a 3 a - 1 + tan 2 a !"# 'XAMPLE!hrg Two loads of 20 kN each are supported by a flexible cable ACDB as shown in Fig. 6.17a. The span of the cable between supports is 15 m and the sag of both the loads is 2.5 m. Determine the tensile force developed in all the three segments of the cable. 5m 5m 5m A B 2.5 m C D 20 kN 20 kN (a) Cable carrying symmetrical loading RA H RB a1 a1 A TAC C TCD TAC a1 H B 2.5 m C TDB D 20 kN 20 kN 20 kN TCD (b) Free body diagram of cable (c) FBD of point C Gjh/!7/28 5OLUTION! * ++ $ $ + $ ", - ",#." 4 S&64 ,7 4 * $ ", /! 5 +4 - 1 + tan 2 a1 40 1 + 0.52 4 - 1 + tan 2 a 0 40 1 + 02 - 1 + tan 2 a1 40 1 + 0.52 $ $ fi a& 4 - $ $ ! 20 ¥ 5 2.5 $ :, / $ +4 $ 55/83!lO 51!lO 55/83!lO < 5 " !9'; $ 0 8 ++ ¥ 9 - - ¥ "!9 , fi a, ,;#! a& "!9.9 ,!9! Bmufsobujwfmz ! # a& ,!9 a&7 % ! !&' ! % = 4 % ! !&' ! > 0 ! ɏ ŵżŹ 4¹¼ºº¬ºɏ¨µ«ɏ#¨©³¬ºɏ + < = 4 AC CD sin 90∞ sin (90∞ + a1 ) 20 sin (180∞ - a1 ) * 55/83!lO % 7/6/3! +4 fi 55/83!lO< +4 51!lO 4 Dbcmf!Tvckfdufe!up!Vojgpsnmz!Ejtusjcvufe!Mpbe! 4 $ . $ % ! !&? ! 4 !< * @A! < * - / D* / D* * $ @ A! D ! @A $ * $ D E + $ B @A! wDx V x H h B Dx y !&? $ /D $ w A $ !% P Dx P q Dy ds Dy Q Q H + DH V + DV L (a) The cable AB (b) FBD of the element PQ Gjh/!7/29! / 5 * $ 5 S# S# ,7 ,7 S&6A ,7 * @A 8 - - - - D-# , fi D- , - * 8 * - D*# 8 D , fi D* - D , D , fi -¥D 8*¥D , -¥D 8*¥D - D ¥ " 5 D #" D 5 D Æ, Ï DH ¸ lim Ì = 0˝ fi x Æ 0 Ó Dx ˛ dH dx , !F# Ï DV ¸ lim Ì + w = 0˝ fi Dx ˛ dV dx 8 !:# xÆ0 Ó BG * # * / D* $ /D ! . * $ H * $ ŵżźɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº Ï H Dy ¸ lim Ì - V = 0˝ x Æ 0 Ó Dx ˛ fi dy dx H * fi * - q dy dx q ! " # $ H% +V% - 1 + tan 2 q Nbyjnvn!Ufotjpo!jo!uif!Dbcmf/! 0 nbyjnvn nbyjnvn & " Njojnvn!Ufotjpo!jo!uif!Dbcmf/! ' * ! {fsp ) ! Uif!Dbcmf!Frvbujpo! * $ ' d%y dx % dV dx 4 - dy dx ! #! ! ! njojnvn ! ( ( ! ! $ / ( fi 3 + d%y - 3 dx % 5 $ ,-. 4 6 % - ' 6 $ 3 % 5 6 5 7 % . % " $ " $ $8 7 8 < 7 " $ @ qbsbcpmjd 9: 9; - ¥ 9 .: ; -¥ 3 3 02 5 2 .% 5 % 6 6 ¥95 ¥.5 % % fi fi % 6 9 Hh 1 + L 2 . # 3 x 2 È Hh 1 ˘ + + wL ˙ 5 9 fi 2 ÍÎ L 2 ˚ dbcmf!frvbujpo $ $ y= wx ( L - x) hx + 2H L ! @ + $ ,-. ɏ ŵżŻ 4¹¼ºº¬ºɏ¨µ«ɏ#¨©³¬ºɏ * ! ( 9 $ wx ( L - x) 2H 69 69 .A% !$ "!! wL2 8H w( L /2){L - ( L /2)} 2H C Mfohui! pg! uif! Dbcmf/! G H = dx 1 + (dy /dx)2 - @ ! * ª ! ! J $ 66 ! dx % + dy % I w h ( L - 2 x) + 2H L ' ( $ ( " ! L Ú ds Ú dx 5 9 9 L x0 fi L+ lc = L + ( " ! ! ! 5 $ J L 1/2 2 ÏÔ Ê w h ˆ ¸Ô ( L x ) 1 + 2 + Ì Á ˜ ˝ L ¯ Ô˛ ÔÓ Ë 2 H 2 $ ! 2 ÏÔ 1 Ê w h ˆ ¸Ô ( L x ) 1 + 2 + Ì ÁË ˜ ˝ L ¯ Ô˛ ÓÔ 2 2 H $ wL2 8 yC - ! hˆ Ê w dx 1 + Á ( L - 2 x) + ˜ Ë 2H L¯ B fi B 67 dy dx < $ $ ! 6 < ( " ! H ! 1 Ê w2 wh h2 ˆ 2 2 ( L + 4 x 4 Lx ) + ( L 2 x ) + ˜ dx 2 ÁË 4 H 2 HL L2 ¯ ! !9 . L 1 ÏÔ w2 wh h 2 ¸Ô 2 2 L 4 x 4 Lx L 2 x ( + ) + ( ) + Ì ˝ dx 2 Ú0 ÔÓ 4 H 2 HL L2 Ô˛ 1 w2 Ê 2 4 x3 x 2 ˆ wh h2 2 + + 4 + ( ) + L x L Lx x x 2 4 H 2 ÁË 3 2 ˜¯ HL L2 L 0 1 È w2 Ê 3 4 L3 4 L3 ˆ wh 2 h2 ˘ 2 L + + ( L L ) + ◊ L˙ Í 2 ÍÎ 4 H 2 ÁË 3 2 ˜¯ HL L2 ˚˙ w2 L3 h2 + 2 2L 24 H # 6% $ ! ! !$ - C − .% A 7 = 9 ŵżżɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº H $ $ .5 7/6/4! $ w2 L3 24 ( wL2 /8 yC ) 2 ! L+ ( 9 .%A7 - $ 6% C 8 yC2 3L 6 Dbcmf!Tvckfdufe!up!jut!Pxo!Xfjhiu ' ! ! $ + $ C " " J $ $ $ . 8 + H 6@ ( ,-. M C < " ! !"$ $ ' $ $ @ 1 ! N @ < ! ( $$ $ $ # $ we A P weDx Q B Ds g x O y Dx P Ds g L Dy Q l1 gDs l2 (a) The cable AB (b) Cable element PQ Gjh/!7/2:! Uif!Dbcmf!Qspgjmf! . H 6@ H ! $ ! D <J fi + Ds Dx ! ! " ! g Dx % + Dy % Dx Ê dy ˆ g 1+ Á ˜ Ë dx ¯ D Æ9 ! < ! Ê Dy ˆ g 1+ Á ˜ Ë Dx ¯ I D 2 2 6/ 4 - < D $ ! " ( d%y dx % $N $ dy dx fi - 3 $ d%y dx % Ê dy ˆ -g 1+ Á ˜ Ë dx ¯ 2 6 ! 6 ɏ ŵżŽ 4¹¼ºº¬ºɏ¨µ«ɏ#¨©³¬ºɏ ( < d dx fi 6 - g 1 + (sinh u ) 2 du dx du H dx ◊ - fi ' ! 3 3 % O∵ du dx fi - % q3 q 6P g H ( - gx + C6 H 64 Q 6 Ê gx ˆ + C6 ˜ ÁË ¯ H dy dx ' 6 ( - 67 # ( Ê gx ˆ + C1 ˜ + C2 ÁË ¯ H H g 6@ % < ! ! $ 6 " % R H 9 ! 9 R 9 - ( H g fi $ 67 $ 6@ R $ fi % % H g 6@ % H Ê gx ˆ H cosh Á - ˜ + Ë H¯ g g + fi y=- ( HÏ gx ¸ - 1˝ Ìcosh g Ó H ˛ %9 O∵ 67 dbufobsz < 6@ C 9 6 39 5 9 5 6 - 6 9 6 $ ) 67 95 J $ " 6@ ( $ , Mfohui!pg!uif!Dbcmf! G ! I ( dx % + dy % Myth H 6@ Ê dy ˆ dx 1 + Á ˜ Ë dx ¯ 2 Dbufobsz!qspÝmf!jt! tbnf!bt!qbsbcpmjd! qspÝmf/ Fact! 3q qP ŵŽŴɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº !" #$% & gx gx Êg ˆ - 0˜ ' - sinh ÁË sinh ¯ H H H dy H 'dx g !" #*% & gx ˆ Ê 1 + Á - sinh ˜ Ë H¯ ' + 2 ' 1 + sinh 2 gx gx gx ◊ dx ' cosh ◊ dx ◊ dx ' cosh 2 H H H , l2 Ú -2 l2 gx Ú cosh H dx ds ' - l1 - l1 l2 fi #& ' H sinh ( gx /H ) ' g g /H -l 1 lc = H g gl2 gl ˆ Ê + sinh 1 ˜ ÁË sinh H H¯ Tmpqf!pg!uif!Dbcmf! 4 q' gl2 - gl1 ˆ Ê - sinh ÁË sinh ˜ H H ¯ .∵ !- q% ' - !" ##% q3 !" #*%& dy ' dx Ê gx ˆ ÁË - ˜¯ ' H gx H !" #5% Gpsdft!jo!uif!Dbcmf Wfsujdbm!Dpnqpofou!pg!Dbcmf!Gpsdf-!W/! 4 *'- q ' -- !" 6%& gx H !" #7% Ufotjmf!Gpsdf!jo!uif!Dbcmf-!U/! ! '! H # + V # !' - 1 + tan 2 q !' H 1 + sinh 2 gx H '4 !" #$% gx H !" #6% !" #6%& '- H g '-- gx H gx H T H Ï ¸ - 1˝ ' - cosh + =- + Ìcosh g H g g g H Ó ˛ !" #"% ɏ ŵŽŵ 4¹¼ºº¬ºɏ¨µ«ɏ#¨©³¬ºɏ 0% ! nbyjnvn!ufotjmf!gpsdf , & , 8 , 'XAMPLE!hrh A uniformly distributed load of 9 kN/m acts over a length of 4 m from the left-hand support A of a cable (assumed weightless), whose both supports are at the same level, as shown in Fig. 6.20a. At the support B, the cable runs over a frictionless pulley of negligible dimensions and suspends a weight of 20 kN. Determine (a) the support reactions, (b) the horizontal component of cable tension, (c) the sag of the point C and maximum sag in the cable, and (d) the maximum and minimum tensile forces in the cable. \MP!7/5^ " #$ 5OLUTION! !9 ! % 8 S&;: ' $< :& - +8 ¥ " > !? ¥ 7% ¥ !# > #% ' $ fi +8 ' 35!lO & +: & S# ' $< +8 > +: - ? ¥ 7 - #$ ' $ fi @: & @: ' #$ A * @ ! % S# ' $ * ' $ H# +V# fi : @: 8@ 202 - 122 2 fi - ' TCB -V2 ' RA HA B x & * ' - #7 > 5" ' *# 9 kN/m A +: ' 43!lO - ' 27!lO RB 9 kN/m A y fi B HB x yC yD y C C D 20 kN 4m 20 kN xD 2m (a) The cable AB (b) FBD of the cable Gjh/!7/31 ! % B E , F & @& G S&;@ ' &@ ' $< - +8 ¥ 7 > - ¥ @ @ > !? ¥ 7% ¥ 4 2 '$ fi H @ ' & 24 ¥ 4 - 72 ' 2/6!n 16 A H & H A G @: ŵŽŶɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº @ 8 H * ' $< - +8 > ? ¥ '$ fi & ' # ""I H - +8 > # ""I > - ¥ S&; ' & ' $< ! % 8 > ? ¥ # ""I ¥ , , ' 8' # H + RA# 8& ' , , ' 3/1!n & 2 H ' fi ' 16 + 24 ' 39/95!lO 2 8 ' 2.667 '$ 2 & H 2 + VD2 ' & & - 2 + 02 ' - ' 27!lO 'XAMPLE!hri A cable of 150-m span and 30-m sag has its supports A and B at the same level as shown in Fig. 6.21a. The cable is carrying only its dead weight of 0.15 kN/m. Determine (a) the horizontal force in the cable, (b) the maximum tensile force in the cable, (c) the support reactions, and (d) the length of the cable. \MP!7/5^ RB RA 150 m H B A A B 30 m H 30 m O x 0.15 kN/m 0.15 kN/m y 75 m 75 m Tmax RA A H (a) The cable AB under its own weight Cable (b) FBD of cable and the joint A Gjh/!7/32 J 5OLUTION! !A 9 " #* ! % H : ! 8%& '- 9: - & ! HÏ gx ¸ - 1˝ Ìcosh g Ó H ˛ B !% : ! ' I6 - 5$ ' - " #$% - Ï 0.15 ¥ 75 ¸ - 1˝ Ìcosh 0.15 Ó ˛ fi & ' - 5$ 1+ 4.5 ' - % ! %& 11.25 - ! % ɏ ŵŽŷ 4¹¼ºº¬ºɏ¨µ«ɏ#¨©³¬ºɏ - ! %& B %& & * G G $# "* -& G *K !*A% L* G & &-'*$ ! !*K% *A !L* ' *K - *A% -'$ 6 & Ubcmf!7/2 Tufq!tj{f!>!6!lO - Tufq!tj{f!>!2!lO *S *M ș* Tufq!tj{f!>!1/3!lO - *M *S ș* - *M *S ș* * 6 6$$ 5M 77$ >5M 756 ** * 7$? * 6I$ >$ *"* *7 # * 5*I * 55$ >$ $*5 6 * ?$$ 7 I?$ ># M?$ *# * 5I6 * 7I" >$ *$* *7 7 * 5*# * 5#* >$ $$? *$ * 76$ * I$# >$ #"# *5 * 57" * 5?M >$ $6# *7 " * 5$M * 5*# >$ $$7 *6 * 5$$ * #?6 N$ $$6 *7 * 5#* * 57$ >$ $*? 25/9 2/415 2/415 !!1/111 *6 * 5$$ * #?6 N$ $$6 *6 $ * 5$$ * #?6 N$ $$6 O & I!>!25/9!lO ! % ! % !" #"% '-- ! % , :% P , ! ! Difdl;! ! % , , ' - 5$ ! ! 8% & ! %& 8 ' *7 M - $ *6 ¥ !- 5$% ' 2:/4!lO , , ' H# +V# ' 14.82 + 12.37 2 ' 2:/4! lO& H H & -8 ' -: ' - ' 25/9!lO +8 Q +: & 8! :% F !" #7% & *& +8 ' +: ' * ' - ! % A # ' I6 0.15 ¥ (- 75) ' 23/48!lO 14.8 gx ' -*7 M H !" ##% & - 3 *7 M & ' $ *6 R & *' N I6 & ' H g gl2 gl ˆ 14.8 Ê 0.15 ¥ 75 0.15 ¥ 75 ˆ Ê sinh + sinh + sinh 1 ˜ ' Á ÁË sinh ˜ ' 275/98!n H H¯ 0.15 Ë 14.8 14.8 ¯ ŵŽŸɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº ! #&&+6+10#.!51.8'&!':#/2.'5 'XAMPLE!hrj For the truss given in Fig. 6.22a, find the forces in members BC, CD and BD. \MP!7/4^ S 5OLUTION! !@ S @& 8 : @ S 9 & H " ## 8 & #:@ ' #@ ' 1 : S S# ' $< - #: - 6$ ' $ fi #: ' - 6$ , : ^ 8@& ' 61!lO!)D* 50 kN A C B 50 kN D C FBC B FAB FBD FCD E (a) Truss carrying 50-kN load (b) FBDs of joints B and C Gjh/!7/33 'XAMPLE!hrk Find the axial forces in all the members of the truss shown in Fig. 6.23a. Also, compute the support reactions.! ! ! \MP!7/4^ RF F F B 5m B C C FBC 30° 5m A 30 kN 5m 5m 5m 5m E 20 kN 5m D FAB A 60° 30 kN FAE FBE FCE 60° 60° E HF FCF 90° 60° RD FCD FDE 60° D HD 20 kN (b) FEBs of all joints (a) The truss Gjh/!7/34 5OLUTION! Bttvnjoh!bmm!uif!nfncfs!gpsdft!bt!ufotjmf-!uif!gsff!cpez!ejbhsbnt!pg!bmm!uif!kpjout!bsf!tipxo!jo! Gjh/!7/34 /!Mfu!vt!opx!dpotjefs!uif!frvjmjcsjvn!pg!kpjout!pof!cz!pof/ Kpjou!B;! 30 !>!45/75!lO!)U* S# ! >!1;! #BC!tjo!71¡!Ï!41!>!1! fi! #BC!>! sin 60∞ ɏ ŵŽŹ 4¹¼ºº¬ºɏ¨µ«ɏ#¨©³¬ºɏ S# ! >!1;! #BC!dpt!71¡!,!#BF!>!1! fi! #BF!>!Ï!45/75!¥!1/6!>!Ï28/43!lO!>!28/43!lO!)D* Kpjou!C;! S# ! >!1;! Ï!#BC!tjo!71¡!Ï!#CF!tjo!71¡!>!1! fi! #CF!>!Ï!#BC!>!Ï!45/75!lO!>!45/75!lO!)D* S# ! >!1;! Ï!#BC!dpt!71¡!,!#CF!dpt!71¡!,!#CD!>!1 fi! #CD!>!45/75!¥!1/6!Ï!)Ï!45/75*!¥!1/6!>!45/75!lO!)U* Kpjou!F; S# ! >!1;! #CF!tjo!71¡!,!#DF!tjo!71¡!Ï!31!>!1 20 - (- 34.64) ¥ 0.866 !>!68/83!lO!)U* 0.866 S# ! >!1;! Ï!#BF!Ï!#CF!dpt!71¡!,!#DF!dpt!71¡!,!#EF!>!1 fi! #EF!>!Ï28/43!Ï!45/75!¥!1/6!Ï!68/83!¥!1/6!>!Ï!74/6!lO!>!74/6!lO!)D* fi! #DF!>! Kpjou!D;! Bu!uijt!kpjou-!uxp!volopxo!gpsdft!#DE!boe!#DG!bsf!qfsqfoejdvmbs!up!fbdi!puifs/!Tp-!ju!jt!bewjtbcmf! up!dpotjefs!uif!frvjmjcsjvn!pg!gpsdft!jo!uif!ejsfdujpo!pg!uiftf!gpsdft!pof!cz!pof/!UivtS#}DG! >!1;! #DG!Ï!#DF!dpt!41¡!Ï!#CD!dpt!41¡!>!1 fi! #DG!>!68/83!¥!1/977!,!45/75!¥!1/977! fi! #DG!>!91!lO!)U* S#}DE! >!1;! #DE!,!#DF!dpt!71¡!Ï!#CD!dpt!71¡!>!1 fi! #DE!>!Ï!68/83!¥!1/6!,!45/75!¥!1/6!>!Ï22/65!lO!>!22/65!lO!)D* Kpjou!E;! S# ! >!1;! +E!,!#DE!tjo!71¡!>!1! fi! +E!>!Ï!)Ï22/65*!¥!1/977!>!21!lO!)≠* S# ! >!1;! -E!Ï!#EF!Ï!#DE!dpt!71¡!>!1! fi! -E!>!Ï!74/6!Ï!22/65!¥!1/6 fi! -E!>!Ï!7:/39!lO!>!7:/39!lO!)¨* Kpjou!G; S# ! >!1;! +G!Ï!#DG!dpt!71¡!>!1! fi! +G!>!91!¥!1/6!>!51!lO!)≠* S# ! >!1;! -G!Ï!#DG!tjo!71¡!>!1! fi! -G!>!91!¥!1/977!>!7:/39!lO!)Æ* 'XAMPLE!hrcl Find the forces in all members of a truss as shown in Fig. 6.24a which carries a horizontal load of 12 kN at the point D and a vertical load of 18 kN at the point C. \MP!7/4^ D 12 kN HA q A RA q 2m C 18 kN B 2m FCD FAD 1.5 m q HA A (a) Truss carrying two point loads q FAC RA RB FBD C FBC 18 kN B RB (b) FBDs of all the joints Gjh/!7/35 5OLUTION! Gspn!hfpnfusz-!q!>!uboÏ2)2/603*!>!47/98¡/!Dpotjefsjoh!uif!frvjmjcsjvn!pg!uif!usvtt!bt!b!xipmf-! xf!ibwf S# ! >!1;! Ï!-B!,!23!>!1! fi! -B!>!23!lO S&}B! >!1;! +C!¥!5!Ï!29!¥!3!Ï!23!¥!2/6!>!1! fi! +C!>!24/6!lO ŵŽźɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº S# ! >!1;! +B!,!+C!Ï!29!>!1! fi! +B!>!29!Ï!+C!>!5/6!lO Gjhvsf!7/35 !tipxt!gsff!cpez!ejbhsbnt!pg!uif!kpjout!B-!D!boe!C/!Mfu!vt!dpotjefs!uif!frvjmjcsjvn!pg!uiftf! kpjout!pof!cz!pof!up!dbmdvmbuf!uif!nfncfs!gpsdft/ Kpjou!B; - 4.5 -+A !>!Ï!8/6!lO!>!8/6!lO!)D* S# ! >!1;! #BE!tjo!q!,!+B!>!1! fi! #BE!>! !>! sin q sin 36.87∞ S# ! >!1;! #BD!,!#BE!dpt!q!Ï!-B!>!1 fi! #BD!>!-B!Ï!#BE!dpt!q!>!23!Ï!)Ï!8/6*!¥!dpt!47/98¡!>!29!lO!)U* Kpjou!D;! S# ! >!1;! #DE!Ï!29!>!1! fi! #DE!>!29!lO!)U* S# ! >!1;! #CD!Ï!#BD!>!1! fi! #CD!>!#BD!>!29!lO!)U* Kpjou!C;! - +B S# ! >!1;! #CE!tjo!q!,!+C!>!1! fi! #CE!>! !>!Ï!33/6!lO!>!33/6!lO!)D* sin 36.87∞ 'XAMPLE!hrcc For the truss shown in Fig. 6.25a, find the forces in the members. Given that all the horizontal and vertical members are of the same length.! \MP!7/4^ B B D FBD q FAB A C E RA F A RF 100 N q FBC FAC C D q q FCD FDE FDF E q F CE RA 100 N FEF q F RF 100 N 100 N (b) FBDs of all joints (a) The truss Gjh/!7/36 5OLUTION! Mfu! cf!uif!mfohui!pg!fbdi!ipsj{poubm!boe!wfsujdbm!nfncfs/!Gjhvsf!7/36 !tipxt!uif!GCEt!pg!bmm! kpjout/!Gspn!uif!hfpnfusz!pg!uif!usvtt-!xf!ibwf q!>!56¡/!Uif!tvqqpsu!sfbdujpot!+B!boe!+G!bsf!dbmdvmbufe!cz! dpotjefsjoh!frvjmjcsjvn!pg!uif!foujsf!usvttS&}G! >!1;! Ï!+B!¥!4 !,!211!¥!3 !,!211!¥! !>!1! fi! +B!>!211!O S# ! >!1;! +B!,!+G!Ï!211!Ï!211!>!1! fi! +G!>!311!Ï!+B!>!211!O Opx-!mfu!vt!dpotjefs!uif!frvjmjcsjvn!pg!kpjout!pof!cz!pof/ Kpjou!B; S# ! >!1;! #BC!tjo!56¡!,!+B!>!1! fi! #BC!>! -100 !>!Ï252/53!O!>!252/53!O!)D* sin 45∞ S# ! >!1;! #BD!,!#BC!dpt!56¡!>!1! fi! #BD!>!Ï!)Ï252/53*!dpt!56¡!>!211!O!)U* ŵŽŻ 4¹¼ºº¬ºɏ¨µ«ɏ#¨©³¬ºɏ Kpjou!C; S# ! >!1;! Ï!#BC!tjo!56¡!,!#CE!>!1 fi!!!#CE!>!Ï252/53!tjo!56¡!>!Ï211!O ! ! ! ! ! !!!!!!!!!!!!!!!!!>!211!O!)D* S# ! >!1;! Ï!#BC!dpt!56¡!Ï!#CD!>!1 ! fi! #CD!>!Ï!)Ï252/53*!dpt!56¡! ! ! ! >!211!O!)U* ! ! ! ! 100 B 0 100 141.42 D C A 141.42 100 ɏ E 100 100 100 N 100 N F 100 100 N 100 N DC is a zero-force member Kpjou!D;! ) Gjh/!7/37! S# ! >!1;! Ï211!,!#CD!,!#DE!tjo!56¡!>!1! O* 100 - 100 !>!1 sin 45∞ S# ! >!1;! Ï!#BD!,!#DE!dpt!56¡!,!#DF!>!1! fi! #DF!>!#BD!>!211!O!)U* Cz!tznnfusz-!xf!ibwf #EF!>!#CD!>!211!O!)U*<!#FG!>!#BD!>!211!O!)U*<!#EG!>!#BC!>!Ï252/53!O!>!252/53!O!)D* Gjhvsf!7/37!tipxt!uif!bobmztfe!usvtt!xjui!nfncfs!gpsdft!nbslfe/ ! fi! #DE!>! !! 'XAMPLE!hrcd Determine support reactions and forces in all the members of the truss shown in Fig. 6.27a. \MP!7/4^ 15 kN B B q1 4m FAB q2 FAC 4m D 4m FBC C A 15 kN q1 HA A q1 q2 C q2 FBD FCD q2 4m q1 D RA 4m 4m 4m RD 4m (b) FBDs of all joints (a) Truss Gjh/!7/38 5OLUTION! Gsff!cpez!ejbhsbnt!pg!bmm!uif!kpjout!bsf!tipxo!jo!Gjh/!7/38 xjui!nfncfs!gpsdft!boe!tvqqpsu! sfbdujpot!+B-!-B!boe!+E/!Dpotjefsjoh!frvjmjcsjvn!pg!uif!usvtt!bt!b!xipmf-!xf!ibwf S&}B! >!1;! +E!¥!9!,!26!¥!9!>!1! fi! +E!>!Ï26!lO!>!26!lO!)Ø* S# ! >!1;! +B!,!+E!>!1! fi! +B!>!Ï!+E!>!26!lO!)≠* S# ! >!1;! -B!Ï!26!>!1! fi! -B!>!26!lO!)Æ* Bohmft!q2!boe!q3 bsf!efufsnjofe!bt!! q2!>!uboÏ2!)905*!>!74/54¡! boe-! q3!>!uboÏ2!)505*!>!56¡ Mfu!vt!opx!dpotjefs!uif!frvjmjcsjvn!pg!kpjout!pof!cz!pof/ ŵŽżɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº Kpjou!B;! S# ! >!1;! #BC!tjo!q2!,!#BD!tjo!q3!,!+B!>!1! fi! #BC!tjo!74/54¡!,!#BD!tjo!56¡!,!26!>!1 )* fi 1/9:!#BC!,!1/818!#BD!>!Ï26! S# ! >!1;! #BC!dpt!q2!,!#BD!dpt!q3!,!-B!>!1! fi! #BC!dpt!74/54¡!,!#BD!dpt!56¡!,!26!>!1 fi! 1/558!#BC!,!1/818!#BD!>!Ï26! ) * Tpmwjoh!Frt/!) *!boe!) *-!xf!hfu!#BC!>!1! boe! #BD!>!Ï!32/33!lO!>!32/33!lO!)D* Kpjou!C;! S# ! >!1;! Ï!#BC!dpt!q2!,!#CE!dpt!q2!Ï!26!>!1! fi! #CE!>!44/65!lO!)U*! S# ! >!1;! Ï!#CD!Ï!#BC!tjo!q2!Ï!#CE!tjo!q2!>!1 15 kN B fi! #CD!>!Ï!#CE!tjo!74/54¡ ! !! ! >!Ï!41!lO!>!41!lO!)D* 30 Zero-force member Kpjou!E; S# ! >!1;! Ï!#DE!dpt!q3!Ï!#CE!dpt!q2!>!1 33.54 0 C -#BD cos q1 !!>!Ï!32/33!lO!! fi! #DE!>! cos q 2 21.22 21.22 ! ! >!32/33!lO!)D* A Gjhvsf!7/39!tipxt!uif!usvtt!xjui!nfncfs!gpsdft!nbslfe/ 'XAMPLE!hrce For the bridge truss shown in Fig. 6.29a, calculate the forces in members DF, CE, CF, EF and FH using the method of sections.! \MP!7/4^ D 15 kN 15 kN Gjh/!7/39! 15 kN ) lO* 5OLUTION! Mfu!vt!Ýstu!efufsnjof!uif!tvqqpsu!sfbdujpot!cz!dpotjefsjoh!frvjmjcsjvn!pg!uif!usvtt!bt!b!xipmf/! S&}B! >!1;! Ï211!¥!7!Ï!66!¥!:!,!+N!¥!29!>!1! fi! +N!>!71/94!lO S# ! >!1;! +B!,!+N!Ï!211!Ï!66!>!1! fi! +B!,!+N!>!266!lO fi! !!+B!>!266!Ï!71/94!>!:5/28!lO Mfu!vt!dpotjefs!uxp!tfdujpot!2 2!boe!3 3!dvuujoh!uif!usvtt!uispvhi!nfncfst!EG-!DG-!DF!boe!GI-!FG-!DF-! sftqfdujwfmz-!bt!tipxo!jo!Gjh/!7/3: /!!Gjhvsft!7/3: boe! !tipx!gsff!cpez!ejbhsbnt!pg!uif!mfgu iboe!qpsujpot! pg!uif!usvtt/ Bqqmzjoh!frvbujpot!pg!frvjmjcsjvn!up!uif!tfdujpo!2 2!pg!usvtt-!tipxo!jo!Gjh/!7/47 S&}D! >!1;! Ï!#EG!¥!4!Ï!:5/28!¥!4!>!1! fi! #EG!>!Ï!:5/28!lO!>!:5/28!lO!)D* S# ! >!1;! :5/28!,!#DG!tjo!56¡!>!1! fi! #DG!>!Ï244/2:!lO!>!244/2:!lO!)D* S# ! >!1;! #EG!,!#DG!dpt!56¡!,!#DF!>!1! fi! Ï!:5/28!,!)Ï244/2:*!dpt!56¡!,!#DF!>!1 fi! #DF!>!296/46!lO!)U* Dpotjefs!opx!uif!frvjmjcsjvn!pg!uif!tfdujpo!3 3!pg!uif!usvtt-!tipxo!jo!Gjh/!7/3: S# ! >!1;! :5/28!Ï!211!,!#FG!!1! fi! #FG!>!6/94!lO!)U* S# ! >!1;! #GI!,!#DF!>!1! fi! #GI!,!296/46!>!1 fi! #GI!>!Ï296/46!lO!>!296/46!lO!)D* ɏ ŵŽŽ 4¹¼ºº¬ºɏ¨µ«ɏ#¨©³¬ºɏ 100 kN D B F J H L N 3m M A C E 3m G 55 kN 3m 3m I 3m K 3m 3m (a) The bridge truss 100 kN 1 B D 2 F J H L N 3m A C 94.17 kN 3m 2 E G 55 kN 1 I M K 60.83 kN 3m 3m 3m 3m 3m (b) Truss being cut by Sections 1-1 and 2-2 100 kN 1 B D B FDF FCF D 2 F FFH FFE A A FCE C C 1 94.17 kN 3m FCE 2 94.17 kN 3m (c) FBD of LHS portion to Section 1-1 3m (d) FBD of LHS portion to Section 2-2 Gjh/!7/3: 30 kN B 60° 30 kN C 60° 5m 4m A F 15 kN 5m Gjh/!7/41! + 4m D E 4m ŶŴŴɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº 'XAMPLE!hrcf Figure 6.30 shows an industrial roof truss carrying loads due to wind. Find the forces in members BC, CF, EF, CE and DE. \MP!7/4^ 5OLUTION! Mfu!vt!sftpmwf!uif!41.lO!gpsdft!joup!ipsj{poubm!boe!wfsujdbm!dpnqpofout!bt!41!dpt!71¼!>!26!lO! boe!41!tjo!71¼!>!36/:9!lO-!sftqfdujwfmz/!Uif!41.lO!gpsdft!bsf!sfqmbdfe!cz!uifjs!dpnqpofout!bt!tipxo!jo! Gjh/!7/42 /!Dpotjefsjoh!gsff!cpez!ejbhsbn!pg!uif!usvtt-!xf!xsjuf S# ! >!1;! -B!Ï!26!Ï!26!>!1! fi! -B!>!41!lO S&}B! >!1;! E!¥ ¥ ¥ ¥ ¥ ¥ fi + S# + + fi + – ! Ê ˆ ÁË ˜¯ "# –$ % –$ % 25.98 kN ! Ê ˆ ÁË ˜¯ " !& q ! (5 - 4) 4 " ./ & &, 0 1 25.98 kN B 15 kN 60° 15 kN C q 5m 4m 45° A 45° 45° F HA D E 15 kN RA RD 1 5m 4m 4m (a) Section 1-1 cutting through the truss 1 25.98 kN FBC C q 60° 15 kN FCE 45° 4m FCF 45° FEF E D FEF E FDE RD 1 4m (b) FBD of right-hand part (c) FBD of the joint E Gjh/!7/42 * $' * '++ , * + - , ! . * ' * , * .* !& 1 0 * '++ + +* ! ,! , / + $ $ !& % !& - !+,& * 0 ɏ ŶŴŵ 4¹¼ºº¬ºɏ¨µ«ɏ#¨©³¬ºɏ %2' ,! * +' 0 ! + 0 33 * 0 - + / ' $ 6 S&7$ #% ¥ + ¥ fi #% + 34/47!lO!)U* 113 ,! 2' , !+ 0 2',3,/ ,' ,! 8 ,- 3 !& * ,6 ! 3 &, - , !+ S# # $ +,! q #$ - + " + fi # $ ; ;#$ fi # $ # $ - + q #$ +,! " #% S# fi ;# $ ; ;#$ fi # $ < 38,! %2+ 9 : !& 9 : #$ 24/94!lO!)D* !& # $ = & ,! 0 - + ,! / + $% !& % - !+,& * 0 / & &, , S# # % #% fi # % #% 34/47!lO!)U*! #$% 1 S# 'XAMPLE!hrcg ; 9∵ p #$ ; ": 9: #$ 9 : 3:/57!lO!)D* 0 * > ,! % + ,8 ! ,! For the simply supported truss shown in Fig. 6.32, find forces in members BD, DE, EG and CE. \MP!7/4^ F 10 kN q D 10 kN 2.25 m q B q A H C 1m E 1m = '++ - G 1m 1m ,! 3 &+ Gjh/!7/43 5OLUTION! !$ !+,& ,! S# S&7 S# * 0 / & &, -? +? ¥ ¥ + +? 0 * '++ +* ¥ fi fi + ! ,! , +? ; ; F 10 kN q D 10 kN 1 10 kN FBD 2.25 m q B A H q C RA 1m E 1m q B 1 1m G RH HH q A RA C 1m (a) Section 1-1 cutting through the truss FDE 1m E FEG q FCE FDE E FEG 1m (b) FBD of left-hand part Gjh/!7/44 FBE (c) FBD of joint E ŶŴŶɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº ! q $' * 2',3,/ ,' 2.25 = 36.87∞ 3 tan -1 '++ , * + - , ! 0 * 3 0 * !& 1 * ' * 0 * '++ / + S&7 # S# # +,! q fi # #%@ # fi #%@ 21!lO!)U* S# = & =*'+ FG AG ,! 0 - ,! S# % ¥ ¥ fi #$% % ! ,! , !& - !+,& 6!lO!)U*# fi # +,! ;" 23/6!lO!)D* fi - +q # #%@ fi #$% # + % / + $% - !+,& # % +,! q fi # % S# # % !& %@ + +* #%@ * 0 / & &, fi # % +,! 9/44!lO!)D* % # % - +q - + fi ;" 0 * > ,! % + ,8 ! ,! , ;" #$% 9 :- + ;" 34/44!lO!)U* 'XAMPLE!hrch Two weights of 15 kN and 40 kN are supported by a wire, which is fixed at the two points A and D as shown in Fig. 6.34a. Inclinations of segments AB and CD with the vertical are 25º and 45º, respectively. Determine tensions in all segments of the wire and inclination of BC with horizontal. \MP!7/5^ 5OLUTION! A 0 / * ,!-3,! , ! 0 * + - ! 113 A ,B+ * * 1 ,! + 8 3' + q =*'+ 0 + 0 1 ,! ! $ , * * * ,6 ! 3 <,!- * + + ,+ ,! 2',3,/ ,' !& $ ,! + 0 ! 3 q !& * ! - 1 * $ !& $ 9 , : * 8 Qpjou!C; BC sin(90∞ + 65∞) AB sin(90∞ - q ) fi $ 15 sin(180∞ - 65∞ + q ) 15 ¥ sin 155∞ sin(115∞ + q ) 6.34 sin(115∞ + q ) 9: 9 : Qpjou!D; BC sin(90∞ + 45∞) CD sin(90∞ + q ) fi $ 40 sin(180∞ - 45∞ - q ) 40 ¥ sin 135∞ sin(180∞ - 45∞ - q ) 28.28 sin(135∞ - q ) 9 : 9 !: ɏ ŶŴŷ 4¹¼ºº¬ºɏ¨µ«ɏ#¨©³¬ºɏ D A TAB 45° 25° 65° B q B TBC q 15 kN TCD TBC C 15 kN 45° q C 40 kN 40 kN (a) Concentrated loads supported by wire (b) FBDs of points B and C Gjh/!7/45 - 1 ,! %2+ 9 : !& 9 !: fi +,! fi fi C' ,! "- +q 6.34 sin(115∞ + q ) 28.28 sin(135∞ - q ) - + 28.28 9+,! 6.34 ; ;- +q 9 - +q * / 8 8 3' " +,! q ; ;: +,! q +,! q 6.34 sin(115∞ + q ) 'XAMPLE!hrci " +,! q: - +q +,! q q 63/26¡ 15 ¥ sin(90∞ - 52.15∞) sin(180∞ - 65∞ + 52.15∞) 6.34 sin(115∞ + 52.15∞) 40 ¥ sin(90∞ + q ) sin(180∞ - 45∞ - q ) ! - + 0 q ,! %2+ 9 : 9 : !& 9 : 15 ¥ sin(90∞ - q ) sin(180∞ - 65∞ + q ) $ fi "- +q 52/49!lO 39/62!lO 40 ¥ sin(90∞ + 52.15∞) sin(180∞ - 45∞ - 52.15∞) 35/85!lO A cable is suspended between the two points at the same level 270 m apart. The cable carries a deck, which supports a uniformly distributed load of 100 N per metre of horizontal length as shown in Fig. 6.35a. If the sag is 50 m, determine the horizontal thrust, support reactions and the maximum cable force \MP!7/5^ ŶŴŸɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº A 0.1 kN/m B 270 m HA 50 m Cable C HB x A B yC = 50 m RA y 100 N/m Deck RB C 270 m (a) Cable carrying a deck (b) FBDs of cable Gjh/!7/46 . 5OLUTION ! - ,- , " ! /7 :& wL2 8 yC < + 7 ,? , " $ %! &' * + , " ! 29/336!lO " -= - 7 , > = $ A B+ ¥ += D $ + ¥ C + H 2 + RA2 - ,! += ! & fi + wL B+ ¥ " += 0.1 ¥ 270 2 F A + 24/6!lO # 18.2252 + 13.52 . C- ¥ , 33/79!lO " - ,D - ,- , "F D D F - ?, " 24/6!lO FF - fi B ' 'XAMPLE!hrcj 270 2 - ?, # Bmufsobujwfmz-! , " ! ¥ 24/6!lO -- \ $ 7 0.1 ¥ 2702 8 ¥ 50 "= - ? S&@= % D E # " # ; - ! ¥ . C ¥ ' C- ¥ , FG ¥ C " ? " + $ . H ¥ 270 270 ¥ 2 4 fi - 29/336!lO Figure 6.36a shows the sectional view of a shelter. The vertical load on the 3-m long ridge cable is 125 N/m. Determine the sag of the ridge and horizontal component of cable force, if the tension in each guy cable is 600 N. \MP!7/5^ ɏ ŶŴŹ 4¹¼ºº¬ºɏ¨µ«ɏ#¨©³¬ºɏ 125 N/m 125 N/m Ridge cable T T T H Guy cable Guy cable 1.5 m RA T RB A B yC C H 1.5 m (a) Section of the shelter with guy cables (b) FBD of the ridge cable Gjh/!7/47 . ' $ 5OLUTION! %! & '& # ", " ! $ + $ # "! - ?, * "= #3& - ! D - ?, % D D F - ?, > ! H 2 + RA2 # * $ 125 ¥ 3 2 wL += & F F ? "D " ! "! - ?, ! ! " wx ( L - x) 2H Bmufsobujwfmz-! " ? ! fi - D , ! - ?, 7 :/7 & 125 ¥ 1.5 ¥ (3 - 1.5) 2 ¥ 570 ? - ,- , 6002 - 187.52 2 Tmax - RA2 " ?D 7 ; 681!O " : ; 1/357!n ! F FF - - ! , FG + S&@ & A B+ ¥ . fi B > 'XAMPLE!hrck C-¥ ¥ C C ¥ ¥ . . ¥ 2 4 C: ¥ ;¥ fi 1/357!n A cable AB spanning 75 m supports a horizontal deck as shown in Fig. 6.37a. The deck carries a uniformly distributed load of 30 kN/m throughout its span. The support A is 15 m above the support B. The lowest point C of the cable is 5 m below the support B. Determine the horizontal component of the cable force, support reactions and the maximum cable force. \MP!7/5^ ŶŴźɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº RA 75 m A A l1 HA 15 m RB 20 m B C 5m C 30 kN/m H HB C 30 kN/m (a) Cable AB carrying the deck B l2 5m 30 kN/m (b) FBDs of the parts AC and CB Gjh/!7/48 A. 5OLUTION " ! ' " ? F - ,- /7 + ,, ! F " " = $ %! &' I F -? ! F F ,, F - S&@ I , ,D !:; - wl22 10 - = ? F ? "D - ?, $ wl12 40 :; =! fi C: ¥ ;¥ 30 ¥ 502 40 - - wl22 10 : ; * " # " S# " &# '& &# H 2 + RA2 ( % ) & ! fi + +, ¥ &# '& ( + +, ¥ % &. $ % + +, ¥ $ % * 2986!lO 2986!lO " # " S# &. 2 - + " , F fi wl12 40 ! -% fi 1 ,D + ": ; $ ! - ! B- ¥ A wl12 40 ( F - - ?, + ,- ? B: ¥ ;¥ 7 ,? S&@= /7 -¥ A = % ) 18752 + 15002 fi + 2611!lO fi + 861!lO & ! fi + +, ¥ " & % ' %&) $ &. " 3512/28!lO ' % ' % & -%" ɏ ŶŴŻ 4¹¼ºº¬ºɏ¨µ«ɏ#¨©³¬ºɏ 'XAMPLE!hrdl Figure 6.38a shows a chain connecting a boat with its anchor at the sea bottom, 35 m below the surface of water. The boat exerts a horizontal force of 5 kN on the chain. Self-weight of the chain is 18 N/m. The buoyant force on the chain in water is 2 N/m. Determine the horizontal span of the chain and the maximum force in the chain. \MP!7/5^ RB Water surface Boat B B 5 kN Chain 35 m 35 m Anchor A RA q A x H L L y (a) Chain connecting the boat and anchor (b) FBD of the chain AB Gjh/!7/49 5OLUTION! !/ ) * , 6 9 6 : ; 6 #" <& $ " = ,: % % $ # !< ! & & $ % %& -% ' > "& $ % " % $ $ #" <& < -%" $ % %& 9 -&9 ' & & !" - H Ê gx ˆ H cosh Á - ˜ + Ë H¯ g g = $ & ' % %& % B Bmufsobujwfmz-! % -+ . $ fi % - &. % &. " " $ gx H & ' % fi $ & + = ¥ +, &. ) = $ & gx H - gy H % :? . % ' 5000 Ê 16 ¥ (- 35) ˆ cosh -1 Á1 ˜ Ë 16 5000 ¯ fi . - = ; = ?, ; %& & -% % @ %&) H gy ˆ Ê cosh -1 Á1 - ˜ Ë q H¯ fi 257/67!n % #& %& ¥ % ' !& &. " 16 ¥ 146.56 5000 #< = = -%" 6/67!lO ) & % ' = % -%" " 6/67!lO "& $ % ' & ŶŴżɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº ! 57//#4; M& &9 & difdl! zpvs!voefstuboejohN!O & % &!G& PD ! .1!hrc! Define and identify various types of trusses. usvtt " % & &!C# & & C! & " " " C! -% & ' & " )& " <' & "& & & ! $ 455 6 ( 5 5278 .1!hrd! Test the stability of a given truss. jefbmjtfe kvtuCsjhje " % % % E '& " pwfs.sjhje!usvttft Joefufsnjobdz $ & # & &< !" < % "& $ " # " D " " )& $ ) & &" # &9 $% % " & ! % % &' & .1!hre! Determine support reactions and forces in members of a truss. -% # %& & ! {fsp.gpsdf! F % ! $G F % ! $ & < ! $ & & ! tusvut # ! % " # " $ % G &9 ! ) !"& < ) ) &'' < " # " "& '& $ % $ & " & ! ujft , & " .1!hrf! Evaluate shape, support reactions, and forces at the various points in a cable carrying the point loads, uniformly distributed load and its self-weight. Dbcmft & % % %&) ) < & # ! & ! < Nbyjnvn ufotjmf! gpsdf! & &# ' !" ! & % ' $ Njojnvn!ufotjmf!gpsdf! & &# " & % ' % ' 1 -% &# ' > " ! HI & ! " ! $C % ! ' -% < & $ ' $ % &# % &! % " % & % "'' +/2146#06!(14/7.#' J -% J & &# & &# & < & < HI & K > qbsbcpmjd L & $ @ I J & &# " ! K > & % - 1 + tan 2 a \MP!7/5^ \MP!7/5^ wx ( L - x) hx + 2H L wL2 8 yC . % $ &# & ! &! @ & '& * $C ! dbufobsz $ "'' w2 L3 h 2 + L 24 H 2 & & & % ) % ! $$ ) $ "'' %* - H g gx ¸ Ï - 1˝ Ìcosh H Ó ˛ \MP!7/5^ ɏ I HÊ gl2 gl ˆ + sinh 1 ˜ Á sinh gË H H¯ % $ &# * - ! ŶŴŽ 4¹¼ºº¬ºɏ¨µ«ɏ#¨©³¬ºɏ gx H - gx H - 5*146 #059'4!37'56+105 ! ! 2/! Xibu!bsf!uif!tjnjmbsjuz!boe!ejggfsfodf!cfuxffo!b!usvtt!boe!b!dbcmf@! ! ! 3/! Xiz!bsf!uif!kpjout!pg!b!usvtt!dpotjefsfe!qjoofe@! ! \MP!7/2^ ! " ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! 4/! Obnf!uif!nbjo!qbsut!pg!b!usvtt/! ! # 5/! Xibu!jt!uif!ejggfsfodf!cfuxffo!b!kvtu.sjhje!boe!bo!pwfs.sjhje!usvtt@! $ % % ! " " 6/! EfÝof!uif!efhsfft!pg!joufsobm!boe!fyufsobm!joefufsnjobdz!pg!usvttft/! ! \MP!7/2^ \MP!7/3^ $ % \MP!7/3^ & \MP!7/3^ 7/! Dbo!xf!dbmm!b!usvtt!up!cf!dpnqmfufmz!efufsnjobuf!jg!uif!tvn!pg!jut!joufsobm!boe!fyufsobm!joefufsnjobdjft!jt! {fsp@! \MP!7/3^ ! ' * + 8/! Xibu!jt!uif!nbyjnvn!ovncfs!pg!nfncfst!pg!b!qmbof!usvtt-!xijdi!tipvme!cf!dvu!cz!b!tfdujpo!jo!uif!nfuipe! pg!tfdujpot@! \MP!7/4^ ! 9/! Xibu!jt!uif!ejggfsfodf!cfuxffo!frvjmjcsjvn!frvbujpot!vtfe!jo!uif!nfuipe!pg!kpjout!boe!nfuipe!pg!tfdujpot! xijmf!dbmdvmbujoh!gpsdft!jo!nfncfst@! \MP!7/4^ ! & ,# ,# -! & ,& -! :/! Xibu!jt!uif!qspÝmf!pg!b!dbcmf!pg!ofhmjhjcmf!tfmg.xfjhiu!dbsszjoh!b!ovncfs!pg!qpjou!mpbet@! \MP!7/5^ ! / 21/! Xibu!bsf!uif!qspÝmft!pg!b!dbcmf!voefs!jut!pxo!xfjhiu!boe!uibu!dbsszjoh!VEM!po!b!ipsj{poubm!tqbo@! \MP!7/5^ ! / ŶŵŴɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº -';!6'4/5 .1!hrc .1!hrd 7 $ .1!hre % ! .1!hrf 9 9 # # 4'8+'9!37'56+105 ; ! 2/!: \MP!7/5^ ;# ! 3/!: \MP!7/2^ ! 4/!< \MP!7/3^ ! 5/!< \MP!7/3^ ;$ ! 6/!$ ; \MP!7/3^ ! 7/!< * \MP!7/5^ < ! 8/!: \MP!7/3^ : ! 9/!# ; \MP!7/4^ & ! :/!< ! ! * ! \MP!7/5^ 1$,'%6+8'!37'56+105 #! /ULTIPLE %HOICE!3UESTIONS = ! 2/! > \MP!7/2^ ! ! ! ! ! 3/! ? \MP!7/3^ ! ! @ ! ! ! 4/! $ \MP!7/3^ ! ! @ ! ! 5/! $ ! * \MP!7/5^ ! ! % ! ! ! 6/! / \MP!7/3^ ! ! ! ! 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Tmjejoh!gsjdujpo!jt! joefqfoefou!pg!bsfb!pg!dpoubdu-! xifsfbt!uif!spmmjoh!gsjdujpo! jodsfbtft!xjui!jodsfbtf!jo!bsfb! pg!dpoubdu/! Fact! Gps!nptu!pg!uif!tvsgbdft-!uijt! jt!usvf/!Cvu-!gps!tpnf!tvsgbdft-! dpfgÝdjfout!dbo!cf!npsf!uibo! vojuz-!f/h/-!gps!dpqqfs.dpqqfs! tvsgbdft-!ju!jt!2/32!boe!svccfs!jo! dpoubdu!xjui!puifs!tvsgbdft!dbo! zjfme!gsjdujpo!dpfgÝdjfout!gspn! 2!up!3/ ŶŶŴɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº Fmax N f m a f f !" m " f £f #! Dpof!pg!Gsjdujpo! $ ! " " !! ( % " ) ) % *f % ( dpof!pg!ljofujd!gsjdujpo ' ' *f ( ! % ( ! ! ) , m " m . / . . . .. ' .0 ./ 1 . + . +0 )2 . ./ . ./ . *. . - " " - % " 3 . . . /. . +. . . 5. . 6. . 5. ! . /. . +. % . * $. . . *. % . / . . % 3 . ./ 7 . 6/ . /. $ . . . .+ . ./ . ./ * 3 4 !! % 4 !! 4 !! " '! ( % ( % )2 1 " % " 2 *f % $ ! ( - " ' ! ! ' % & + ) ! ! % ! ! ! & ( $ ( dpof!pg!tubujd!gsjdujpo $ ! ! !" % % ɏ ŶŶŵ &¹°ª»°¶µɏ Cone of static friction R R W 90° Cone of kinetic friction F= a 2fk 2fs mN N f (b) Angle of repose (a) Cone of friction Bohmf!pg!Sfqptf! % ' & & ' & + % ! " " ! , ) a% a % " ! 8 f ' 9 % f : ( ; ( ( m m ) f f f f % 8/3! .1!ird $ ,°º»ɏ»¯¬ɏ³¨¾ºɏ¶ɏ¹°ª»°¶µȍ & * ) + = �! MBXT!PG!GSJDUJPO 5 1 ! mbxt!pg!gsjdujpo< % % " " : / & ! % > ! ! % ! % 0 & ! " ? 8/4! .1!ire �! DPOEJUJPOT!PG!GSJDUJPO!EFWFMPQNFOU < 2/!Dpoejujpo!pg!Tubujd!Frvjmjcsjvn/ ) ! ( 3/!Dpoejujpo!pg!Jnqfoejoh!Npujpo/ ) ( " ) %´·³¶Àɏ»¯¬ɏª¶µª¬·»ɏ ¶ɏ¹°ª»°¶µɏ»¶ɏ·¹¨ª»°ª¨³ɏ ·¹¶©³¬´ºȍ m ) : m ( '! m " & ) ) @ ( ) " m ŶŶŶɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº 4/!Dpoejujpo!pg!Sfmbujwf!Npujpo! $ ! m' ' ( 8 ! " ) ! A ! ! " !" "! % ! )j*!!Gps!Ipsj{poubm!Qmbof; $ ( B )jj*!Gps!Jodmjofe!Qmbof; B S S . . $ ( 8 S S . $ ( . $ ( 'XAMPLE!irc A body of 350-N weight is resting on a rough horizontal surface (Fig. 7.4a). Find the magnitude of the pulling force acting on the body at an angle of 25° with the horizontal, which will just make the body move. Take m = 0.3. ! " 5OLUTION! 7 < ' ! % & S S - + .9 m .+ m / % .< 3 .< E % . +9 ' * D * D " ! . fi .+ 3.6 + . fi E . /* &?C . + . W = 350 N P 25° N P 25° F = mN F = mN W = 350 N N (b) Force diagram (a) Block on horizontal surface n 20° n 50 N n 50 N N 50 N t 14° t 14° mN t 14° mN 20° 100 N N 100 N 100 N 20° (a) (b) (c) Force diagram ɏ ŶŶŷ &¹°ª»°¶µɏ A block weighing 100 N is resting on a rough plane inclined at 20° to the horizontal. It is acted upon by a force of 50 N directed upwards at an angle of 14° above the plane as shown in Fig 7.5a. If the block is about to move up the plane, calculate friction and coefficient of friction. ! " 'XAMPLE!ird 5OLUTION! Hjwfo;! !>!211!O-! !>!61!O/ Bt!uif!cmpdl!jt!bcpvu!up!npwf!vqxbset-!gsjdujpo!bdut!epxo!uif!qmbof!bt!tipxo!jo!Gjh/!8/6 /!Frvbujpot!pg! frvjmjcsjvn!bsf!bqqmjfe!up!uif!gpsdft-!xijdi!bsf!sftpmwfe!opsnbm!boe!ubohfoujbm!up!uif!jodmjofe!qmbof/ S ! >!1;! !,!61!tjo!25¡!Ï!211!dpt!31¡!>!1! fi! !>!92/98!O S ! >!1;! 61!dpt!25¡!Ï!211!tjo!31¡!Ï!m !>!1! fi! m!>! ! Gsjdujpo-! >!m !>!1/28!¥!92/98!>! # A body resting on a horizontal rough surface requires a pull of 150 N inclined at 30° to the horizontal to initiate the motion, as shown in Fig. 7.6a(i). Also, it requires a push of 250 N inclined at 40° to the horizontal to just start the motion, as shown in Fig. 7.6a(ii). Calculate the weight W of the body and the coefficient of friction. ! " 'XAMPLE!ire 5OLUTION! $ %j&'! ( 2 /!Uif!261$O!gpsdf!jt!sftpmwfe!joup!uxp!dpnqpofout!>!261!dpt!41¡!>!23:/:!O<! ! 2 !>!261!tjo!41¡!>!86!O 160.7 N 75 N 30° 30° 40° 129.9 N 191.5 N F2 = mN2 F1 = mN1 (i) N1 W N1 75 N 40° N2 P1 30° 129.9 N F1 = mN1 W F2 = mN2 P2 160.7 N 40° W 191.5 N (i) (a) Block on horizontal surface N2 W P2 = 250 N (ii) P2 P1 P1 = 150 N (ii) (b) Free body diagrams and force diagrams ( Bqqmzjoh!uif!frvbujpot!pg!frvjmjcsjvn!)tff!uif!GCE!jo!Gjh/!8/7 ) **-!xf!hfu S ! >!1;! 23:/:!Ï!m 2!>!1! fi! m 2!>!23:/: boe! S ! >!1;! 2!,!86!>! ! fi! 2!>! !Ï!86 ŶŶŸɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº Ejwjejoh!uif!uxp-! m!>! 129.9 ! - 75 !) * %jj&;! ( !Uif!361.O!gpsdf!jt!sftpmwfe!joup!uxp!dpnqpofout!>!361!dpt!51¡!>!2:2/6!O<! ! ! 3 !>!361!tjo!51¡!>!271/8!O 3 Bqqmzjoh!uif!frvbujpot!pg!frvjmjcsjvn!)tff!uif!GCE!pg!Gjh/!8/7 ) **-!xf!hfu $ S ! >!1;! m 3!Ï!2:2/62!>!1! fi! m S ! >!1;! 3!Ï! !,!271/8!>!1! fi! boe! Ejwjejoh!uif!uxp-! m!>! F3 !>! 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B! bu! uif! foe!B/! Tjodf! uif!mbeefs!ibt!b!ufoefodz!up!tmjq!bxbz!gspn!uif!xbmm-!uif! gsjdujpo! B!bdut!upxbset!uif!xbmm/! Uif! qspcmfn! pg! mbeefs! gsjdujpo! jowpmwft! b! tztufn! pg! opo. dpodvssfou!gpsdft/!Uifsfgpsf-!xf!vtf!gpmmpxjoh!uisff!frvbujpot! pg!frvjmjcsjvn/ S !>!1<! S !>!1! boe! S)!>!1 Fact! Op/!Uifsf!jt!pomz!pof!jodmjofe! sfbdujpo-!xiptf!dpnqpofout! bsf!opsnbm!sfbdujpo!boe!uif! gsjdujpo/! FB = mNB NB B Rough wall ! Tvctujuvujoh! G q A FA = mNA W Rough floor NA % C ɏ ŶŶŹ &¹°ª»°¶µɏ 'XAMPLE!irf Determine the minimum angle q at which a uniform ladder can be placed against a wall without slipping under its own weight. The coefficient of friction for all surfaces is 0.2. ! " 5OLUTION! Sfgfs!up!Gjh/!8/8/!Bqqmzjoh!uif!frvbujpot!pg!frvjmjcsjvn-! S ! >!1;! C!>!m B! fi! boe! S !>!1;! B!,!m )m B*!>! ! fi! B!,!m C!>! B!>! (1 + m 2 ) Jg! !jt!uif!mfohui!pg!uif!vojgpsn!mbeefs-!ubljoh!npnfout!bcpvu!uif!foe!C-!xf!hfu S)}C! >!1;! Ï dpt!q!,! !¥! !dpt!q!,!m B !tjo!q >!1 3 Wl ! !tjo!q >!1 ! !dpt!q!,! !dpt!q!,!m! fi! Ï! 1 + m2 3 1 + m2 8/5/3! B fi!! ÏÔ 2 - 1 - m 2 ¸Ô ÏÔ 1 1 ¸Ô !tjo!q!>! !dpt!q!>! !dpt!q ˝ Ì Ì 2 ˝ 2 2 ˛Ô 1 + m2 ÓÔ1 + m ÓÔ 2 (1 + m ) ˛Ô fi ubo!q!>! m 2 Ê1 - m2 ˆ 1 - m2 Ï2 Ê 1 - 0.2 ˆ ! fi! q!>!uboÏ2! Á !>!ubo ! ˜ Á 2 ¥ 0.2 ˜ !>!( 2m Ë 2m ¯ Ë ¯ )* Xfehf!Gsjdujpo B!xfehf!jt!pof!pg!uif!tjnqmftu!boe!nptu!vtfgvm!nbdijoft/!Ju!jt!b!xppefo!ps!nfubmmjd!cmpdl!xjui!usjbohvmbs!ps! usbqf{pjebm!dsptt!tfdujpo!bt!tipxo!jo!Gjh/!8/9/!Ju!jt!vtfe!up!sbjtf!ifbwz!cmpdlt!ps!up!dibohf!uifjs!qptjujpot/! Uif!xfjhiu!pg!b!xfehf!jt!wfsz!tnbmm!bt!dpnqbsfe!up!uif!xfjhiu!mjgufe/!Ifodf-!ju!jt!vtvbmmz!ofhmfdufe!jo!uif! dbmdvmbujpot/!B!xfehf!efqfoet!po!gsjdujpo!gps!jut!xpsljoh/ Gjhvsf!8/9 a $ !"# ! $% " sin(90∞ + 2f + a ) & !1 !2 & sin(180∞ - a - f ) sin(90∞ - f ) ! $% !2 !3 & & sin(180∞ - 2f - a ) sin(90∞ + f ) sin(90∞ + f + a ) '$ *+ !"# , - '$ *+ '$ $+ a '$ $+ ŶŶźɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº Block R1 f R2 a f W a f R3 f R2 R2 Block (90° + 2f + a) a Wedge P (180° – a – f) (90° – f) R1 (90° + f) R3 P (180° – 2f – a) R2 (90° + f + a) W (a) P Wedge (b) (c) ) 'XAMPLE!irg A 1500-N block overlying a 10° wedge on a horizontal floor and leaning against a vertical wall, is to be raised by applying a horizontal force P as shown in Fig. 7.9a. Determine the force P necessary to just start the motion, if the coefficient of friction is 0.3. ! " 1500 N R2 R1 f 10° f O P O1 f f R3 10° 1500 N f = 16.7° 10° P R2 f = 16.7° R2 R3 (90° + 2f + 10°) (90° + f) (180° – 10° – 2f) P (90° – f) R1 1500 N (a) Wedge pushed by P R2 (c) FBD of wedge (b) FBD of block # f& 5OLUTION! !"# ' /4 m& /4 !4 !: ! 5 6 & 4* $7 !6 $; + 1500 !2 & sin(90∞ + 2 ¥ 16.7∞ + 10∞) sin(90∞ - 16.7∞) fi !: & 4;$$ < = $; ɏ ŶŶŻ &¹°ª»°¶µɏ sin(180∞ - 10∞ - 2 ¥ 16.7∞) .1!irg 8/6! & !2 sin(90∞ + 16.7∞) fi & 5 $: ¥ 4;$$ < & �! SPMMJOH!GSJDUJPO! % 5µ«¬¹³°µ¬ɏ»¯¬ɏ¨ª»¶¹ºɏ »¯¨»ɏ¨Ĺ¬ª»ɏ»¯¬ɏ¹¶³³°µ®ɏ ¹°ª»°¶µȍ W spmmjoh!gsjdujpo/ r > O ! P f $ 45 ? A B ' @ ! + p @ ! A R a B ! fi f& m D a & r cosf &m dpfgÝdjfou!pg!spmmjoh!gsjdujpo" ' E + - '$ %+ m " ? fi a a P ª & m & r cosf r W ' f B ? 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Ju! jt! jnqpttjcmf! up! bdijfwf! 211&! fgÝdjfodz! pg! b! nbdijof! evf! up! gsjdujpo!jo!uif!npwjoh!qbsut/ Jo!tqjuf!pg!uiftf!ejtbewboubhft-!gsjdujpo!jt!offefe!up!qfsgpsn!wbsjpvt!bdujpot/!Uivt-!gsjdujpo!jt!b!ofdfttbsz!fwjm/ Uifsf!bsf!b!ovncfs!pg!nfuipet!up!sfevdf!gsjdujpo/!Tpnf!bsf!ejtdvttfe!cfmpx/ 2/! B!dpnnpo!xbz!up!sfevdf!gsjdujpo!jt!cz!vtjoh!b!mvcsjdbou-!tvdi!bt!pjm-!xbufs!ps!hsfbtf/!Uif!mvcsjdbou!jt! bqqmjfe!cfuxffo!uif!uxp!tvsgbdft/!Uijt!esbnbujdbmmz!sfevdft!uif!dpfgÝdjfou!pg!gsjdujpo/ ! 3/! Efwjdft!tvdi!bt!xiffmt-!cbmm!cfbsjoht-!spmmfs!cfbsjoht!boe!bjs!dvtijpot!ps!puifs!uzqft!pg!Þvje!cfbsjoht! dbo!dibohf!tmjejoh!gsjdujpo!joup!b!nvdi!tnbmmfs!spmmjoh!gsjdujpo/ ! 4/! Gsjdujpo!dbo!cf!sfevdfe!cz!dibohjoh!uif!eftjho!pg!gbtu$npwjoh!pckfdut/!Uif!gspout!pg!wfijdmft!boe! bjsdsbgu!bsf!nbef!pcmpoh!up!njojnjtf!bjs!gsjdujpo/ ! ɏ ŶŶŽ &¹°ª»°¶µɏ ! #&&+6+10#.!51.8'&!':#/2.'5 A body of 400-N weight is lying on a rough plane inclined at an angle of 25° with the horizontal. It is supported by a force P parallel to the plane as shown in Fig. 7.11a. Calculate maximum and minimum values of P, for which the equilibrium can exist, if the angle of friction is 20°. ! " 'XAMPLE!irh n n P P t 25° 25° F 25° F 25° 400 N N 400 N N n N t n 25° N t t 25° P P 25° F 25° F 400 N (a) Block tending to slip upwards ( 400 N (b) Block tending to slip downwards ' 5OLUTION! Hjwfo;! !>!511!O-!q!>!36¡-!f >!31¡/ Uif!dpfgÝdjfou!pg!gsjdujpo-!m!>!ubo!f!>!ubo!31¡!>!1/47 ) *! . / . . Q'!Uif!cpez!jt!po!uif!wfshf!pg!tmjejoh!vqxbset/!Uif!gsjdujpo!bdut!epxoxbset!bt!tipxo! jo!Gjh/!8/22 /!Gpsdft!bsf!sftpmwfe!opsnbm!boe!ubohfoujbm!up!uif!jodmjofe!qmbof/!Bqqmzjoh!frvbujpot!pg! frvjmjcsjvnS ! >!1;! !Ï!511!dpt!36¡!>!1! fi! !>!473/63!O S ! >!1;! !Ï! !Ï!511!tjo!36¡!>!1! fi! !Ï!1/47 !Ï!27:!>!1 fi! !>! ## ) * . . . Q '!Uif!cpez!jt!po!uif!wfshf!pg!tmjejoh!epxoxbset/!Uif!gsjdujpo!bdut!vqxbset-!bt!tipxo!jo! Gjh/!8/22 /!!Uif!wbmvft!pg! !boe! !sfnbjo!tbnf!bt!jo!) *!bcpwf/!Bqqmzjoh!uif!frvbujpot!pg!frvjmjcsjvn-! S ! >!1;! 'XAMPLE!iri !,! !Ï!511!tjo!36¡!>!1 fi! !,!1/47 !Ï!27:!!>!1! fi! !>! ) A horizontal force P is applied on a block of 2000-N weight kept on a rough inclined plane, as shown in Fig. 7.12a. Determine whether the block is in equilibrium, if q = 30° and P = 400 N. Take m s = 0.3 and m k = 0.2. ! " ŶŷŴɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº 2000 N n 2000 N n P t N t 30° 400 N F 30° F 400 N q 2000 N N (a) Block on an inclined plane (b) FBD of block (c) Force diagram 5OLUTION! Uif!cmpdl!dbo!ibwf!b!ufoefodz!fjuifs!up!hp!epxo!ps!up!hp!vq!uif!qmbof/!!Gjstu-!mfu!vt!bttvnf! uibu!uif!cmpdl!jt!jo!frvjmjcsjvn!boe!ibt!b!ufoefodz!up!tmjef!epxo/!Sfgfs!up!uif!GCE!pg!uif!cmpdl!tipxo!jo! Gjh/!!8/23 /! Sftpmwjoh!uif!gpsdft!bmpoh! .byjt!boe! .byjt-!boe!bqqmzjoh!frvbujpot!pg!frvjmjcsjvn!)Gjh/!8/23 *-! xf!hfu S ! >!1;! !Ï!3111!dpt!41¡!Ï!511!tjo!41¡!>!1! fi! !>!2:43!O S ! >!1;! !,!511!dpt!41¡!Ï!3111!tjo!41¡!>!1! fi! !>!764/69!O!>! fr Cvu-! nby!>!m !>!1/4!¥!2:43!>!68:/7!O/!Tjodf!uif!gsjdujpo! fr!dbo!ofwfs!cf!hsfbufs!uibo! nby-!uif! ./!Ju!jt! 1 1 2 3 Ifodf-!uif!bduvbm!wbmvf!pg!uif!gsjdujpobm!gpsdf!jt!hjwfo!bt '!>!m' !>!1/3!¥!2:43!>! )( Tjodf!ju!jt!dpoÝsnfe!uibu!uif!cmpdl!jt!tmjejoh!epxo!uif!qmbof-!uifsf!sfnbjot!op!offe!up!dpotjefs!uif!puifs! qpttjcjmjuz!pg!uif!cmpdl!hpjoh!vq!uif!qmbof/ 0 'XAMPLE!irj A uniform ladder of 5-m length and 20-N weight is placed against a smooth vertical wall with its lower end 4 m away from the wall. If the ladder is just to slip, determine the coefficient of friction between the ladder and floor, and the frictional force acting on the ladder at the point of contact with the floor. ! " 5OLUTION! Wbsjpvt!gpsdft!bdujoh!po!uif!mbeefs!bsf!tipxo!jo!Gjh/!8/24/ ! DB!>!5!n<!!!!BC!>!6!n! fi! DC!>! 52 - 42 !>!4!n! fi tjo!q!>!406!boe!dpt!q!>!506 Sftpmwjoh!bmm!uif!gpsdft!bmpoh!ipsj{poubm!boe!wfsujdbm!ejsfdujpot!B!>! !>!31!O! boe! !C!>! Ubljoh!npnfout!bcpvu!uif!foe!B- B!>!m!B !C!¥!DC!>!31!¥!BH!dpt!q! fi! !C!>! boe!uivt! m!>! RC 13.33 ! >! W 20 >!m 20 ¥ 2.5 ¥ (4 / 5) !>!24/44!O 3 (( m! ¥ ɏ Ŷŷŵ &¹°ª»°¶µɏ B FB = 0.2RB RB B 20 N 30 5mG 250 N x q FA = mRA A 800 N C Rough floor Rough wall G RB ° Smooth wall 5m RA FA = 0.2RA 4m A Rough RA floor C 'XAMPLE!irk A 5-m long ladder and of 250-N weight is placed against a rough vertical wall in a position where its inclination to the vertical is 30°. A man weighting 800 N climbs the ladder. At what position will he induce slipping? The coefficient of friction for both the contact surfaces of the ladder, viz., with the wall and the floor is 0.2. ! " 5OLUTION! % $ S S 9 ! 8 9 <= $ & ! ' !* & ! !* ' 6 ' 7 # : ; !* :; < fi ! - # !* & !* ¥ 6 ? @!* ¥ 6 ? @'7 ¥ fi ??!* - 6!* ' '? 6 fi : ¥ ?? - 6 ¥ :' '? 9 $ $ 'XAMPLE!ircl # $ +, + , 6 q 6 S)> S)>* # S ? @' 6 ¥ 6 6 fi ? @ (( . Determine the force P required to move the uniform 50-kg plank from its position of rest (Fig. 7.15a). Take the coefficient of friction at both the contact surfaces as 0.5. ! " 6 ¥:7 '!! " " $ $ S)> + ! ! ' : ; $ 6 + 7A , ? 7 @ * 2.42 + 1.82 ? $ *B = <= $ $ & ' : ¥ ? 7 @¥? fi ? ? & : ¥ +? ' , ? 7 @- 6 ¥? ? 7 @' fi 6 7; # $ = & ' ' : ? 7 @- 6 - 6 ? 7 @fi #( * ; ¥? ? 7 @ ? 7 @ ŶŷŶɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº P P C 36.87° C 490 N B 4.8 m B 1.8 m F1 = 0.5N1 2.4 m 1.8 m A 3m F2 = 0.5N2 q A 2.4 m N1 36.87° N2 (a) (b) 'XAMPLE!ircc Find the least horizontal force P required to just start the motion of any part of the system resting on one another as shown in Fig. 7.16a. The weights of blocks A, B and C are 3000 N, 1000 N and 2000 N, respectively. Coefficient of friction between the blocks A and B is 0.3, between B and C is 0.2 and between the block C and ground is 0.1. ! " A WA P A B A WA P A WA P WB B FA NA WB B FB C P WC C NB FC NC NC NB FC NA FA FB A P A (b) Block A A WA WB WA (a) Block A being pulled P P WA WB WC (c) Block A and B (d) Block A, B and C ( 5OLUTION C9 & $ ? ; !% &'! * * ' ; * ; m D = * D $ = $ # S S ? m m * $ & & ? m ; ?¥? = $ : ; 9 * < m E E # $ ɏ Ŷŷŷ &¹°ª»°¶µɏ % &'! ( * $ ' * $ S S % &'! + $ 'D D m * * & $ * & * * - * m * ? - # S S F % 7 = D $ & & m D 9 * - 9 - m? D - ¥ D $ E $ ? D * # ; $ D = # ; ¥ ' * < $ E E $ E D = # ; ; $ 9 ( = 'XAMPLE!ircd Two blocks A and B, each of weight W, are placed on a rough inclined plane. The blocks are connected by a light string. If mA = 1/2 and m B = 1/3, show that both the blocks will be on the point of impending motion when the plane is inclined at an angle q = tan–1 (5/12). ! " T A B FB A q W NB n B NA FB T q B =* < = W $ = $ 8$ E E S S q q ' m* * % + & & = m* =*+ 9 * $ & & (c) $ 8 = * x A (b) m* G m FA T (a) S S q q W 5OLUTION 8 q W NA n x NB q q FA T m * $ 9 I H =* =* 9 J , q ' m* q K q ' + A?, qL +, q' q K+ A , q' qL + , , q % m ' q m ŶŷŸɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº M# M# + , + , q ' + A?, fi 5 12 q q fi q q' + A , 4 q fi q ' K+ A?, - + A ,L q 5 12 'XAMPLE!irce A 108-N block is held on a 40° incline by a bar attached to a 150-N block on a horizontal plane as shown in Fig. 7.18a. The bar which is fastened by smooth pins at each end is inclined at 20° to the horizontal. The coefficient of friction between each block and the planes is 0.325. For what horizontal force P applied to the 150-N block, will the motion to the left be impending? ! " N1 C 20° P 2 F2 N2 C 108 N 1 F 150 N 1 2 N1 C 108 N F2 = 0.325N2 1 20° 40° 40° N2 40° P P 20° F1 = 0.325N1 20° C 1 C 108 N 150 N (a) 150 N (c) (b) ) $ 5OLUTION $ = = $ 9 E 8 9 9 E S # & S & M# + , S S + , & & 'XAMPLE!ircf < *B $ # = @' ? 6 @fi :? 76 @- ? 6 @' fi 7: ' : 7; E # ' @' 6 fi ? 6 @' fi ? 6 ¥ 7: 7¥ ? 6 9 E = @' ? 6 7 = N 7 +, @ + , $ # 7: ; : = ( A block of weight W1 = 1000 N rests on a horizontal surface and supports, on top of it, another block of weight W2 = 250 N as shown in Fig. 7.19a. The 250-N block is attached to a vertical wall by the inclined string AB. Find the magnitude of the horizontal force P applied to lower block that will be necessary to cause slipping to impend. The coefficient of friction for all contact surfaces is 0.3. ! " ɏ ŶŷŹ &¹°ª»°¶µɏ q T N2 F2 = 0.3N2 N1 A P 4 q 3 q F2 = 0.3N2 F1 = 0.3N1 250 N W2 B 250 N W1 1000 N T P q N1 N2 F2 = 0.3N2 P F2 = 0.3N2 F1 = 0.3N1 250 N (a) 1000 + N2 1000 + N2 (b) (c) # ' 5OLUTION! q S S 8 9 # J; =+ + A?, 6? ?@ E # $ # -% 6? ?@ ' 6 ? '% 6? ?@ +, + , 7; : , S S *B 6 J; =+ : , & & & & '+ '+ ? - , - ? , +, + , % 6? ; fi E # $ # 7; fi 'XAMPLE!ircg Two identical blocks A and B are connected by a rod and rest against vertical and horizontal planes, respectively, as shown in Fig. 7.20a. If sliding impends when q = 45°, determine the coefficient of friction m, assuming it to be same at the floor and wall. ! " F1 = mN1 A A N1 C W 45° q 45° N2 C (b) B B F2 = mN2 W (a) 5OLUTION N # # S S = * $ *B = + & ' 6@ fi & m ' 6@ fi (c) $ 9 E , + - m, +, Ŷŷźɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº Bqqmzjoh!frvbujpot!pg!frvjmjcsjvn!up!GCE!pg!uif!cmpdl!C!)Gjh/!8/31 *-!xf!hfu S ! >!1;! !tjo!56¡!Ï!m 3!>!1! fi! 3!>!1/818 0m S ! >!1;! 3!Ï! !Ï! !dpt!56¡!>!1! fi! !>!1/818 |)20m*!Ï!2~! Frvbujoh!Frt/!) *!boe!) *-!xf!hfu ! )2!,!m*!>!|)20m*!Ï!2~! fi! m3!,!3m!Ï!2!>!1 Tpmwjoh!uijt!rvbesbujd!frvbujpo-!xf!hfu!m!>! ! ! )uif!ofhbujwf!wbmvf!jt!jhopsfe* ) * 'XAMPLE!irch Two boxes are placed on an inclined plane, in contact with each other as shown in Fig. 7.21a, and released from rest. The coefficient of static friction between the box A and the plane is 0.4 and that between box B and the plane is 0.3. Describe what happens. ! " B A 20 kg 10 kg 20° RAB A B mA N A mBNB RAB 98 N NB 196 N 20° NB 20° mBNB (a) 20° B RAB NA RAB NA 20° 20° mA N A 20° A 98 N 196 N (b) (c) 5OLUTION! Hjwfo;! B!>!21!¥!:/9!>!:9!O-! C!>!31!¥!:/9!>!2:7!O-!mB!>!1/5- mC!>!1/4/ Mfu!vt!Ýstu!Ýoe!xibu!ibqqfot!up!uif!uxp!cpyft-!jg!uifz!bsf! !qmbdfe!po!uif!jodmjofe!qmbof/! ( !B;! Uif!opsnbm!sfbdujpo-! ! >! !dpt!31¡!>!:9!dpt!31¡!>!:3/1:!O B B ! Uif!mjnjujoh!gsjdujpo!bdujoh!vq!uif!qmbof-! B! >!mB B!>!1/5!¥!:3/1:!>!47/95!O ! Uif!gpsdf!bdujoh!epxo!uif!qmbof-! B! >! B!tjo!31¡!>!:9!tjo!31¡!>!44/63!O ! Tjodf! B!=! B-!uif!cpy!B!epft!opu!tmjef!epxo/ ( !C;! Uif!opsnbm!sfbdujpo-! C! >! C!dpt!31¡!>!2:7!dpt!31¡!>!295/29!O ! Uif!mjnjujoh!gsjdujpo!bdujoh!vq!uif!qmbof-! C! >!mC C!>!1/4!¥!295/29!>!66/36!O ! Uif!gpsdf!bdujoh!epxo!uif!qmbof-! C! >! C!tjo!31¡!>!2:7!tjo!31¡!>!78!O ! Tjodf! C!?! C-!uif!cpy!C!tmjeft!epxo!uif!qmbof/ Uivt-!xf!Ýoe!uibu!jg!qvu!uphfuifs-!uif!cpy!C!qvtift!uif!cpy!B!boe!uif!uxp!cpyft!uphfuifs!tmjef!epxo! uif!qmbof/!Mfu!!BC! cf!uif!sfbdujpo!cfuxffo!uif!cpyft!B!boe!C/!Uif!GCEt!pg!uif!uxp!cpyft!bsf!tipxo!jo! Gjh/!8/32 !boe! ɏ ŶŷŻ &¹°ª»°¶µɏ Bqqmzjoh!frvbujpot!pg!frvjmjcsjvn!up!uif!cpy!C!)Gjh/!832 *-!xf!hfu S ! >!1;! C!Ï!2:7!dpt!31¡!>!1! fi! C!>!295/29!O S ! >!1;! !BC!,!mC C!Ï!2:7!tjo!31¡!>!1! fi! !BC!>!22/89!O Uif!sfbdujpo!gpsdf!!BC!jt!uif!qvtijoh!gpsdf!bqqmjfe!po!uif!cpy!B!cz!uif!cpy!C/!Uifsfgpsf-!uif!ofu!gpsdf!bdujoh! po!uif!cpy!B!epxo!uif!qmbof!jt!hjwfo!bt )ofu*!>! B!tjo!31¡!,!!BC!Ï! B!>!:9!tjo!31¡!,!22/89!Ï!47/95!>!9/57!O Uijt!jt!uif!ofu!gpsdf!bdujoh!po!uif!tztufn!pg!uxp!cpyft!epxo!uif!qmbof-!uif!tztufn!tmjeft!xjui!bo!bddfmfsbujpo! hjwfo!bt P( net ) 8.46 !>! !>! ) .8 ! >! mA + mB 10 + 20 'XAMPLE!irci Two rectangular blocks of weights W1 and W2, connected by a flexible cord, rest upon a horizontal and inclined plane, respectively, with the cord passing over a pulley as shown in Fig. 7.22a. In a particular case, where W1 = W2 and the coefficient of static friction m is same for the continuous surface, find the angle a of inclination of the inclined plane at which motion of the system will impend. Neglect friction in the pulley. ! " T W mN1 A T B T N1 B N2 mN2 a T W N2 A mN1 a N1 T a a A T W W (a) B mN2 (b) 5OLUTION! Mfu! 2!>! 3!> /!!Uif!GCEt!pg!uif!uxp!cmpdlt!bsf!tipxo!jo!Gjh/!8/33 !boe! ( '!B;!Sftpmwjoh!uif!gpsdft!opsnbm!boe!ubohfoujbm!up!uif!qmbof-!xf!hfu S ! >!1;! 2!Ï! !dpt!a!>!1! fi! 2!>! !dpt!a S ! >!1;! %!Ï! !tjo!a!,!m 2!>!1! fi! %!>! !)tjo!a!Ï!m!dpt!a*! (c) )* Ŷŷżɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº ( '!C;!Sftpmwjoh!gpsdft!jo!ipsj{poubm!boe!wfsujdbm!ejsfdujpot-!xf!hfu S ! >!1;! 3!Ï! !>!1! fi! 3!>! ! S ! >!1;! m 3!Ï!%!>!1! fi! %!>!m ! Frvbujoh!Frt/!) *!boe!) *-!xf!hfu m!! >!)tjo!a!Ï!m!dpt!a* fi! m! >! ) * sin a 2 sin(a / 2) cos(a / 2) !>! !>!ubo!)a03*! fi! a!>! (1 + cos a ) 2 cos 2 (a / 2) 4 m 'XAMPLE!ircj Two blocks, connected with a horizontal link AB, are supported on two rough planes as shown in Fig. 7.23a. If fC = 15° and mA = 0.4, find the smallest value of W required for the equilibrium of the system. ! " 400 N W 400 N A W FB C A B 30° NB 30° B NA FB NA C 400 N (a) Two blocks connected with a tie rod FA NB B 30° C 30° (b) FBD of block B C A FA W (c) FBD of block A 5OLUTION Hjwfo;!mB!>!1/5-!mC!>!ubo!fC!>!ubo!26¡!>!1/378/ Cmpdl!C!ibt!b!ufoefodz!up!tmjef!epxo/!Mfu! !cf!uif!dpnqsfttjwf!gpsdf!jo!uif!dpoofdujoh!mjol/!Jg!uif!cmpdl!B! jt!mjhiufs!uibo!uif!njojnvn!xfjhiu! !offefe!gps!frvjmjcsjvn-!ju!xjmm!cf!qvtife!upxbset!mfgu/!Dpotjefs!uif! tztufn!jo!frvjmjcsjvn!cvu!bu!uif!qpjou!pg!jnqfoejoh!npujpo/!Uif!GCEt!pg!C!boe!B!bsf!hjwfo!jo!Gjh/!8/34 ! boe! -!sftqfdujwfmz/ Uif!gsjdujpobm!gpsdft!gps!uif!uxp!cmpdlt!bsf!hjwfo!bt C! >!mC C!>!1/378 C! boe! B!>!mB B!>!1/5 B Bqqmzjoh!frvbujpot!pg!frvjmjcsjvn!gps!uif!cmpdl!C!)Gjh/!8/32 *S ! >!1;! C!dpt!41¡!,! C!tjo!41¡!Ï!511!>!1 fi! 1/378 C!dpt!41¡!,! C!tjo!41¡!Ï!511!>!1! fi! C!>!658/13!O S ! >!1;! C!tjo!41¡!Ï! C!dpt!41¡!,! !>!1! fi! !>!511/8!O Bqqmzjoh!frvbujpot!pg!frvjmjcsjvn!up!uif!cmpdl!B!)Gjh/!8/34 *400.7 !>!2112/9!O S ! >!1;! B!Ï! !>!1! fi! mB B!>! ! fi! B!>! !>! mB 0.4 S ! >!1;! B!Ï! !>!1! fi! !>! B!>! ) ɏ ŶŷŽ &¹°ª»°¶µɏ 'XAMPLE!irck A semicircular right cylinder of radius r and weight W rests on a horizontal surface and is pulled at right angles to its geometric axis by a horizontal force P applied at the point B of the front edge as shown in Fig. 7.24a. Find the angle a that the flat face will make with the horizontal plane just before the sliding begins, if the coefficient of friction at the point of contact A is m. The weight W must be considered as acting at the centre of gravity G of the cylinder. ! " O 4r 3p O r a a G B P W A B¢ G¢ a B P A mN N (a) (b) 5OLUTION! Uif!dfousf!pg!hsbwjuz!H!jt!bu!b!ejtubodf!pg!5 04ʪ!gspn!uif!qpjou!P/!Sfgfs!up!uif!GCE!pg!uif!dzmjoefs! tipxo!jo!Gjh/!8/35 -!Gspn!DHPH¢HH¢!>!)5 04p*!tjo!a! boe!!!BC¢!>!PB!Ï!PC¢!>! !Ï! !tjo!a!>! !)2!Ï!tjo!a* Bqqmzjoh!frvbujpot!pg!frvjmjcsjvn!up!uif!GCE!pg!uif!tfnjdzmjoefs!bt!tipxo!jo!Gjh/!8/35 -!xf!hfu S ! >!1;! ! !>! ! boe! S !>!1;! S)}B! >!1;! !¥!HH¢!>! !¥!BC¢ fi! fi ! !>!m !>!m !¥!)5 04p*!tjo!a!>! !)2!Ï!tjo!a*!>!m !)2!Ï!tjo!a* 3ol 4 m !tjo!a!,!m!tjo!a!>!m! fi! tjo!a!>! !>! 3p (4 / 3p ) + m 4 + 3ol 'XAMPLE!irdl Referring to Fig. 7.25a, determine the least value of the force P to cause motion to impend rightward. Assume the coefficient of friction under the blocks to be 0.2 and the pulley to be frictionless. ! " 5OLUTION! Gsff!cpez!ejbhsbn!pg!uxp!cmpdlt!bsf!tipxo!jo!Gjh/!8/36 !boe! Uif!gsjdujpobm!gpsdft!1/3 1/3 3!bdu!pqqptjuf!up!uif!ejsfdujpo!pg!jnqfoejoh!npujpo/ 2!boe! ŶŸŴɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº T 1000 N A P T B q 0.2N1 1000 N T N1 60° 0.2N2 T N1 A 1500 N N2 0.2N2 T 60° P q B A 0.2N1 1000 N 60° ( N2 1500 N T (a) Two blocks connected with a tie P q B 1500 N (b) FBD of the block A (c) FBD of the block B '!B;! Sftpmwjoh!uif!gpsdft!opsnbm!boe!ubohfoujbm!up!uif!qmbof-!xf!hfu S ! >!1;! 2!Ï!2611!dpt!71¡!>!1! fi! 2!>!861!O S ! >!1;! %!Ï!1/3 2!Ï!2611!tjo!71¡!>!1 fi! %!>!1/3!¥!861!,!23::/15!>!255:/15!O '!C;! Sftpmwjoh!gpsdft!jo!ipsj{poubm!boe!wfsujdbm!ejsfdujpot-!xf!hfu S ! >!1;! !dpt!q!Ï!1/3 3!Ï!%!>!1! fi! 6 !dpt!q!Ï! 3!>!6%! S ! >!1;! !tjo!q!,! 3!Ï!2111!>!1! fi! !tjo!q!,! 3!>!2111! ( )* ) * Beejoh!Frt/!) *!boe!) *-!boe!tvctujuvujoh!uif!wbmvf!pg!%-!xf!hfu !>! 1000 + 5% 8245.2 !>! 5 cos q + sin q 5 cos q + sin q Pcwjpvtmz-! !xjmm!cf!njojnvn!xifo!uif!efopnjobups!)6!dpt!q!,!tjo!q*!jt!nbyjnvn-!j/f/-!xifo! ! q fi \! !)6!dpt!q!,!tjo!q*!>!1! fi! Ï!6!tjo!q!,!dpt!q!>!1 ubo!q!>!206! fi! q!>!22/4¡ njo!>! 'XAMPLE!irdc 8245.2 5 cos q + sin q 8245.2 5 cos 11.3∞ + sin 11.3∞ ( A smooth circular cylinder of weight Q and radius r is supported by two semicircular cylinders each of the same radius r and weight Q/2 as shown in Fig. 7.26a. If the coefficient of static friction between the flat faces of the semi-circular cylinder and the horizontal plane on which they rest is μ = 0.5 and the friction between the cylinders themselves is neglected, determine the maximum distance b between the centres B and C for which equilibrium will be possible without the middle cylinder touching the horizontal plane. ! " ɏ ŶŸŵ &¹°ª»°¶µɏ A A q q q q Q 2 N1 N1 Q D B N3 N2 C q Q B Q 2 Q 2 b (a) (b) mN3 (c) ( 5OLUTION! S S % & ' ! ! ( q# q$ q q , , " 2 cos q + , , , , $ " $ q , q 2 cos q , q fi q fi -, " 2 cos q " " ) * ) S ! S ! ) \ , m , q (& 7 ! fi " q /-6 /-6 ¥7 + ( q ) s 57//#4; ( ( & ( ( ( ( Tubujd!gsjdujpo ) ) ) (& ( ( & Ljofujd! gsjdujpo ) ( Spmmjoh!gsjdujpo 9: ) ( Gsjdujpo ) ( 9 ) ( ( & (> ( 1 1 CD -.. / 2 B% ( ' ( . .012 ( List the laws of friction. .1!ird ( & ( .1!irg A( ; ( <( Define friction, state its types and explain its theory. .1!irc ( :: : < (( (:: = : : ( > ) ( dpfgÝdjfou!pg!gsjdujpo ; ( ( : : ( ( ( ( Underline the factors that affect the rolling friction. Spmmjoh!gsjdujpo ? : ( ( ( ) ( ( ( ) ( ( @ 9 ) ( & ( : ) ( ) ( = ( ( ( ŶŸŶɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº ! +/2146#06!(14/7.#' � E � F � � A m m> ) & 3 (> P m W (9 > ( � � m> I m I m ! m dpfgÝdjfou!pg!tubujd!gsjdujpo m> dpfgÝdjfou!pg!ljofujd!gsjdujpo/! ( a G &( f>H ( f> m> a r ) ! ! ! ! " " " " ! " ! " ! ( " ! " ! " ! " 5*146 #059'4!37'56+105 <2 %( ( > , J > , = 2 1 1 & , 3 1 3 : ( <2 J = ( C3 2 Ȼ= 2 M ) <2 = ( ( ( ( ? ( ( 1 1 ) (9 & :( ( ( ( ( 1 (:: & F D ( > : ( & ( 2 = ( . & ( ( ) O ( 1 , ! " ! " ! " ! 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( S RRRRRRRRRRRR ) ( ( ( ( ( ( ( ( ( ( ( ( RRRRRRRRR ! " ! " ! " ! " ! " ! " ! " ! " RRRRRRRR ( ( %! /ATCH %OLUMNS!3UESTIONS ) ! $ . E % $ ( & T 7 : C 7 D & 7 & . EE :: A ) (& 9: > ( ( % ( = M ( ( ( ( :( ( ( #NSWERS!TO!1BJECTIVE!3UESTIONS %C& 3 F$2 G ( %H& %$& FH G ( ( @ > ( ( ) ( : : ( 2F 4D # . G 4% 4= ! & #T & #C 24#%6+%'!241$.'/5 > ( L* ) ( > : : &( ( ( :( ( ( @ -6 ( ( (:: m , UC " ( " *V ) ! " ɏ ŶŸŹ &¹°ª»°¶µɏ P b 400 N 1 kN P P F 30° q 200 N 25° ) 7 ) ( W& ( ? ( : ! > ) ,-6 ( 6/ q ) Q * =(> m ) &( &( # ( ,( / L* ) b/!Ublf!mt!>!1/4!boe!ml!>!1/3/! ? ( ( m> UC >( :( / *V ! - > : : \C 389/17-!25/15¡^! " P ( ! " Bo!jodmjofe!qmbof!jt!vtfe!up!vompbe!b!cmpdl!pg!611.O!xfjhiu!gspn!b!ifjhiu!pg!3!n!bt!tipxo!jo!Gjh!8/41/!Tubuf! xifuifs!ju!jt!ofdfttbsz!up!qvti!uif!cpez!epxo!uif!jodmjofe!qmbof!ps!ipme!ju!cbdl!gspn!tmjejoh!epxo/!Xibu!jt! uif!njojnvn!wbmvf!pg!gpsdf!sfrvjsfe!gps!uijt!qvsqptf@!Ublf!m!>!1/4!cfuxffo!uif!cmpdl!boe!jodmjofe!qmbof \C Op!qvti!sfrvjsfe- 263/73!O^! ! " B W2 6m G b W1 2 3 W A 500 N a O a B!711.O!cmpdl!jt!jo!frvjmjcsjvn-!xifo!lfqu!po!b!qfsgfdumz!tnppui!jodmjofe!qmbof!boe!bdufe!vqpo!cz!b!gpsdf! pg!211!O!bu!bo!bohmf!pg!41¡!up!uif!jodmjofe!qmbof/!Xifo!uijt!cmpdl!jt!lfqu!po!b!spvhi!jodmjofe!qmbof!pg!tbnf! jodmjobujpo-!efufsnjof!uif!gpsdf! !sfrvjsfe!up!cf!bqqmjfe!qbsbmmfm!up!uif!qmbof!up!npwf!ju!vq!uif!qmbof/!Ublf! m!>!!1/36!gps!uif!spvhi!qmbof/! \C !!346!O^! ! " ! ( B! vojgpsn! mbeefs! pg! 6.n! mfohui! boe! 411.O! xfjhiu! mfbot! bhbjotu! b! tnppui! wfsujdbm! xbmm/! Jut! mpxfs! foe! jt! qmbdfe!3!n!bxbz!gspn!uif!xbmm!po!spvhi!Þpps/!Uif!dpfgÝdjfou!pg!gsjdujpo!cfuxffo!uif!mbeefs!boe!Þpps!jt!1/4/! Dbmdvmbuf!uif!gsjdujpobm!gpsdf!bdujoh!bu!uif!qpjou!pg!dpoubdu!pg!uif!mbeefs!boe!Þpps!boe!tipx!uibu!uif!mbeefs!jt!jo! frvjmjcsjvn!jo!uijt!qptjujpo/! \C !76/6!O^! ! " B!7.n!mbeefs!BC mfbot!bhbjotu!b!xbmm-!tvdi!uibu!uif!sbujp! 0 !jt!frvbm!up!3-!bt!tipxo!jo!Gjh/!8/42/!Bttvnjoh!uibu! uif!dpfgÝdjfou!pg!tubujd!gsjdujpo!mt!jt!uif!tbnf!bu!uif!xbmm!boe!uif!Þpps-!efufsnjof!uif!tnbmmftu!wbmvf!pg!mt!gps! xijdi!frvjmjcsjvn!jt!nbjoubjofe/!! \C 1/35^! ! " ŶŸźɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº ) Uxp!cmpdlt!pg!xfjhiu! 2!boe! 3!sftu!po!b!spvhi!jodmjofe!qmbof/!Uif!cmpdlt!bsf!dpoofdufe!cz!b!tipsu!qjfdf!pg! tusjoh!bt!tipxo!jo!Gjh/!8/43/!!Jg!m2!>!1/3!boe!m3!>!1/4-!sftqfdujwfmz-!Ýoe!uif!bohmf!pg!jodmjobujpo!pg!uif!qmbof!gps! xijdi!tmjejoh!xjmm!jnqfoe/!Bttvnf! 2!>! 3!>!6!O/! \C !a!>!25/15¡^! ! "! # Gjoe!uif!ipsj{poubm!gpsdf! sfrvjsfe!up!kvtu!tubsu!uif!npwfnfou!pg!b!611.O!cpy!po!bo!jodmjofe!tvsgbdf!bt!tipxo! jo!Gjh/!8/44/!!Hjwfo!uibu!q 6 31¡!boe!m!>!1/36/! \C 63/36!O^! ! " C P q A 50 N B q ! ! Bo!9.lh!cmpdl!B- buubdife!up!uif!mjol!BD- sftut!po!uif!23$lh!cmpdl!C!bt!tipxo!jo!Gjh/!8/45/!!Lopxjoh!uibu!uif! dpfgÝdjfou!pg!tubujd!gsjdujpo!jt!1/3!cfuxffo!bmm!tvsgbdft!pg!dpoubdu!boe!ofhmfdujoh!uif!nbtt!pg!uif!mjol-!efufsnjof! uif!wbmvf!pg!q!gps!xijdi!npujpo!pg!uif!cmpdl!C jt!jnqfoejoh/! \C 54¼^! ! " B!311$O!cmpdl!jt!qvmmfe!vq!bo!jodmjofe!qmbof!cz!b!gpsdf! -!bt!tipxo!jo!Gjh/!8/46/!Efufsnjof!xifuifs!uif!cmpdl! jt!jo!frvjmjcsjvn-!boe!Ýoe!uif!nbhojuvef!boe!ejsfdujpo!pg!uif!gsjdujpo!gpsdf!xifo! >!61!O!boe!q!>!36¡/!Bttvnf! uif!qvmmfz!up!cf!gsjdujpomftt-!boe!ublf!m!>!1/4/! \C ' Jo frvjmjcsjvn-!34!O^! ! " q W P 30° A 20° P q B P = 100 N ( Uxp!cmpdlt!B!boe!C!pg!6$lO!boe!21$lO!xfjhiut-!sftqfdujwfmz-!bsf!jo!frvjmjcsjvn!bt!tipxo!jo!Gjh/!8/47/!Jg!uif! dpfgÝdjfou!pg!gsjdujpo!cfuxffo!bmm!tvsgbdft!jo!dpoubdu!jt!1/4-!Ýoe!uif!gpsdf !sfrvjsfe!up!npwf!uif!cmpdl!C/! \C 6/67!lO^! ! " Efufsnjof!uif!tnbmmftu!wbmvf!pg!bohmf!q!gps!xijdi!npujpo!pg!uif!cmpdl-!tipxo!jo!Gjh/!8/48-!up!uif!sjhiu!jt! jnqfoejoh!xifo!) *! >!211!O-!) *! >!411!O-!boe!) *! >!511!O-!dpotjefsjoh!pomz!uif!wbmvft!pg!q!mftt!uibo! :1¡/!Ublf!m!>!1/36/! \C !) *!39/17¼<!) *!71/79¼<!) *!9:/9:¼^! ! " B!vojgpsn!mbeefs!pg!21.n!mfohui!xfjhijoh!311!O!jt!qmbdfe!bhbjotu!b!tnppui!wfsujdbm!xbmm!xjui!jut!mpxfs!foe! 5!n!bxbz!gspn!uif!xbmm!bt!tipxo!jo!Gjh/!8/49/!Uif!dpfgÝdjfou!pg!gsjdujpo!cfuxffo!uif!mbeefs!boe!Þpps!jt!1/4/! Difdl!xifuifs!uif!mbeefs!xjmm!sfnbjo!jo!frvjmjcsjvn!jo!uijt!qptjujpo!ps!opu/! \C Jo!frvjmjcsjvn^! ! " ! B!26¼!xfehf!ibt!up!cf!esjwfo!epxoxbset!gps!npwjoh!uif!cmpdl!C!pg!861.O!xfjhiu!bt!tipxo!jo!Gjh/!8/4:/!Jg!uif! bohmf!pg!gsjdujpo!gps!bmm!tvsgbdft!jt!21¡-!Ýoe!uif!gpsdf! sfrvjsfe!up!npwf!uif!xfehf/ \C :3/69!O^! ! 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Cmpdl!B!jt!sftusbjofe!gspn!npwjoh!cz!b!dbcmf!buubdife!up!b!Ýyfe!tvqqpsu/!Efufsnjof!uif!njojnvn!wbmvf!pg! -! xijdi!kvtu!qsfwfout!uif!jnqfoejoh!npujpo/! \C 5:/15!O^! ! " A C C P P A B ms = 0.3 W P ms = 0.4 B 35° ms = 0.45 ŶŸżɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº #¯¨·»¬¹ 9 Tjnqmf!Mjgujoh!Nbdijoft .EARNING!1BJECTIVES ɏ!ļ¬¹ɏª¶´·³¬»°µ®ɏ»¯°ºɏª¯¨·»¬¹ȎɏÀ¶¼ɏ¾°³³ɏ©¬ɏ¨©³¬ɏ»¶ɏ ! .1!jrcɏ ɏ ɏ ɏ ɏ ɏ .1!jrdɏ .1!jreɏɏ .1!jrfɏ .1!jrgɏɏ .1!jrhɏ ɏ .1!jri! ɏ .1!jrjɏ ɏ .1!jrkɏɏ ɏ .1!jrclɏ $°º»°µ®¼°º¯ɏ©¬»¾¬¬µɏº°´·³¬ɏ¨µ«ɏª¶´·¶¼µ«ɏ´¨ª¯°µ¬ºȎɏ¨µ«ɏ®°½¬ɏ¬¿¨´·³¬ºɏ¶ɏº°´·³¬ɏ³°ļ°µ®ɏ ´¨ª¯°µ¬ºȍ $¬ķµ¬ɏª¶´´¶µɏ»¬¹´ºɏ¹¬³¨»¬«ɏ»¶ɏº°´·³¬ɏ³°ļ°µ®ɏ´¨ª¯°µ¬ºȍ 5µ«¬¹³°µ¬ɏ»¯¬ɏ¹¬¨º¶µºɏ¶¹ɏ³¶ºº¬ºɏ°µɏ¨ɏ´¨ª¯°µ¬ɏ¨µ«ɏ«¬»¬¹´°µ¬ɏ°»ºɏ¬ĺª°¬µªÀȍ $°Ĺ¬¹¬µ»°¨»¬ɏ©¬»¾¬¬µɏ¹¬½¬¹º°©³¬ɏ¨µ«ɏº¬³Ɋ³¶ª²°µ®ɏ´¨ª¯°µ¬ºȍɏ )µ¬¹ɏ»¯¬ɏ³¨¾ɏ¶ɏ´¨ª¯°µ¬ºȍ %º»¨©³°º¯ɏ»¯¬ɏ¹¬³¨»°¶µɏ©¬»¾¬¬µɏ´¬ª¯¨µ°ª¨³ɏ¨«½¨µ»¨®¬ɏ¨µ«ɏ³¶¨«Ȏɏ¨µ«ɏ©¬»¾¬¬µɏ¬ĺª°¬µªÀɏ¨µ«ɏ ³¶¨«ȍ #³¨ºº°Àɏ³¬½¬¹ºɏ¨µ«ɏ«¬»¬¹´°µ¬ɏ´¬ª¯¨µ°ª¨³ɏ¨«½¨µ»¨®¬ɏ¶ɏ¬¨ª¯ɏª³¨ººȍ $¬»¬¹´°µ¬Ȏɏ »¯¬ɏ ½¬³¶ª°»Àɏ ¹¨»°¶ɏ ¨µ«ɏ ¬ĺª°¬µªÀɏ ¶ɏ ¨µɏ °µª³°µ¬«ɏ ·³¨µ¬Ȏɏ ¾¬«®¬Ȏɏ ¨µ«ɏ º°´·³¬ɏ ¨µ«ɏ «°Ĺ¬¹¬µ»°¨³ɏºª¹¬¾ɏ±¨ª²ºȍ $¬»¬¹´°µ¬ɏ»¯¬ɏ½¬³¶ª°»Àɏ¹¨»°¶ɏ¨µ«ɏ¬ĺª°¬µªÀɏ¶ɏ³¬½¬¹ºȎɏ¨µ«ɏº°´·³¬ɏ¨µ«ɏ«°Ĺ¬¹¬µ»°¨³ɏ·¼³³¬ÀɏºÀº»¬´ºȍ #¨³ª¼³¨»¬ɏ½¬³¶ª°»Àɏ¹¨»°¶ɏ¨µ«ɏ¬ĺª°¬µªÀɏ¶ɏș°Țɏ¾¯¬¬³Ɋ¨µ«Ɋ¨¿³¬Ȏɏș°°Țɏ¾¯¬¬³ɏ¨µ«ɏ«°Ĺ¬¹¬µ»°¨³ɏ¨¿³¬Ȏɏ ș°°°Țɏ¾¶¹´ɏ¨µ«ɏ¾¶¹´Ɋ¾¯¬¬³Ȏɏ¨µ«ɏș°½Țɏ¾°µª¯ɏª¹¨©ºȍ .1!jrc 9/2! �! JOUSPEVDUJPO Ju!jt!wfsz!ejgÝdvmu!up!votdsfx!b!ovu!ps!cpmu!xjui!cbsf!iboet-!cvu!ju!dbo!fbtjmz!cf!epof! vtjoh!b!xsfodi!ps!b!tdsfx!esjwfs/!Uif!xsfodi!boe!tdsfx!esjwfs!bsf!tjnqmf!nbdijoft/! B! tjnqmf! nbdijof! nblft! b! ejgÝdvmu! ubtl! fbtjfs! cz! dibohjoh! uif! nbhojuvef! boe! ejsfdujpo!pg!uif!bqqmjfe!gpsdf/!Ju!ibt!b!gfx!ps!op!npwjoh!qbsut/!Uiftf!nbdijoft!bsf!tp! dpnnpo!jo!pvs!ebjmz!mjgf!uibu!xf!epoÔu!bqqsfdjbuf!ipx!wbmvbcmf!uifz!bsf/!Epps!lopct-! dbo!pqfofst-!qfodjm!tibsqfofst!boe!tubqmfst!bsf!tpnf!tvdi!tjnqmf!nbdijoft/! tjnqmf! nbdijof dpnqpvoe nbdijof- dpnqmfy!nbdijof-!f/h/-!ifbwz! nbdijoft!)ep{fst-!cbdlipf-!dsboft-!fud/*!boe!bvupnpcjmft!)tuffsjoh!xiffm-!kbdl-!fohjof-!ejggfsfoujbm!bymf-! fud/*/ tjnqmf!mjgujoh!nbdijof-!f/h/-! !vtfe!gps!mjgujoh! xbufs!gspn!b!xfmm!boe! !vtfe!gps!mjgujoh!b!npups!dbs/!Uifsf!bsf!tjy!uzqft!pg!tjnqmf!mjgujoh!nbdijoft; $°º»°µ®¼°º¯ɏ©¬»¾¬¬µɏ º°´·³¬ɏ¨µ«ɏª¶´·¶¼µ«ɏ ´¨ª¯°µ¬ºȎɏ¨µ«ɏ®°½¬ɏ ¬¿¨´·³¬ºɏ¶ɏº°´·³¬ɏ ³°ļ°µ®ɏ´¨ª¯°µ¬ºȍ ɏ ŶŸŽ 3°´·³¬ɏ,°ļ°µ®ɏ-¨ª¯°µ¬ºɏ ! ! 2/! Jodmjofe!Qmbof/! 3/! Xfehf/! 4/! Tdsfx!Kbdl/ 5/! Mfwfs/! 6/! Qvmmfz/! 7/! Xiffm boe Bymf/!! Jodmjofe!qmbof!boe!mfwfs!bsf!uif!tjnqmftu!pg!uiftf!nbdijoft/!Npsf!dpnqmjdbufe!nbdijoft!bsf!qvmmfz-! xiffm boe bymf-!xfehf-!boe!tdsfx/!Tjnqmf!nbdijoft!gbmm!joup!uxp!dbufhpsjft;! ! ! 2/! Uiptf!efqfoefou!po!uif wfdups!sftpmvujpo!pg!gpsdft tvdi!bt 3/! Uiptf!cbtfe!po uif frvjmjcsjvn!pg!npnfout- tvdi!bt .1!jrd $¬ķµ¬ɏª¶´´¶µɏ»¬¹´ºɏ ¹¬³¨»¬«ɏ»¶ɏº°´·³¬ɏ³°ļ°µ®ɏ ´¨ª¯°µ¬ºȍ 9/3! - boe boe �! UFSNJOPMPHZ Uif!mjof!ejbhsbn!pg!b!tjnqmf!nbdijof!jt!tipxo!jo!Gjh/!9/2/!Uif!gpmmpxjoh!ufsnt!bsf! dpnnpomz!vtfe!xijmf!ejtdvttjoh!tjnqmf!mjgujoh!nbdijoft/! Input, Px } Effort P P Load W Machine W } x y Distance moved by effort Distance moved by load Output, Wy Gjh/!9/2 2/!Mpbe0Sftjtubodf!)X*! Ju!jt!uif!mpbe!mjgufe!ps!uif!sftjtubodf!pwfsdpnf!cz!uif!nbdijof/!Ju!ibt!uif!voju!pg! gpsdf!)O!ps!lO*/ 3/!Fggpsu!)Q*! Ju!jt!uif!gpsdf!sfrvjsfe!cz!b!nbdijof!up!mjgu!uif!mpbe!ps!up!pwfsdpnf!uif!sftjtubodf/!Jut!voju!jt! O!ps!lO/ 4/!Nfdibojdbm!Bewboubhf!)NB*! Ju!jt!efÝofe!bt!uif!sbujp!pg! uif!mpbe!mjgufe!up!uif!fggpsu!bqqmjfe-!! ! W !>! ! P )9/2* Uivt!jt!b! !cz!xijdi!b!nbdijof!nvmujqmjft!uif! fggpsu/! Nptu! pg! uif! nbdijoft! bsf! vtfe! up! mjgu! b! mbshf! mpbe! cz! bqqmzjoh!b!dpnqbsbujwfmz!tnbmm!fggpsu-!nbljoh! !hsfbufs!uibo! pof/!Uif!nfdibojdbm!bewboubhf!dbo!cf!dbmdvmbufe!cz!vtjoh!uif! frvbujpot!pg!frvjmjcsjvn/!Ju!ibt!op!vojut/! �! 9ORTHY!OF!0OTE Tpnf! nbdijoft! nbz! ibwf! nfdibojdbm!bewboubhf!mftt!uibo! pof/! Tvdi! nbdijoft! offe! bo! fggpsu!npsf!uibo!uif!mpbe!mjgufe/! Uiftf! nbdijoft! bsf! vtfe! up! qspwjef!hsfbufs!dpowfojfodf!pg! epjoh!b!ubtl-!f/h/-!b!qbjs!pg!upoht/ 5/!Joqvu! Uif!xpsl!epof!!cz!uif!fggpsu!jt!lopxo!bt!joqvu!up!uif!nbdijof/!Ju!jt!frvbm!up!uif!qspevdu!pg!fggpsu! !!boe!uif!ejtubodf! !uispvhi!xijdi!ju!npwft/!Uivt-!uif!joqvu!up!b!nbdijof!jt!! ¥! /!Jut!voju!jt!K!ps!lK/ 6/!Pvuqvu! Ju!jt!uif!vtfgvm!xpsl!epof!cz!uif!nbdijof-!xijdi!jt!tbnf!bt!uif!xpsl!epof!po!uif!mpbe/!Ju!jt!hjwfo! cz!uif!qspevdu!pg!mpbe!"!mjgufe!boe!uif!ejtubodf! uispvhi!xijdi!ju!npwft/!Uivt-!uif!pvuqvu!pg!b!nbdijof!jt! " ¥! /!Jut!voju!jt!K!ps!lK/ 7/!Wfmpdjuz!Sbujp!)WS*! Ju!jt!uif!sbujp!pg!uif!wfmpdjuz!pg!fggpsu!!!up!uibu!pg!mpbe!"/!Ipxfwfs-!bt!uif!fggpsu! npwft!cz!b!ejtubodf! !boe!uif!mpbe!npwft!cz!b!ejtubodf! !jo!uif!tbnf!ujnf! -!uif!wfmpdjuz!sbujp!jt!hjwfo!bt Velocity of the effort x 0 t x !>! !>! ! )9/3* ! #$!>! Velocity of the load y 0t y ŶŹŴɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº Uivt-!uif! !pg!b!nbdijof!jt!tjnqmz!hjwfo!bt! �! 9ORTHY!OF!0OTE h ' - /!Uif!#$ pg!b!nbdijof!efqfoet! pomz!po!jut!hfpnfusjdbm!gfbuvsft+/!Ju!jt!b! !gps!b!hjwfo! nbdijof-!jssftqfdujwf!pg!boz!fyusb!fggpsut!sfrvjsfe!evf!up!gsjdujpo/! Ipxfwfs-! !pg!b!nbdijof!dibohft!xjui!dibohf!jo!gsjdujpo/!! ! ! #$ ' 8/!Fggjdjfodz!)g*! Ju!jt!uif!sbujp!pg!uif!pvuqvu!up!uif!joqvu/!Ju!jt!b!nfbtvsf!pg!ipx!nvdi!pg!uif!xpsl!epof!po! uif!nbdijof!jt!dpowfsufe!joup!vtfgvm!xpsl/!Uif!fgÝdjfodz!pg!b!nbdijof!jt!bmtp!hjwfo!cz!uif!sbujp!pg!jut! #$-!bt!tipxo!cfmpx/ Output work W ¥ y W 0P MA !>! !>! ! !>! Input work P¥x VR x0y h!>! ! !up! )9/4* Uif!fgÝdjfodz!pg!b!nbdijof!jo!qsbdujdf!jt!bmxbzt!mftt!uibo!211&-!bt!tpnf!fofshz!jt!mptu!evf!up!gsjdujpo!jo! uif!gpsn!pg!ifbu-!tpvoe-!fud/!Uijt!jnqmjft!uibu!gps!b!nbdijof-!jut! #$ 9/!Jefbm!Nbdijof! �! 9ORTHY!OF!0OTE h #$ ! . ! / " # $ % $ ! gsjdujpomftt!nbdijof # # # :/!Jefbm!Fggpsu!)Qjefbm*! & ' ( Myth $# W P #$ \ W VR ! & #$ )* 21/!Jefbm!Mpbe!)Xjefbm*! & ( $# " .1!jre 5µ«¬¹³°µ¬ɏ»¯¬ɏ¹¬¨º¶µºɏ ¶¹ɏ³¶ºº¬ºɏ°µɏ¨ɏ´¨ª¯°µ¬ɏ ¨µ«ɏ«¬»¬¹´°µ¬ɏ°»ºɏ ¬ĺª°¬µªÀȍ ,& Fact! Uif!wfmpdjuz! sbujp!pg!bo!bduvbm! nbdijof!jt!mftt! uibo!uibu!pg!uif! jefbm!wfstjpo!pg! uibu!nbdijof/ " #$ ! $ & $# 0 ! ' # " ! ¥ #$ 9/4! & )+ �! MPTTFT!JO!NBDIJOFT! ! ( # $! " # ( & ! $# ! " ! " ( !! $ # " $ ' ɏ MA VR h - ŶŹŵ 3°´·³¬ɏ,°ļ°µ®ɏ-¨ª¯°µ¬ºɏ $ 3' )* # ! 4! Wactual ¥ y P¥x $ 3' ) + " 9/4/2! !4 )2 $ W VR )5 ! $# ( Wactual P ¥ VR h " W h ¥ VR fi ! )2 ! 6 3' ) 7 W/Pactual VR fi " ( ! h ¥ #$ $# " ! ¥ #$ 4 " 4" )) Fggjdjfodz!jo!Ufsnt!pg!Jefbm!boe!Bduvbm!Fggpsut!boe!Mpbet 3 $# fi h Output Input Input - Losses Input Pactual ¥ x - Pfriction ¥ x Pactual ¥ x h !ideal !actual ∵ ! 4! 6 ( fi h Output Input h "actual "ideal ! !actual - !friction !actual 0! Output Output + Losses ∵" " Wactual ¥ y Wactual ¥ y + Wfriction ¥ y "actual "actual + "friction 4" 'XAMPLE!jrc !µɏ¬Ĺ¶¹»ɏ¶ɏżŴɏ.ɏ°ºɏ¹¬¸¼°¹¬«ɏ©Àɏ¨ɏ´¨ª¯°µ¬ɏ»¶ɏ³°ļɏ¨ɏ³¶¨«ɏ¶ɏżŴŴɏ.ȍɏ4¯¬ɏ¬Ĺ¶¹»ɏ´¶½¬ºɏ ©Àɏ¨ɏ«°º»¨µª¬ɏ¶ɏŹŴŴɏ´´ɏ¨µ«ɏ»¯¬ɏ³¶¨«ɏ´¶½¬ºɏ©ÀɏŸżɏ´´ȍɏ#¨³ª¼³¨»¬ɏ»¯¬ɏ´¬ª¯¨µ°ª¨³ɏ ¨«½¨µ»¨®¬Ȏɏ½¬³¶ª°»Àɏ¹¨»°¶ɏ¨µ«ɏ¬ĺª°¬µªÀɏ¶ɏ»¯¬ɏ´¨ª¯°µ¬ȍɏɏ7¯¨»ɏ°ºɏ»¯¬ɏ¬Ĺ¬ª»ɏ¶ɏ¹°ª»°¶µɏ¶µɏ¬Ĺ¶¹»ɏ¨µ«ɏ ³¶¨«ȓ \MP!9/3-!MP!9/4^ 5OLUTION! 8 # 9 ! ) : " : + = #$ 3 h W P x y MA VR . # ) $ *) 800 21 80 500 21/53 48 10 C2 10.42 :7!& ŶŹŶɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº 3 D ( 9/5! .1!jrf 80 - !4 " ! ¥ #$ 4 " 800 10.42 ) ¥ 4/33!O *E 4 ) 47/7!O �! SFWFSTJCMF!BOE!TFMG.MPDLJOH!NBDIJOFT !( D $°Ĺ¬¹¬µ»°¨»¬ɏ©¬»¾¬¬µɏ ¹¬½¬¹º°©³¬ɏ¨µ«ɏº¬³Ɋ ³¶ª²°µ®ɏ´¨ª¯°µ¬ºȍ W VR ! !! "( $ ! & # $ 2/! Uif! mpbe! mjgufe! gbmmt! epxo! " # ( 6 sfwfstjcmf!nbdijof ! ! " ! $ )E 3/!Uif!mpbe!mjgufe!epft!opu!gbmm!epxo! " jssfwfstjcmf! !tfmg.mpdljoh!nbdijof 0 ! $# F & ! 4G ! ! 4" & $ # $ ( # ( # ! ( # ( # ( " # ( HG ! fi ! 4" fi ! H E" fi Wy 1 J Px 2 fi hJ " 1 2 ( ! # ( 0 ! # ( # & # 6 ( # ! ( ! $ ( ! & ! " ! $ # H" hJ+ + G Effort, P + Actual machine B m = tan q A q DW DP Ideal machine G P W W ! Gjh/!9/3! ( c O Load, W Gjh/!9/4! % ɏ ŶŹŷ 3°´·³¬ɏ,°ļ°µ®ɏ-¨ª¯°µ¬ºɏ 9/6! .1!jrg �! MBX!PG!NBDIJOFT " )µ¬¹ɏ»¯¬ɏ³¨¾ɏ¶ɏ ´¨ª¯°µ¬ºȍ # ! & " mbx!pg!nbdijof ( 0! !( $ )7 ! ' ! ! ! 0 $ $ ( $ ! $ $ )7 9/7! " )C ! ) )C ( !! �! NBYJNVN!NB!BOE!NBYJNVN!g " " ( W P ( # ( 1 m + c/W W mW + c ! N" ( $ 3' ) " M N" 0 " M " !! ( M N" # $ $# ) 0 " "( # $ )* h MA 1 MA = m + c /W 0 0 MAmax = 1 m h= 1 1 ¥ VR m + c /W h max = 0 0 W 1 1 ¥ VR m W (b) Variation of h with load W (a) Variation of MA with load W Gjh/!9/5 " & ( h MA VR " ( 1 1 ¥ VR m + c/W ( ( $ 3' )7 h # # ! # $ $ Opuf %º»¨©³°º¯ɏ»¯¬ɏ¹¬³¨»°¶µɏ ©¬»¾¬¬µɏ´¬ª¯¨µ°ª¨³ɏ ¨«½¨µ»¨®¬ɏ¨µ«ɏ³¶¨«Ȏɏ ¨µ«ɏ©¬»¾¬¬µɏ¬ĺª°¬µªÀɏ ¨µ«ɏ³¶¨«ȍ F ! DP DW / G8 q ! ! ( .1!jrh !! ! 4! $ $ "K ! G # ' ! ) ! "" ! ŶŹŸɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº h $ VR % m !"& !h h #! ! # ¥ ' ! ( ' & ! ! # # " & # ! 'XAMPLE!jrd !ɏº°´·³¬ɏ³°ļ°µ®ɏ´¨ª¯°µ¬ɏ³°ļºɏŶŴŴɏ.ɏ¨µ«ɏŷŴŴɏ.ɏ©Àɏ¼º°µ®ɏ¬Ĺ¶¹»ºɏ¶ɏŹŴɏ.ɏ¨µ«ɏźŴɏ.ɏ ´¨®µ°»¼«¬Ȏɏ ¹¬º·¬ª»°½¬³Àȍɏ ɏ )ɏ °»ºɏ ½¬³¶ª°»Àɏ ¹¨»°¶ɏ °ºɏ ŶŹȎɏ «¬»¬¹´°µ¬ɏ ș¨Țɏ »¯¬ɏ ³¨¾ɏ ¶ɏ ´¨ª¯°µ¬Ȏɏș©Țɏ°»ºɏ¬ĺª°¬µªÀɏ¨»ɏ©¶»¯ɏ»¯¬ɏ³¶¨«ºɏ¨µ«ɏª¯¬ª²ɏ»¯¬ɏ¹¬½¬¹º°©°³°»Àɏ¶ɏ»¯¬ɏ´¨ª¯°µ¬ȎɏșªȚɏ»¯¬ɏ¬Ĺ¶¹»ɏ ³¶º»ɏ°µɏ©¶»¯ɏ»¯¬ɏª¨º¬ºȎɏ¨µ«ɏș«Țɏ»¯¬ɏ´¨¿°´¼´ɏ¬ĺª°¬µªÀɏ¶ɏ»¯¬ɏ´¨ª¯°µ¬ȍ \MP!9/4-!MP!9/7^ 5OLUTION! ) "$ *!! $ + , ," ! $ - , ," #$ $ + *!$ "0 * !$+ , "$ , fi + $ 0 * !$- , "$% , fi - $% 0 & # " $ ! $% " * Q!>!1/2X!,!41 * !$+ , "$ , W 200 MA 4 $ $( ! h$ $ $ - $ 27!& $ P 50 VR 25 2 h + 7" jssfwfstjcmf/ !. ' ' 2 ' ' * !$- , "$% , W 300 MA 5 $ $+ ! h$ $ $ $ 31!& $ P 60 VR 25 2 h + 7" jssfwfstjcmf/ !' * !$+ , "$ , " ' * !$- , "$% $ VR ¥ m $ 1 1 ¥ $ 25 0.1 ! $ P- W 200 $ 50 $ 53!O VR 25 ! $!: W 300 $- : $ 59!O VR 25 , " h "$% ( $ 51!& �!0OTE 2 # .1!jri #³¨ºº°Àɏ³¬½¬¹ºɏ¨µ«ɏ «¬»¬¹´°µ¬ɏ´¬ª¯¨µ°ª¨³ɏ ¨«½¨µ»¨®¬ɏ¶ɏ¬¨ª¯ɏ ª³¨ººȍ + 7" 9/8! & !" �! MFWFST . ! < !& . . ! # ! & # ! ! ! !& ! # ! = ! & " & # # ! ! < ɏ ŶŹŹ 3°´·³¬ɏ,°ļ°µ®ɏ-¨ª¯°µ¬ºɏ ' ! # ! & ! + < ! ! ! ! ? ! " ! # & # "¥ :!¥ S A@ $ * !" mpbe!bsn < & fggpsu!bsn ! @" W a Effort arm $ $ load arm P b fi $ > ! # Load W Effort P O Pivot B A a b Gjh/!9/6! & = & mfwfsbhf ( bewboubhf C 9/8/2! D mbx!pg!mfwfs B & ! # ! ! ! nfdibojdbm! !" Dmbttjgjdbujpo!pg!Mfwfst ! # 2/! Dmbtt.J! # F ! < # " # # # ! !& ! & & " 3/!Dmbtt.JJ! B ? GBB # ! ! & G ? !& !& # ! " ! ' !& # # ! GB # ! ! ! ' # # H & ! # = 4/!Dmbtt.JJJ! !& # # ! ! ' - */!Uif! !jt!bmxbzt!mftt! uibo!pof/!Uiftf!mfwfst!jodsfbtf!uif!tqffe!pg!xpsl!xjuipvu!boz!hbjo!jo!gpsdf/!Gjsf!upoht-!uxff{fst-!lojwft!boe! tqbeft!bsf!tpnf!fybnqmft!pg!Dmbtt.JJJ!mfwfst/ Effort Load Load Fulcrum (a) Class-I lever Fulcrum Load Effort (b) Class-II lever Fulcrum Effort (c) Class-III lever Gjh/!9/7 B!dpnqpvoe!mfwfs!jt!b!tztufn!jo!xijdi!uxp!ps!npsf!mfwfst!bdu!vqpo!fbdi!puifs/!Fybnqmft!pg!dpnqpvoe! mfwfst!bsf!objm!dmjqqfst-!ujo!tojqt!boe!hbsefo!tifbst/! ŶŹźɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº 'XAMPLE!jre !ɏª¶´·¶¼µ«ɏ³¬½¬¹Ȏɏ¨ºɏº¯¶¾µɏ°µɏ&°®ȍɏżȍŻ¨Ȏɏ°ºɏ¼º¬«ɏ»¶ɏ³°ļɏ¨ɏ³¶¨«ɏ¶ɏŷŴɏ².ɏ¨ª»°µ®ɏ¨»ɏ »¯¬ɏ·¶°µ»ɏ&ȍɏ#¨³ª¼³¨»¬ɏ»¯¬ɏ¬Ĺ¶¹»ɏ»¶ɏ©¬ɏ¨··³°¬«ɏ¨»ɏ»¯¬ɏ·¶°µ»ɏ!ɏ»¶ɏ±¼º»ɏ³°ļɏ»¯¬ɏ³¶¨«ȍɏ !³º¶Ȏɏª¨³ª¼³¨»¬ɏ»¯¬ɏ´¬ª¯¨µ°ª¨³ɏ¨«½¨µ»¨®¬ɏ¶ɏ»¯¬ɏª¶´·¶¼µ«ɏ³¬½¬¹ɏ°ɏ³ŵɏǛɏŵŸŴɏ´´Ȏɏ³ŶɏǛɏŹŴɏ´´Ȏɏ³ŷɏǛɏŶŴŴɏ ´´ɏ¨µ«ɏ³ŸɏǛɏŸŴɏ´´ȍ \MP!9/8^ Effect P B Pivot l1 A Load W E Pivot D l2 C F l3 l4 (a) Compound lever Effect P Pivot B E A l1 l2 R Load W R C Pivot D F l3 l4 (b) FBD of each lever Gjh/!9/8 5OLUTION! Hjwfo;!"!>!41!lO-! 2!>!251!nn-! 3!>!61!nn-! 4!>!311!nn-! 5!>!51!nn/ Xifo!mfwfst!BC!boe!DE!bsf!tfqbsbufe-!uifz!fyfsu!b!sfbdujpo!$!)xijdi!jt!b!gpsdf!jo!uif!mjol!CD*!po!fbdi!puifs! bt!tipxo!jo!Gjh/!9/8 /!Gps!uif!mfwfs!DE-!ubljoh!npnfout!bcpvu!uif!qpjou!E-!xf!hfu Wl4 ! )* $!¥!) 4!,! 5*!Ï!"!¥! 5!>!1! fi! $!>! l3 + l4 Gps!uif!mfwfs!BC-!ubljoh!npnfout!bcpvu!uif!qpjou!F-!xf!hfu !!¥! 2!Ï!$!¥! 3!>!1 Tvctujuvujoh!uif!wbmvf!pg!$!gspn!Fr/!) *!jo!uif!bcpwf!frvbujpo-!xf!hfu Wl4 Wl4 l2 30 ¥ 40 ¥ 50 !¥! 3!>!1! fi! !!>! !>! !>!2/8:!lO !!¥! 2!Ï! l3 + l4 l1 (l3 + l4 ) 140 (200 + 40) W 30 \! Nfdibojdbm!bewboubhf-! >! !>! !>!27/87 P 1.79 .1!jrj $¬»¬¹´°µ¬Ȏɏ½¬³¶ª°»Àɏ ¹¨»°¶ɏ¨µ«ɏ¬ĺª°¬µªÀɏ ¶ɏ¨µɏ°µª³°µ¬«ɏ·³¨µ¬Ȏɏ ¾¬«®¬Ȏɏ¨µ«ɏº°´·³¬ɏ¨µ«ɏ «°Ĺ¬¹¬µ»°¨³ɏºª¹¬¾ɏ±¨ª²ºȍ 9/9! 9/9/2! �! NBDIJOFT!CBTFE!PO!JODMJOFE!QMBOF Tjnqmf!Jodmjofe!Qmbof! Bo!jodmjofe!qmbof!jt!pof!pg!uif!tjnqmftu!nbdijoft!vtfe!gps!btdfoejoh!ps!eftdfoejoh! pckfdut/!Uif!gpsdf!sfrvjsfe!up!npwf!bo!pckfdu!pwfs!)vq!ps!epxo*!bo!jodmjofe!qmbof!jt! rvjuf!mftt!dpnqbsfe!up!uibu!sfrvjsfe!up!npwf!ju!wfsujdbmmz-!cvu!uif!ejtubodf!npwfe!cz! ɏ ŶŹŻ 3°´·³¬ɏ,°ļ°µ®ɏ-¨ª¯°µ¬ºɏ uif!fggpsu!jodsfbtft/!Tpnf!bqqmjdbujpot!pg!jodmjofe!qmbof!bsf!sbnqt-!nfubm!boe!xbufs!tmjeft-!bjsqmbof!sftdvf! tmjeft-!ipqqfst!boe!gvoofmt/!Uif!gpmmpxjoh!uisff!gpsdft!bdu!po!uif!cmpdl!qmbdfe!po!bo!jodmjofe!qmbof!bt!tipxo! jo!Gjh/!9/9/ ! )j*! Xfjhiu!"!pg!uif!cmpdl!bdujoh!wfsujdbmmz!epxoxbse/! ! )jj*! Gsjdujpo!(!bdujoh!qbsbmmfm!up!uif!qmbof/ ! )jjj* Opsnbm!sfbdujpo!)!bdujoh!qfsqfoejdvmbs!up!qmbof/! Dpotjefs!b!mpbe!)cmpdl*!pg!xfjhiu!"!qmbdfe!po!b!qmbof!jodmjofe!bu!bo!bohmf!a!xjui!ipsj{poubm-!bt!tipxo!jo! Gjh/!9/9/!Uif!cmpdl!jt!qvmmfe!vq!cz!bo!jofyufotjcmf!tusjoh!qbttjoh!pwfs!b!qvmmfz/!Xifo!bo!fggpsu!!!jt!bqqmjfe! bu!uif!gsff!foe!pg!tusjoh-!uif!cmpdl!npwft!gspn!uif!qptjujpo!D!up!uif!qptjujpo!E!cz!b!ejtubodf! /!Tjodf!uif! tusjoh!jt!jofyufotjcmf-!uif!fggpsu!!!bmtp!npwft!cz!tbnf!ejtubodf! /!Uif!mpbe!jt!mjgufe!vq!wfsujdbmmz!cz!b!ejtubodf! !tjo!a/!Uif!wfmpdjuz!sbujp!jt!hjwfo!bt Distance moved by the effort #$!>! !>!dptfd!a )9/25* !>! Distance moved by the load sina Uif!nfdibojdbm!bewboubhf!jt!hjwfo!bt W !>! P B B P1 x P D W F A ! P1 W 90° C f a Gjh/!9/9! * 90° C x N f P A 35° N1 F = mN1 Gjh/!9/:! 'XAMPLE!jrf !µɏ¬Ĺ¶¹»ɏ¶ɏŻŴɏ.ɏ°ºɏ¹¬¸¼°¹¬«ɏ»¶ɏ´¶½¬ɏ¨ɏ³¶¨«ɏ¶ɏŵŶŴɏ.ɏ¼·ɏ¨µɏ°µª³°µ¬«ɏº´¶¶»¯ɏ·³¨µ¬ɏ ¾°»¯ɏ ¨µɏ °µª³°µ¨»°¶µɏ ¶ɏ ŷŹǸɏ ¾°»¯ɏ »¯¬ɏ ¯¶¹°Á¶µ»¨³ȍɏ ș¨Țɏ #¨³ª¼³¨»¬ɏ »¯¬ɏ ½¬³¶ª°»Àɏ ¹¨»°¶Ȏɏ ´¬ª¯¨µ°ª¨³ɏ¨«½¨µ»¨®¬ɏ¨µ«ɏ¬ĺª°¬µªÀɏ¶ɏ»¯¬ɏºÀº»¬´ȍɏ)ºɏ»¯¬ɏ°µª³°µ¬«ɏ·³¨µ¬ɏ¹¬½¬¹º°©³¬ɏ¶¹ɏ°¹¹¬½¬¹º°©³¬ȓɏș©Țɏ )ɏ»¯¬ɏª¶¬ĺª°¬µ»ɏ¶ɏ¹°ª»°¶µɏ©¬»¾¬¬µɏ³¶¨«ɏ¨µ«ɏ°µª³°µ¬«ɏ·³¨µ¬ɏª¯¨µ®¬ºɏ»¶ɏŴȍŷȎɏ¾¯¨»ɏ¾°³³ɏ©¬ɏ»¯¬ɏ¬Ĺ¶¹»ɏ³¶º»ɏ «¼¬ɏ»¶ɏ»¯¬ɏ¬¿»¹¨ɏ¹°ª»°¶µȓɏșªȚɏ7°»¯ɏ»¯¬ɏ¨©¶½¬ɏ¹°ª»°¶µȎɏ«¶¬ºɏ»¯¬ɏ°µª³°µ¬«ɏ·³¨µ¬ɏ©¬ª¶´¬ɏ°¹¹¬½¬¹º°©³¬ȓɏ)ɏ µ¶»Ȏɏ¾¯¨»ɏ°ºɏ»¯¬ɏ´¨¿°´¼´ɏ¨µ®³¬ɏ¶ɏ°µª³°µ¨»°¶µɏ¶¹ɏ»¯¬ɏ°µª³°µ¬«ɏ·³¨µ¬ɏ»¶ɏ¹¬´¨°µɏ°¹¹¬½¬¹º°©³¬ȓ \MP!9/9^ 5OLUTION! Hjwfo;!"!>!231!O-!!!>!81!O/!a!>!46¡-!m!>!1/4/ ! ) *! Wfmpdjuz!sbujp-! #$ >!dptfd!a!>!dptfd!46¡!>!2/85 W 120 !>!2/8 ! ! Nfdibojdbm!bewboubhf-! + !>! P 70 MA 1.7 ! ! FgÝdjfodz-! h! >! !>! !>!1/:8!>!:8!& VR 1.74 ! ! Tjodf!h!jt!hsfbufs!uibo!61!&-!uif!jodmjobujpo!jt!sfwfstjcmf/ ! ) *! Mfu!uif!sfrvjsfe!fggpsu!cf!!2-!xifo!uif!dpfgÝdjfou!pg!gsjdujpo!dibohft!up!1/4/!Sftpmwjoh!gpsdft!ubohfoujbm! boe!opsnbm!up!uif!jodmjofe!qmbof!boe!bqqmzjoh!frvbujpot!pg!frvjmjcsjvn!)sfgfs!Gjh!9/:*-!xf!hfu S( !>!1;! )2!Ï!"!dpt!a!>!1! fi! )2!>!231!dpt!46¡!>!:9/3:!O ŶŹżɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº S(!!>!1;! !2!Ï!m)2!Ï!"!tjo!a!>!1 fi! !2!>!1/4!¥!:9/3:!,!231!tjo!46¡!>!:9/43!O ! ! ! ! Fggpsu!mptu!jo!fyusb!gsjdujpo!>!!2!Ï!!!>!:9/43!Ï!81!>!39/43!O MA W/P1 120 / 98.32 !>! 1/8 81!& ! ) *! Uif!fgÝdjfodz-!h >! VR 1.74 1.74 h sfwfstjcmf ! ! ! ≥" (¥ ≥"¥ a fi m) ≥ " %& fi a£m fi a £ # '$ fi 9/9/3! a$ ≥ " a Xfehf! ! ( ( , - a fi m #" a £ 27/8¡ # )$ # * ( $ �! 9ORTHY!OF!0OTE ( * ! " +& +& ) ( +& * ) ) 5 ) * ) 6 ( ) ) #$ # +: $ l h 1 tan a #+ & $ #! #$ # +& $ l h ) 2{ ◊ tan(a /2)} #! ) " $ 1 2 tan(a /2) #+ &=$ " Block ) $ W W W a Wedge (a) A wedge used to lift a load a/2 h a P W P l l (b) Single wedge (c) Double wedge Gjh/!9/21! " 5> ) ( ? P h ɏ ŶŹŽ 3°´·³¬ɏ,°ļ°µ®ɏ-¨ª¯°µ¬ºɏ * ) @ ! : A6 #: =$ #: :$ " ! " fi ! sin(180∞ - 2f - a ) sin(90∞ + 2f + a ) sin(2f + a ) ! " fi ! " #Bf C a$ cos(2f + a ) fi " sin(180∞ - 2f - a ) sin(90∞ + 2f + a ) #+ &:$ 'XAMPLE!jrg #¨³ª¼³¨»¬ɏ»¯¬ɏ¬Ĺ¶¹»ɏ¹¬¸¼°¹¬«ɏ»¶ɏ³°ļɏ¨ɏŵŶŴŴɊ.ɏ©³¶ª²ɏ¼º°µ®ɏ¨ɏº°µ®³¬ɏ¾¬«®¬ɏ¶ɏŵŹǸɏ ¨µ®³¬ȍɏ4¯¬ɏª¶¬ĺª°¬µ»ɏ¶ɏ¹°ª»°¶µɏ¶µɏ¨³³ɏº¼¹¨ª¬ºɏ°µɏª¶µ»¨ª»ɏ°ºɏŴȍŵŶȍɏ!³º¶ɏª¨³ª¼³¨»¬ɏ»¯¬ɏ ´¬ª¯¨µ°ª¨³ɏ¨«½¨µ»¨®¬Ȏɏ»¯¬ɏ½¬³¶ª°»Àɏ¹¨»°¶ɏ¨µ«ɏ»¯¬ɏ¬ĺª°¬µªÀɏ¨µ«ɏª¯¬ª²ɏ¾¯¬»¯¬¹ɏ»¯¬ɏºÀº»¬´ɏ°ºɏ¹¬½¬¹º°©³¬ɏ ¶¹ɏ °¹¹¬½¬¹º°©³¬ȍɏ $¬»¬¹´°µ¬ɏ »¯¬ɏ ´¨¿°´¼´ɏ ¨µ®³¬ɏ ¶ɏ »¯¬ɏ ¾¬«®¬ɏ ¼·ɏ »¶ɏ ¾¯°ª¯ɏ »¯¬ɏ ºÀº»¬´ɏ ¹¬´¨°µºɏ °¹¹¬½¬¹º°©³¬ȍɏɏɏ \MP!9/9^ E" f 5OLUTION! D A &B %& 6 H I ) AJ ) h * J , a & F m &B %& m &B = +GF ! " #Bf C a$ &B #B ¥ = +GF C & F$ W 1200 2/94 P 656.44 1 1 4/84 #$ tan a tan 15∞ MA 1.83 h GK 5:!& 3.73 VR jssfwfstjcmf 767/55!O ) MA WLP "/{" tan(2f + a )} (1 - tan 2f tan a ) a 1/tan a VR VR (tan 2f + tan a ) Bf #B ¥ = +GF$ BG'G (1 - tan 2f tan a ) tan a (1 - 0.2434 tan a ) h a (tan 2f + tan a ) (0.2434 + tan a ) A6 # $ J )h a ) ) + && O )hJ P * h M J J , J * fi M * #h £ $ J #$ N K N a J ) ) h tan a (1 - 0.2434 tan a ) (0.2434 + tan a ) A6 # $ fi a a% BG'G aB 71/7¡ B a% a C &B&: 6 6 a& B+' fi a& 26/9¡ a E Q a £ 26/9¡ 71/7¡ £ a Q K F BG'G &B&: C aB & ::B ++ O fi B a ) ŶźŴɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº > J 6 a " J ) ) #B ¥ = +GF C & +F$ #B ¥ = +GF C = =F$ 6 #Bf C a&$ " #Bf C aB$ " Q a £ 26/9¡ " " !& !B J &B &B ¥ ='K =:+ ': , ¥ ' ' GB=' G , 70 60 Efficiency, h (%) 50 40 30 20 10 15.8° 0 10 a1 20 0 30 40 50 60.6° a2 70 80 Wedge angle, a (degrees) f + 7/95¡ )m!>!1/23* Gjh/!9/22! , 9/9/4! Tdsfx!Kbdl! B!tdsfx!kbdl!jt!vtfe!up!mjgu!ifbwz!mpbet!cz!bqqmzjoh!tnbmmfs!fggpsut/!B!tdsfx!kbdl!ps!tjnqmz!b!kbdl!jt!dpnnpomz! vtfe!up!mjgu!wfijdmft!gps!dibohjoh!xiffmt/!Ju!dpotjtut!pg!b!cbtf!jo!uif!tibqf!pg!ipmmpx!gsvtuvn!pg!b!dpof-!b!ovu!bu! uif!upq!pg!uif!gsvtuvn-!b!uisfbefe!tdsfx!xjui!b!dpmmbs-!boe!b!mfwfs!buubdife!up!uif!dpmmbs-!bt!tipxo!jo!Gjh/!9/: /! Uif!qjudi!pg!uif!uisfbet!pg!uif!tdsfx!boe!uibu!pg!uif!ovu!jt!tbnf-!tp!uibu!uif!tdsfx!dbo!nfti!xjui!uif!ovu/ W W l Collar l Lever M=F¥l Screw p F F Screw d Nut O Lever p a P Base (a) Construction d a (b) Top view of screw jack R f (c) Forces acting on screw Gjh/!9/23! - Gps!mjgujoh!ps!mpxfsjoh!uif!mpbe-!uif!mfwfs!jt!spubufe/!Jg! !jt!uif!mfohui!pg!mfwfs!boe! !jt!uif!qjudi!pg!uisfbet! pg!tdsfx-!uifo!jo!pof!spubujpo!uif!fggpsu!npwft!cz!b!ejtubodf!3p !boe!uif!mpbe!jt!mjgufe!cz!b!ejtubodf! /! ɏ Ŷźŵ 3°´·³¬ɏ,°ļ°µ®ɏ-¨ª¯°µ¬ºɏ Uivt-!uif!wfmpdjuz!sbujp!nbz!cf!fyqsfttfe!bt Distance moved by effort 3pl ! )9/29* !>! #$! >! p Distance moved by load Uijt!tipxt!uibu!uif!wfmpdjuz!sbujp!pg!b!tdsfx!kbdl!dbo!cf!jodsfbtfe!cz!efdsfbtjoh!uif!qjudi!pg!uif!tdsfx/ Gjhvsf!9/23 !tipxt!uif!upq!wjfx!pg!uif!tdsfx!boe!mfwfs/!Uif!fggpsu!bqqmjfe!bu!uif!foe!pg!uif!mfwfs!jt!(-! xijdi!jt!usbotgfssfe!bt!jut!frvjwbmfou!fggpsu!! bu!uif!nfbo!sbejvt!pg!tdsfx! 03-!xifsf! !jt!uif! ! pg!uif!tdsfx/!Ubljoh!npnfout!pg!uif!gpsdft!bcpvu!uif!dfousf!pg!uif!tdsfx-!xf!hfu 3Fl S }P!>!1;! 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" G< " " " & # D + "@ # H" H " +G @ D "# " & + 9 " + ! # nfubm!mppq Y !W " " " W " ? : G< :& " < : & : " + " < #< < & + < + & " !" H & ! : : ! + H H + ɏ ŶżŻ 3°´·³¬ɏ,°ļ°µ®ɏ-¨ª¯°µ¬ºɏ Gjh/!9/38 !32/! F %" W " " & !% & ! !33/! F # !34/! D + !9 " + <: WH<" " BB ## I !! " !" # ! H< & + & # !! " + ! :& # " ! > H< & !" #H + !% + " : = " H & ? : ! :& & ! ! = + > < # <#: " ! # " ! !! " ! = ! :& HH & !" TBot/!= !! " !9 !! " ! = = [ !% [ 'U < \MP!9/:^ ! > \MP!9/21^ " 9B H<" ## " "# ! & ! # B J F % TBot/! ' 9 U TMP!9/21U Ŷżżɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº #¯¨·»¬¹ : Dfousf!pg!Hsbwjuz!boe!Dfouspje .EARNING!1BJECTIVES ɏ!ļ¬¹ɏª¶´·³¬»°µ®ɏ»¯°ºɏª¯¨·»¬¹ȎɏÀ¶¼ɏ¾°³³ɏ©¬ɏ¨©³¬ɏ»¶ɏ ɏ ɏ ɏ ɏ .1!krcɏ .1!krdɏ .1!kreɏɏ .1!krfɏ .1!krgɏɏ 5µ«¬¹³°µ¬ɏ»¯¬ɏª¶µª¬·»ɏ¶ɏª¬µ»¹¬ɏ¶ɏ®¹¨½°»ÀȎɏª¬µ»¹¬ɏ¶ɏ´¨ººɏ¨µ«ɏª¬µ»¹¶°«ȍ $¬»¬¹´°µ¬ɏ»¯¬ɏ³¶ª¨»°¶µɏ¶ɏª¬µ»¹¬ɏ¶ɏ®¹¨½°»ÀȎɏª¬µ»¹¬ɏ¶ɏ´¨ººɏ¨µ«ɏª¬µ»¹¶°«ɏ¶ɏ¨ɏ©¶«Àȍ &°µ«ɏ»¯¬ɏ³¶ª¨»°¶µɏ¶ɏª¬µ»¹¶°«ɏ¶ɏ³°µ¬ºȎɏº¼¹¨ª¬ºɏ¨µ«ɏ©¶«°¬ºȍ )«¬µ»°Àɏ»¯¬ɏ³¶ª¨»°¶µɏ¶ɏª¬µ»¹¶°«ɏ¶ɏª¶´·¶º°»¬ɏ©¶«°¬ºȍɏ 3»¨»¬ɏ¨µ«ɏ·¹¶½¬ɏ0¨··¼ºɏ»¯¬¶¹¬´ºȎɏ¨µ«ɏ¨··³Àɏ»¯¬º¬ɏ»¯¬¶¹¬´ºɏ»¶ɏª¶´·¼»¬ɏ»¯¬ɏº¼¹¨ª¬ɏ¨¹¬¨ɏ¨µ«ɏ ½¶³¼´¬ɏ¶ɏ¨ɏ©¶«Àɏ¯¨½°µ®ɏ¨¿°¨³ɏºÀ´´¬»¹Àȍ .1!krc :/2! �! JOUSPEVDUJPO 5µ«¬¹³°µ¬ɏ»¯¬ɏª¶µª¬·»ɏ ¶ɏª¬µ»¹¬ɏ¶ɏ®¹¨½°»ÀȎɏ ª¬µ»¹¬ɏ¶ɏ´¨ººɏ¨µ«ɏ ª¬µ»¹¶°«ȍ ! " .1!krd $¬»¬¹´°µ¬ɏ»¯¬ɏ³¶ª¨»°¶µɏ ¶ɏª¬µ»¹¬ɏ¶ɏ®¹¨½°»ÀȎɏ ª¬µ»¹¬ɏ¶ɏ´¨ººɏ¨µ«ɏ ª¬µ»¹¶°«ɏ¶ɏ¨ɏ©¶«Àȍ :/3! �! MPDBUJPO!PG!DFOUSF!PG!HSBWJUZ !DFOUSF!PG!NBTT! BOE!DFOUSPJE :/3/2! Dfousf!pg!Hsbwjuz # ! $ % ( ) ) + Ú dW $ % ) &' * ɏ ŶżŽ #¬µ»¹¬ɏ¶ɏ'¹¨½°»Àɏ¨µ«ɏ#¬µ»¹¶°«ɏ Myth ( - , ) $ % &' ) . / , 0 Fact! - 1 23 4 z 4 + Ú x dW ¥ fi + dW Ú x dW G W dW 4 + Ú y dW ¥ fi 4 ! 5 O W 4 * 6 4 x + 4 + Ú zdW x x y y * 4 ) / . ¥ W Ú y dW + % y P z z fi z+ Ú zdW W ) x= :/3/3! Ú x dW ; y = Ú y dW ; z = Ú z dW W W ,& '- W Dfousf!pg!Nbtt $ ! ) ! + x= 23 + 23 ,& '- Ú x dm ; y = Ú y dm ; z = Ú z dm m m ,& 7- m ,& 7) ) ŶŽŴɏ :/3/4! %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº Dfouspje % Myth &' Fact! r ) ) 4 r �! 9ORTHY!OF!0OTE 4 ! % ! % &7 ) ! 0 " 8 Dam Water body Tyre + R Road surface R Water pressure distributed on dam Distributed force (b) Resultant R of hydrostatic pressure (a) Concentrated reaction R 9 x= 23 +" # 23 ,& 7- ,& 1- + "# # # Ú x dV ; y = Ú y dV ; z = Ú z dV V V ,& 1- V # : ɏ ŶŽŵ #¬µ»¹¬ɏ¶ɏ'¹¨½°»Àɏ¨µ«ɏ#¬µ»¹¶°«ɏ :/3/5! Tjhojgjdbodf!pg!Dfousf!pg!Hsbwjuz # $!%& ' " % " !% # ( ) ' .1!kre &°µ«ɏ»¯¬ɏ³¶ª¨»°¶µɏ ¶ɏª¬µ»¹¶°«ɏ¶ɏ³°µ¬ºȎɏ º¼¹¨ª¬ºɏ¨µ«ɏ©¶«°¬ºȍ :/4! �! DFOUSPJET!PG!MJOFT !TVSGBDFT!BOE!CPEJFT �! 9ORTHY!OF!0OTE :/4/2! ! Mjoft $ " 2 $ & 2 (x *+ y z) ! ! ' $ #$ /0 $* +& , #- $ . $ & " ' x= Ú x dL = ; y = Ú y dL ; z = Ú z dL L L $* 1& L z z dL O x C P (x, y, z ) dA y C (x, y, z ) y z z y O x y x (a) Centroid of a line segment x (b) Centroid of a surface ŶŽŶɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº :/4/3! Tvsgbdft 3 ( ( #- #- x= :/4/4! ! *+ ! (x, y, z ) $ & . 4 , /0 $* +& Ú x dA = ; y = Ú y dA ; z = Ú z dA A A $* 5& A Tufqt!gps!Efufsnjojoh!Dfouspje!pg!b!Hjwfo!Cpez 6 /0 $* 7& $* 5& 8 9 7 ) ' ! : "( # " 0 $ + ! 1 # 5 3 ' (say, ( x , y , z )) & # ( $ < ! + " ) Myth > 0 ) Fact! 5 A < ? ! 0 :/4/5! Dfouspjet!pg!Tpnf!Tuboebse!Hfpnfusjdbm!Gjhvsft Bsd!pg!b!Djsdmf! ! ! :a 4 3 -@ *1 " *1 ɏ ŶŽŷ #¬µ»¹¬ɏ¶ɏ'¹¨½°»Àɏ¨µ«ɏ#¬µ»¹¶°«ɏ y y y R cos q Rdq dq a 45° q O a x C R O x x C 45° O R (a) Arc of circle x C R (b) Quarter arc (c) Semi-circular arc ! q A " q /0 $* 1& a a Ú xdl = Ú-a x R dq = Ú- a R cos q dq = R 2 (sin a - sin(- a )) = R sin a a a 2Ra a Ú dl Ú-a R dq Ú- a Rdq 2 $* <& )j*!Rvbsufs!Djsdvmbs!Bsd!)Gjh/!:/5c* a - 15B - p81 fi - ! sin(45∞) p /4 o )jj*!Tfnj.djsdvmbs!Bsd!)Gjh/!:/5d* a p fi ! sin(90∞) p /2 fi ! sin 180∞ p o )jjj*!Djsdvmbs!Sjoh a p Usjbohvmbs!Bsfb! ! " # $ %" & ' * + # " , # ' # # " $ b x b (h - y ) fi h (h - y ) h 3 $ $ * $ Ay fi - Ay Ú y dA Ú yx dy fi b Ï h 2 h3 ¸ bh 2 - ˝ Ìh hÓ 2 3˛ 6 + - " $ - # h Ay Ú0 fi bh $ . bh 2 6 " h b Ï y 2 y3 ¸ - ˝ Ìh hÓ 2 3 ˛0 b (h - y ) y dy h ¥y # " fi " = :' ;< " ŶŽŸɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº y y y a C a a O xc R R O dRo Ro dq a x a q O x C y y x O R xc dy h x a Method I (b) Circular strip (a) Circular sector b Method II (c) Radial strip % Djsdvmbs!Tfdups! ! %" ? - - , @ - 2a (p ! 2 ) 2p A B + $ + B - - ! - - " " - a > $ a! - D + A B * " - - - # C + * " # , Nfuipe!J # B> $ - %" fi - - Ú x- dA 2 3 a! !3 sin a Ú0 ! " * ! a ¥ ! ! a ! % EF a a G " EF & $ " Ro sin a 2 Roa dRo a 2 ! sin a 3a fi " ? - ! ¥ R fi a! ¥ , Nfuipe!JJ # # Ax %" ! -* ? # B " 2Ê Á 3Ë :! !< " sin ` ˆ ˜ ` ¯ H " q q ? B * $ # 1 2 , $ R dq !! q - " " G - - " EF !- aR x Ú x- dA 2 3 fi !3 sin a ( R 2a ) x fi - # B Ú-a ( 23 R cos q ) ( 12 R a 2 ! sin a 3a 2Ê Á 3Ë 2 dq ) sin a ˆ ˜ a ¯ )j*!Rvbsufs!Djsdvmbs!Bsfb a p fi " $ A q & $ " Ax fi 2 3 - 2 ! sin 45∞ 3(p /4) 4 2! 3p 2Ê2 2 ˆ 3 ÁË p ˜¯ q a a ɏ ŶŽŹ #¬µ»¹¬ɏ¶ɏ'¹¨½°»Àɏ¨µ«ɏ#¬µ»¹¶°«ɏ )jj*!Tfnj.djsdvmbs!Bsfb a fi 2 ! sin 90∞ 3(p /2) fi 2 ! sin 180∞ 3p p 4! 3p 2Ê2 ˆ Á ˜ 3Ë o ¯ )jjj*!Djsdvmbs!Bsfb a p ! Qbsbcpmjd!Tqboesfm! * $6 . 7 :" : "# * : "# " ". $ ! Ú ?..!+" AB$ a !" "# Ú dA a $ ' ; ". . a 2 Ú0 x y dx a 2 x4 ◊ a3 4 ..!+ $ 3 a /3 "# * ! -" " a a Ú ( ydA ) dA Ú0 ( y /2) ( y dx) Ú0 ( x 2 a 0 /2) ( x 2 dx) O 4 . " :".! AB$ x5 ◊ 2a 3 5 -" $ E" $ ' a 3 2 10 0 dy b C (x, y) y x O a (a) Vertical strip . 2 3 10 3 $ 10 : x x a " "# * b (x, y) y dx # !" #" " " >$ ? 3 4 y = x2 C (x, y) #" + : y x (x, y) : = 3 y = x2 -. 4 3 . 3 a 3 /3 A y F" : " " @ 3 A a /3 : : #" ". " <" $ $& A %&' @ dx a 3 " "# $ Ú0 x ( y dx) Ú0 x ( x dx) Ú x dA # : ! # # + , -" . ! ; " <" $ $& $ # ":= ## . #" : !": : -" ": ! # #" (a – x) (b) Horizontal strip : Ê 3a 3b ˆ $ ÁË , ˜ 4 10 ¯ " ;" 6" #" ! $ : ŶŽźɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº ! # $ ! " ! $ x+ # %& '! (a + x) 2 (a - x) 2 * b b Ú0 (ay - y Ú ydA dA Ú0 y (a - x) dy 3 È ay 2 y 5/2 ˘ Í ˙ 5/2 ˚ a3 Î 2 0 a 3 /3 a 3 /3 A b y ) dy 5 3 2 3 3 È ¥ 4 2( 2 )5/2 ˘ 3 3 È ab 2 2b5/2 ˘ ¥ Í ˙ Í ˙ 3 3 5 ˚ 10 10 10 5 ˚ a3 Î 2 Î 2 b (a + x) b b 2 2 Ú xdA dA Ú0 2 (a - x) dy Ú0 (a - x ) dy Ú0 (b - y) dy A 2a 3 /3 a 3 /3 2a 3 /3 b 3 È 2 b2 ˘ Íb - ˙ 2˚ 2a 3 Î y2 ˘ 3 È by Í ˙ 2˚ 2a 3 Î 0 Sjhiu!Djsdvmbs!Tpmje!Dpof! , - (x, y, z ) $ 6 8 # $ :;! $ PQ OB + 3b 2 4a 3 3 4 4 3 $ $ $ $ $ 7 $ $ $ * < 7= h- y h x r AP AO 3 4 Ê r Á> Ë fi :; yˆ ˜ h¯ 2 ? $ ? @ %& $ " # p # 1 p 3 p yˆ Ê ÁË1 - ˜¯ dy h " C! h Ú y dV V Ú0 2 yˆ ÔÏ Ô¸ Ê y Ìp r 2 Á1 - ˜ dy ˝ Ë h¯ ÓÔ ˛Ô 2 p r h/3 3 È y2 2 y3 ˘ y4 Í + 2 ˙ 3h ˚ h Î 2 4h # " $ $ h 0 $ hÊ y3 2 y 2 ˆ pr2 Ú Á y + 2 dy 0Ë h ˜¯ h p r 2 h/3 4 3Ê 2 2 3ˆ Á 2 + 2 - 3 ˜ 4 Ë ¯ Ê ˆ ÁË 0, , 0˜¯ 4 E# D >F>"G 8 9 ɏ ŶŽŻ #¬µ»¹¬ɏ¶ɏ'¹¨½°»Àɏ¨µ«ɏ#¬µ»¹¶°«ɏ �!0OTEs! $ $ H $ $ # $ $ $ $ $ H $ $ ! y A y x z y2 + z2 = r2 Q P dy h C dy y y r y C O r x B z O x ! Tpmje!Ifnjtqifsf! , H H $ & $ $ # * 9 ? ? I " 8 # & $ $ " $ $ r " - y" p " p 2 p C 3 # # " " ! C! Ú y dV r Ú0 y{p (r V # 8 * $ " %& - (x, y, z ) $ 2 r p Ú ( yr 2 - y 3 ) dy - y 2 ) dy} (2/3)p r 0 3 (2/3)p r 3 Ê 3 ˆ ÁË 0, , 0˜¯ 8 $ E, r 3 È y2 2 y4 ˘ Í r ˙ 4 ˚0 2r 3 Î 2 # 3 8 >F>CG �!0OTEs $ $ H $ # $ $ H $ ! ŶŽżɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº ' ! "# $% - $ $ y $ a a O ; x C R $ $ "!a ! sin a a 8 6 $ C D! r cos 45∞ = 2R p x $ C R r= x 6 L= y r J$ $ pR 2 E6 $ 2 2R p 2R p r sin 45∞ = C DG $ $ C p! "! p 8 R y E6 $ O C DG # h C y C E6 $ b 2 - $ 1 A = bh 2 C DG b 2 $ R a a O x !"a C 2 ! sin a 3a 8 E6 $ C DG ' ɏ ŶŽŽ #¬µ»¹¬ɏ¶ɏ'¹¨½°»Àɏ¨µ«ɏ#¬µ»¹¶°«ɏ ; $ $ C y r Q r= x p R2 4 A= p R2 2 A= 2 ah 3 A= 4 ah 3 r cos 45∞ = 4R 3p r sin 45∞ = 4R 3p 4 2R 3p C R y A= 4! 3p O a C h y 3 8 3 5 O x C h y 3 5 O a a y = kx2 h C ah A= n +1 y Ê n +1 ˆ ÁË 4n + 2 ˜¯ h Ê n + 1ˆ ÁË n + 2 ˜¯ a 3 4 ! 3 10 ! O x " # y c h a x A= C y O b x ( a + b) h 2 2ac + a 2 + cb + ab + b 2 3(a + b) h ( 2a + b) 3(a + b) $% & ' ŷŴŴɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº ' () y 1 V = p r 2h 3 h C y r ) x O ) y C 'XAMPLE!krc y y r O 3 8 2 V = p r3 3 x ,¶ª¨»¬ɏ»¯¬ɏª¬µ»¹¶°«ɏ¶ɏ»¯¬ɏ·¨¹¨©¶³°ªɏ¨¹¬¨ɏ©¶¼µ«¬«ɏ©Àɏ»¯¬ɏ¿Ɋ¨¿°ºɏ¨µ«ɏ»¯¬ɏ³°µ¬ɏ¿ɏǛɏ¨Ȏɏ ¨ºɏº¯¶¾µɏ°µɏ&°®ȍɏŽȍŵŴ¨ȍ & ' ( y x = ky2 x = ky2 b b y/2 x O O x a a (a) Area bounded by a parabola and a line 5OLUTION * + /- ) + (b) Vertical strip considered ) , , fi " ) - + / Ú dA a Ú0 b y dx = Ú y (2ky dy ) 0 a Ú xc dA Ú0 x ( y dx) A .( . ! ) a x dx 2kb3 /3 b Ú0 (ky 2 b Ú0 2 2ky dy y3 2k 3 b )[ y (2ky dy )] 2kb3 /3 Ú0 2k b 0 2 4 2kb3 3 y dy 2kb3 /3 3k È y 5 ˘ Í ˙ b3 Î 5 ˚ b 0 ) (+ ɏ ŷŴŵ #¬µ»¹¬ɏ¶ɏ'¹¨½°»Àɏ¨µ«ɏ#¬µ»¹¶°«ɏ & ' 3kb 2 5 #, *+ 3 5 fi ∵ - * $ . !" #$% !" #$% ' y a x a Ú0 2 ( y dx) Ú0 2k dx Ú y dA 2kb3 /3 A 3(kb 2 ) 2 3 8 2 3 8k b 2kb3 /3 a È x2 ˘ Í ˙ 4k 2 b3 Î 2 ˚ 0 3 3a 2 8k 2 b 3 ∵' Locate the centroid of the area under the curve x = ky3 from x = 0 to x = a as shown in Fig. 9.11a. & ' ( 'XAMPLE!krd y y b x = ky3 (a, b) x = ky3 x b C y C y O yc x a O (a) Area under cubic curve 5OLUTIONs + * 7 " " 1 9 Ú, b 1" Ú, dA Úx + 6 ' * 6 9 1 b y dx b Ú0 2 y (3ky dy ) b a Ú0 x ( y dx) Ú0 (ky dA A 4kb3 7 " * " 1" Ú y dA A ∵ " - 3 È y4 3k Í ÍÎ 4 $ b ˘ ˙ ˙˚ 0 ) y (3ky 2 dy ) 3kb 4 /4 A 4 7 dx (b) Vertical strip 1 1 x x 6 fi 6 ' + ## 7 3kb 4 4 4k 2 kb 4 È y7 ˘ Í ˙ Î7˚ b 0 4k b 7 b4 7 6 ' - "" ' a Ú0 y ( y dx) 2 A b Ú0 y2 (3ky 2 dy ) 2 3kb 4 /4 2k È y 5 ˘ Í ˙ kb 4 Î 5 ˚ b 0 2 5 4 5 2 5 - ŷŴŶɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº 'XAMPLE!kre ,¶ª¨»¬ɏ»¯¬ɏª¬µ»¹¬ɏ¶ɏ´¨ººɏ¶ɏ»¯¬ɏ¯¨³ɊªÀ³°µ«¬¹ɏ¨ºɏº¯¶¾µɏ°µɏ&°®ȍɏŽȍŵŶ¨ȍɏ!ºº¼´¬ɏ»¯¨»ɏ »¯¬ɏ´¨»¬¹°¨³ɏ¶ɏ»¯¬ɏªÀ³°µ«¬¹ɏ°ºɏ¯¶´¶®¬µ¬¶¼ºȍɏ & ' ( y y y 200 mm y x z yc dq (a) Half cylinder h x z 100 mm C r q (b) Determining the centre of mass C 5OLUTION ! $ % $ % $ %% & # '' ' "# ' # # # ' 1 ) q* 2 ' & # + # q◊ # q ( 2 3 ( & . # # dV Ú 23 y dV Ú 2 p [cos q ] 0 3p & q /, )! 0* V & # '# ,# 1p 2 # - Úy ( & ! q p 2 (r 0 3 Ê r2 ˆ sin q ) ◊ Á dq ◊ h˜ Ë 2 ¯ 1 p r 2h 2 V 2 (-1 - 1) 3p - 4 3o 4 ¥ 100 $ 3p ) * * + - %% 2r 3p p Ú0 sin q dq ɏ ŷŴŷ #¬µ»¹¬ɏ¶ɏ'¹¨½°»Àɏ¨µ«ɏ#¬µ»¹¶°«ɏ :/5! .1!krf �! DFOUSPJET!PG!DPNQPTJUF!CPEJFT 8 )«¬µ»°Àɏ»¯¬ɏ³¶ª¨»°¶µɏ¶ɏ ª¬µ»¹¶°«ɏ¶ɏª¶´·¶º°»¬ɏ ©¶«°¬ºȍ ' ' ' ' # 9 Axis of reference x3 ) * x2 ' : ( # & ; ,# # W2 x1 ' C2 '# ( < C1 C : ' # & # W3 ( ' ! 0 9 # # 0 W1 ' X ' ' ( ) ' ' & 0 ' C3 ,# : 0 ' ( ='' ' WX fi W x > W x > W0 x0 fi ) ) W1 x1 + W2 x2 + W3 x3 W1 + W2 + W3 X= SW x SW y SW z , Y = , Z= SW SW SW > > 0* ) W1 x1 + W2 x2 + W3 x3 SWx SW & # 9 ' /, )! A* # # ' ' & r # # #- # ' : ' )! !* ' # ' 'XAMPLE!krf B /, )! A* SV x SV y SV z , Y = , Z= SV SV SV ' ' )r* ' X= 9 )! A* ' # ' ' ) # * & # C # /( # % ' !D ,¶ª¨»¬ɏ»¯¬ɏª¬µ»¹¶°«ɏ¶ɏ»¯¬ɏº¯¨«¬«ɏ¨¹¬¨ɏº¯¶¾µɏ°µɏ&°®ȍɏŽȍŵŸ¨ȍɏ & ' ( ŷŴŸɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº y y y 76.66 mm a 40 mm A1 R = 55 mm R = 55 mm 225 mm = b A2 112.5 mm 75 mm O C2 – C1 x x O 100 mm 130 mm x O 50 mm (b) Full rectangle (a) The plate with a cut (c) Semicircular cut 5OLUTION! ! " # $ # ' % && # % () * )) * )) * & % ) 100 225 % )& # % % % % % 2 2 4r 4 ¥ 55 %a% 100 % (+ ++ # 3p 3p # # ) % () * )) % ,& - ! 6 % 2=- 7 S )%% + ) ¥ && % )%%+ )& )&& p ¥ 552 = - 4751.66 2 Ax )%% + )%%+ ) (+ ++ ) &&& ,& ;,+ (( < , (+& (,( ( >( X , Y ) ) % + , S Ax 760 737.7 % = SA 17 748.34 %% .+/ Ay )%% + ), ;+ ( ( ) < , ), " * % S Ay 1913 534.2 % % SA 17 748.34 )& %% ɏ ŷŴŹ #¬µ»¹¬ɏ¶ɏ'¹¨½°»Àɏ¨µ«ɏ#¬µ»¹¶°«ɏ :/6! .1!krg �! UIFPSFNT!PG!QBQQVT! ! 3»¨»¬ɏ¨µ«ɏ·¹¶½¬ɏ 0¨··¼ºɏ»¯¬¶¹¬´ºȎɏ¨µ«ɏ ¨··³Àɏ»¯¬º¬ɏ»¯¬¶¹¬´ºɏ »¶ɏª¶´·¼»¬ɏ»¯¬ɏº¼¹¨ª¬ɏ ¨¹¬¨ɏ¨µ«ɏ½¶³¼´¬ɏ¶ɏ ¨ɏ©¶«Àɏ¯¨½°µ®ɏ¨¿°¨³ɏ ºÀ´´¬»¹Àȍ ! " �! 9ORTHY!OF!0OTE Uifpsfn! J ! % K L # N $ Qsppg;! > ! ,+&D " " E # ) ! ÚL dA % ÚL % 7" L )((; + , # " " + & ' # $ " % p ¥ $# " ! $ !# &;,)& # #M ! $ E ! NJ ' " # #" ! p y dL % p Ú y dL $ M ! ! L y A 1 6 y 4 L dL 5 C dA C 2 3 y y y y x O x O A dA Path of the centroid dV V (a) Pappus theorem I (b) Pappus theorem II F # Ú y dL $# L 7 # E L # p ! %I Uifpsfn!JJ % GH # % p Ú ydL % 2p ( y L) % ' E p $ > ¥J # ! #" ÚL y dL % y L ŷŴźɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº ! Qsppg; "$ # %& ' $ Ú A dV + ÚA y dA $ " # $ " & * p y dA + p Ú y dA A $ $ $ # # #& " & # & #+ ÚA "$ #+ p ¥ & "$ & ," * " # ÚA y dA + y A $ -. ' / # + p Ú y dA + 2p ( yA) + (2p y ) A A 0 "$ # * p & & ¥3 #+1 'XAMPLE!krg $¬»¬¹´°µ¬ɏ»¯¬ɏ³¨»¬¹¨³ɏº¼¹¨ª¬ɏ¨¹¬¨ɏ¨µ«ɏ½¶³¼´¬ɏ¶ɏ¨ɏ¹°®¯»ɏª°¹ª¼³¨¹ɏª¶µ¬ɏ¶ɏ©¨º¬ɏ¶ɏ ¹¨«°¼ºɏ¹ɏ¨µ«ɏ¯¬°®¯»ɏ¯ɏ¼º°µ®ɏ»¯¬ɏ0¨··¼ºɏ»¯¬¶¹¬´ºȍ & ' ( y y B A x1 h h C x2 O D x A O r r (a) Triangle being revolved 5OLUTION % &" ' & x B (b) Cone of revolution & 14, % $%& ' %& ' ! 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S p7 # )%% + p 7 ¥ 8- +- < +7 " p 7 ¥ +"#*<= 7* )%%+ !, Ay )%% + 7# =>7 7+ 4 2 ¥ 50 = 30.01 3p @ +8 * 7 == #8 7+- # !, 182 450.12 3063.05 p7 4 2 ¥ 80 = 48.02 3p =-<= -+ SAy SA a 4 2 ! 3p +- * * 8! 1 3p 2 8! & ' %% ( ɏ ŷŴŽ #¬µ»¹¬ɏ¶ɏ'¹¨½°»Àɏ¨µ«ɏ#¬µ»¹¶°«ɏ $¬»¬¹´°µ¬ɏ»¯¬ɏª¬µ»¹¶°«ɏ¶ɏ»¯¬ɏ¨¹¬¨ɏº¯¶¾µɏ°µɏ&°®ȍɏŽȍŶŴ¨ȍɏ 'XAMPLE!krk & ' y ( C2 25 mm C1 C y1 y A (a) 5OLUTION !, ¥ + 5 ! ! ! $ ! 4 -¥ + # - 12 p $ Myth Fact! . 5 / x B O 20 mm (b) A B 20 mm 25 mm y2 ! * 4 +-- 2 $ " - 12 ¥ p ¥ " % & % * % ' ' # & + ' , " ! 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" $ 'XAMPLE!clrc $¬»¬¹´°µ¬ɏ»¯¬ɏ´¶´¬µ»ɏ¶ɏ°µ¬¹»°¨ɏ¶ɏ¨ɏ¹¬ª»¨µ®¼³¨¹ɏ¨¹¬¨ɏ¾°»¯ɏ¹¬º·¬ª»ɏ»¶ɏș¨Țɏ°»ºɏ ª¬µ»¹¶°«¨³ɏ¨¿¬ºɏ·¨¹¨³³¬³ɏ»¶ɏ°»ºɏº°«¬ºȎɏ¨µ«ɏș©Țɏ°»ºɏ©¨º¬ȍɏșªȚɏ$¬»¬¹´°µ¬ɏ»¯¬ɏ·¶³¨¹ɏ ´¶´¬µ»ɏ¶ɏ°µ¬¹»°¨ɏ¨©¶¼»ɏ°»ºɏª¬µ»¹¶°«Ȏɏ¨µ«ɏș«Țɏ·¹¶«¼ª»ɏ¶ɏ°µ¬¹»°¨ɏ¾¹»ɏ°»ºɏ©¨º¬ɏ¨µ«ɏ³¬ļɏ¬«®¬ȍɏ 5OLUTION! ! % ) "# $ ' % ) *! ) ! ! % + ! & y Y ( ) ( ( % ) dY h 2 ,- "# " Y ( h/2 ( Ú * h 2 h/2 2 Y dA * - h /2 Ú 2 Y ¥ b dY - h /2 O h/2 È h3 Ê h3 ˆ ˘ bh3 ÈY 3 ˘ = bÍ - Á- ˜˙ * * bÍ ˙ 12 ÍÎ 24 Ë 24 ¯ ˙˚ Î 3 ˚-h / 2 ) ! ! ! + . ,- "# 0 1 6 7 & ( * ( & 7 ) * ' ! * ! 6 ) * 2 3 x b 2 hb 3 12 * 56 ! 3 ( b 2 " % * X C "# $ 1 3 bh bh bh bh Ê hˆ 7! ¥ Á ˜ = + = Ë 2¯ 12 12 4 ! + 9 ,- "# * bh3 hb3 bh + = 12 12 12 + ( *! "# : * ; (¢ ŷŷżɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº y 9 6 ! ¥ ¥ ! 6 * * ! 6 dy h 2 1 % % * Ú dI xy = Ú 1 h b 2 h2 b2 È y 2 ˘ y dy = Í ˙ = 2 2 Î 2 ˚0 4 < hb 0 *# 2 C¢ b 2 y h 2 1 x O "# "5"> b " 'XAMPLE!clrd Derive an expression for moment of inertia of a circular lamina about (a) an axis perpendicular to its plane and passing through centroid, and (b) one of its diametral axes. 5OLUTION! & % + ( "# ? ! * 6 * 6p % * 6p % . @ 9 ,- *# * R Èr4 ˘ o 4 * Ú dI z = Ú 2p r dr = 2p Í ˙ = 2 Î 4 ˚0 0 R ( * 3 y l dy dr y r x x C R x q C x R y (a) Ring-element for determining JC (b) Strip-element for determining Ix , ! ( ! * " A "# ?! & q q 1 % *6 q * q 1 * q ɏ ŷŷŽ -¶´¬µ»ɏ¶ɏ)µ¬¹»°¨ɏ * * 6 * ¥ q ¥ A . 6 *6 6 6 6 q q *6 6 q q q 6*6 q q¥ 6 6 q q q 1 ,p /2 * Ú dI x = Ú -p / 2 p /2 4 p56 q q p /2 2 R 4 cos 2 q sin 2 q dq = 1 4 1 - cos 4q R dq 2 -pÚ/ 2 2 q ∵ 1 4 q * B p56 sin 4q ˘ 1 È = Íq - 4 ˙ Î ˚ -p /2 4 4 q q Ê p sin 2p p sin(-2p ) ˆ + + ÁË ˜¯ 2 4 2 4 o !" # # % $ # $" " JC p R4 / 2 o = = 2 2 fi &' * # + ,-. 'XAMPLE!clre $¬¹°½¬ɏ¨µɏ¬¿·¹¬ºº°¶µɏ¶¹ɏ´¶´¬µ»ɏ¶ɏ°µ¬¹»°¨ɏ¶ɏ¨ɏª°¹ª¼³¨¹ɏ«°ºªɏ¶ɏ´¨ººɏ-Ȏɏ»¯°ª²µ¬ººɏ »Ȏɏ¹¨«°¼ºɏ2ɏ¨µ«ɏ«¬µº°»Àɏrɏ¨©¶¼»ɏș¨Țɏª¬µ»¹¶°«¨³ɏ¨¿°ºɏ·¬¹·¬µ«°ª¼³¨¹ɏ»¶ɏ°»ºɏ·³¨µ¬Ȏɏ¨µ«ɏ ș©Țɏ¶µ¬ɏ¶ɏ°»ºɏ«°¨´¬»¹¨³ɏ¨¿¬ºȍɑ 5OLUTION 0 $ 6 3 # 5 " 5 # 5 ¥ ¥r 6 * &+ p pr 7 # $ 5 # , 5 . - # 8 " R Èr4 ˘ p t r R4 Ú dI z = Ú-p / 2 2p t r r dr = 2p t r Í 4 ˙ = 2 Î ˚0 R $ 0 # (p R 2 t r ) R 2 MR = 2 3 ∵6 # " p r # 3 ŷŸŴɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº y l dy dr y r C R x q x x x C R y (a) (b) , ! $ < # ; # # " " 8 * q + q " " 8 # 5 0 3! q q 6 " " ¥ 6 ¥ ¥r fi # ; # ¥ # + q¥ ¥r¥ " r q q¥ r # q p /2 r q q q r q q # q 2 R 4 t r cos 2 q sin 2 q dq = -p / 2 1 4 1 - cos 4q R tr Ú dq 2 2 -p / 2 q ∵ q q q p /2 1 4 È sin 4q ˘ 1 Ê p sin 2p p sin(-2p ) ˆ R t r Íq = R 4t r Á + + ˜¯ ˙ Ë2 4 4 ˚ -p /2 4 4 2 4 Î > 5 p # Myth 2 r 1 p 4 MR = 4 4 # Fact! ! ' # $ + " #@ # A " 6BD # # " % # " fi # ! # $ r 2 " # q " " r" p p, " p, p /2 Ú dI x = Ú fi q q * #@ # # # ! J C MR 2 / 2 MR 2 = = 2 2 4 &' * # + ,? , . ɏ ŷŸŵ -¶´¬µ»ɏ¶ɏ)µ¬¹»°¨ɏ 'XAMPLE!clrf $¬»¬¹´°µ¬ɏ»¯¬ɏ´¨ººɏ´¶´¬µ»ɏ¶ɏ°µ¬¹»°¨ɏ¶ɏ¨ɏº¶³°«ɏ¹°®¯»ɏª°¹ª¼³¨¹ɏªÀ³°µ«¬¹ɏ¶ɏ¹¨«°¼ºɏ 2ɏ¨µ«ɏ¯¬°®¯»ɏ¯ɏ¨©¶¼»ɏș¨Țɏ°»ºɏ³¶µ®°»¼«°µ¨³ɏ¨¿°ºȎɏ¨µ«ɏș©Țɏ¨µɏ¨¿°ºɏª¶µ»¨°µ¬«ɏ°µɏ¶µ¬ɏ¶ɏ °»ºɏ¨ª¬ºɏ¨µ«ɏ·¨ºº°µ®ɏ»¯¹¶¼®¯ɏª¬µ»¹¬ɏ¶ɏ»¯°ºɏ¨ª¬ȍɏ 5OLUTION < 5 - # ! 8 5 p ¥ " . F + + ¥ G6 # # E $ 0 D# r # # " # # Ú dI y = Ú r 0 r . p r - dm = Ú r 2p r r hdr = 2pr h Ú r dr 0 E # # # # ∵ ! 5 $ ! 6 0 # " # " # 2 dmR % 4 p R dy r ¥ R 4 ¥ . r # r 6 , H p # 2 p 8 * " , R 3 0 &+ Èr4 ˘ pr hR 2 R2 2pr h Í ˙ = = (p R 2 hr ) 2 2 Î 4 ˚0 R 2 MR ! ¥ " R 2 * " 5 " R " # * 2 p r¥ ! h Ê p R 4 dy r ˆ 2 2 dI = Ú x Ú ÁË 4 + p R y dy r˜¯ 0 h È R2 y y3 ˘ + ˙ pR rÍ 3 ˚0 Î 4 2 fi È R 2 h2 ˘ + ˙ Í 3˚ Î 4 h Ê R 2 dy ˆ p R2 r Ú Á + y 2 dy ˜ 4 ¯ 0Ë È R 2 h h3 ˘ + ˙ p R2 r Í 3˚ Î 4 p ∵ p È R 2 h2 ˘ + ˙ r Í 3˚ Î 4 "# r y y R R dr r h h y x (a) Shell element for determining Iz dy O R x (b) Disc element for determining Ix $ %! %&%%' ŷŸŶɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº 'XAMPLE!clrg &°µ«ɏ»¯¬ɏ´¨ººɏ´¶´¬µ»ɏ¶ɏ°µ¬¹»°¨ɏ¶ɏ¨ɏ¯¶³³¶¾ɏªÀ³°µ«¬¹ɏ¨©¶¼»ɏ°»ºɏ¨¿°ºȍɏ4¯¬ɏ´¨ººɏ¶ɏ »¯¬ɏɏªÀ³°µ«¬¹ɏ°ºɏŹɏ²®Ȏɏ°µµ¬¹ɏ¹¨«°¼ºɏ°ºɏŵŴɏª´Ȏɏ¶¼»¬¹ɏ¹¨«°¼ºɏ°ºɏŵŹɏª´ɏ¨µ«ɏ¯¬°®¯»ɏ°ºɏɏ ŶŴɏª´ȍɏ 5OLUTION * + , 56 ! . %! . %5 . # r fi 7 p M r 2 5000 2 p ( R - r )h 2 p (15 - 102 ) ¥ 20 $ 9 ! 89: & %5 %! $ $ 4 4 rp hR rp hr rp h 4 = (R - r 4 ) 2 2 2 'XAMPLE!clrh 0.637 ¥ p ¥ 20 4 (15 - 104 ) 2 ɑ!ɏ¹¬ª»¨µ®¼³¨¹ɏ·¨¹¨³³¬³¬·°·¬«ɏ¯¨ºɏ»¯¬ɏ¶³³¶¾°µ®ɏ«°´¬µº°¶µºȏ length along x-axis = a ; breadth along y-axis = b height along z-axis = h ; density of the material = r. Determine the mass moment of inertia of the parallelepiped about (a) its edge along x-axis, and (b) the centroidal axes. 5OLUTION ; > %! %% < < ¥!¥ << ? ! , ¥ ? < ¥ + $ < + Úm r dm ÚV ( y 2 + z 2 )( r dV ) ∵ hba Ú Ú Ú (y 2 + z 2 ) r(dxdydz ) ∵ . 000 hb hb a r Ú Ú ÈÎ( y 2 + z 2 ) x ˘˚ dydz 0 00 b Èy ˘ ra Ú Í + z 2 y ˙ dz 3 ˚0 0Î h r Ú Ú ÈÎ( y 2 + z 2 )a ˘˚ dydz 3 00 Ê b3 ˆ ra Ú Á + z 2 b˜ dz 3 ¯ 0Ë h ɏ ŷŸŷ -¶´¬µ»ɏ¶ɏ)µ¬¹»°¨ɏ h È b3 z3 ˘ ra Í z + b ˙ 3 ˚0 Î3 È b3 h3 ˘ r abh 2 (b + h 2 ) ra Í h + b˙ = 3 3 3 Î ˚ fi /¥ ∵ ¥!¥ z z X dm z r y h C d h/2 y a b/2 b x (a) Rectangular parallelepiped ! C y X ! (b) Front view of the parallelepiped (yz plane) ( ( ( ( ! "# ""! $ Ê bˆ Ê hˆ ÁË ˜¯ + ÁË ˜¯ \ % ( & ' ! & % ! ÈÊ b ˆ Ê hˆ ˘ ÍÁ ˜ + Á ˜ ˙ Ë ¯ ˙ ÍÎË ¯ ˚ % Ê b2 h2 ˆ Á 4 + 4˜ Ë ¯ Ê1 ÁË 3 ! ) ! ) * 12 %º»°´¨»¬ɏ·¹°µª°·¨³ɏ ´¶´¬µ»ºɏ¶ɏ°µ¬¹»°¨ɏ¶ɏ ¨µɏ¨¹¬¨ȍ 21/8! 12 * 12 *+ .1!clri 1ˆ ˜ 4¯ $ + �! QSJODJQBM!NPNFOUT!PG!JOFSUJB!PG!BO!BSFB 8 9 $ Úy dA: Úx dA Ú xydA "# ","-/ ŷŸŸɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº y s Ê Ix - Iy ˆ ÁË 2 ˜¯ dA A s y cos q q y r q 2q1 x sin q q O 2q2 Ixy x Ê Ix - Iy ˆ -Á Ë 2 ˜¯ y sin q z, t –Ixy x x cos q r (a) Rotated axes system 2 Ê Ix - Iy ˆ 2 ÁË 2 ˜¯ + I xy (b) Coordinate axes rotated by angle q 8 "# " q ; < ! ! q q% q > $ Ús Ú Ú ( y cos q - x sin q ) dA fi Ix + I y Ir = 2 dA q q% + 2 q% q q Ix - I y 2 q q q cos 2q - I xy sin 2q > "# "& q ∵ q q% q q q $ Úr Ú Ú ( x cos q + y sin q ) dA q fi Is = Ix + I y 2 Ix - I y 2 $ Ú rs dA Ú q Ú dA q q - 2 q q @ q q q q cos 2q + I xy sin 2q "# "? ! q q0 q% q q q q q% q ɏ ŷŸŹ -¶´¬µ»ɏ¶ɏ)µ¬¹»°¨ɏ % fi q Ix - I y I rs = 2 !" " % $ ' % ! ! q sin 2q + I xy cos 2q Ix + I y & q% q Ix - I y + fi # ! $ % !" " # '% ! & & ! " % ' ! " * " !" " & & ! " & # ! ! ! $ % $ % ! " Qsjodjqbm!Npnfout!pg!Jofsujb ! & # ! * " # & ' , ! ! " & "%" # ' +' "%" ! ' , " ! # ! ' & dI r dq fi % ! ' q 1 * % q/ ¥ q 1 ( I x - I y )/2 q q # ## ! q # q ' # q q ∓ I xy q ( 2 I xy + ( I x - I y )/ 2 6% ' - I xy ! 5 ## ! Ê Ix - I y ˆ / ¥Á ˜¯ Ë , % ' & & ! ' " " &! " , % # " " # ! ! & ! !" ! ' q % ! % . ! q $ , , % Ix + I y + Ix - I y ¥ ) ( $* 3 4 q " *$ ! $ ' 2 " & "%" ± ( I x - I y )/ 2 ( 2 I xy + ( I x - I y )/ 2 Ix + I y \ 6 " &" , , " $ ( ± ( I x - I y )/ 2 q ( " ) / "%" , % ¥ ( ) ) ÏÔÊ I - I ˆ ¸Ô x y + I ÌÁ xy ˝ ˜¯ ÔÓË Ô˛ 1 ± ! $* ! 2 2 # ) 2 " " # ! ∓ I xy ( 2 I xy + ( I x - I y )/ 2 Ê Ix - I y ˆ ± Á ˜¯ + I xy Ë " & # ! ) 2 + ( I x - I y )/ 2 I xy 2 + ( I x - I y )/ 2 I xy Ix + I y , $ ) 2 8 , , ŷŸźɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº 9 % ' " % ! %' Ix - I y Ix - I y # ! # ! ' q $ , , % ( 2 I xy + ( I x - I y )/ 2 ) ( ± ( I x - I y )/ 2 ¥ 2 ) ( 2 + ( I x - I y )/ 2 I xy fi " *$ % ' q ∓I xy ¥ # # !$ ) 2 ' "$ : ' # " ! &¹¶´ɏ »¯¬ɏ ķ¹º»ɏ ·¹°µª°·³¬ºȎɏ «¬»¬¹´°µ¬ɏ »¯¬ɏ ·¹¶«¼ª»ɏ ¶ɏ °µ¬¹»°¨ɏ ¶ɏ ¨ɏ ¹°®¯»Ɋ¨µ®³¬«ɏ »¹°¨µ®³¬ɏ¶ɏ©¨º¬ɏ©ɏ¨µ«ɏ¨³»°»¼«¬ɏ¯ɏ¾¹»ɏ¿Ɋɏ¨µ«ɏÀɊ¨¿¬ºɏ¨ºɏº¯¶¾µɏ°µɏ&°®ȍɏŵŴȍŵŷȍ 'XAMPLE!clri 5OLUTION 9 % ' ! # ; ; , ! ' #' #' ! ! ! # # ! ! # '- & " " *"" !* #! " #! " & & h Ê ˆ ÁË∵ From geometry, y = ¥ x˜¯ b ! %' # ! # ! y hx = 2 2b ! " ¥ ! dI xy @% % ! ! ' ' ? ! # # %' ! # $ % $ % ! ' ! ' & : ¥ ¥ ! ! ' Ú b b h2 Ú0 2b2 ! & 2 y h h2 = x ¥ dx ¥ 2 x 2 = 2 2 2b 2b ! ! %' # ! = x ¥ dx ¥ #! " dI xy & dI xy ! 2 \ ' ¥ ! ! ! b x3 dx = h2 È x4 ˘ Í ˙ 2b 2 Î 4 ˚ 0 h2 b4 ¥ 4 2b 2 b 2 h2 8 y y 2 mm 2 mm y y= 4 mm A2 h x b 10 mm (x, y) 10 mm y 2 mm x (x, y) dx b C1 A1 h x O 6 mm (a) C2 8 mm 2 mm O 6 mm (b) x ɏ ŷŸŻ -¶´¬µ»ɏ¶ɏ)µ¬¹»°¨ɏ 'XAMPLE!clrj &°µ«ɏ»¯¬ɏ·¹°µª°·¨³ɏ´¶´¬µ»ɏ¶ɏ°µ¬¹»°¨ɏ¨©¶¼»ɏ»¯¬ɏ¶¹°®°µɏ/ɏ¶ɏ»¯¬ɏ¨¹¬¨ɏº¯¶¾µɏ°µɏ &°®ȍɏŵŴȍŵŸ¨ȍɏ 5OLUTION! D , 9 ' ' ! ' # E "" ¥ "" . ! $ ' ! 5 ¥1 #! ' $ ! ' E¥ 1 "" G " # ! # $ % & G " # ! # $ % & # ! # # ! 9 #9 ' ' $ % $ % 6! %' # ! # ! ' & 6! %' # ! # ! ' & ¥ ¥ I ! 6! %' # ! #9 ' / ? 8 ! ' " " Ix + I y ! EI ! 1 ! 18 "" 18 188 "" "" b 2 h 2 62 ¥ 102 = 4 4 81E "" # ' & ' / 81E # " &" ' ! 3 " 3 Ê 10 ¥ 63 ˆ Ê 8 ¥ 43 2ˆ Á 3 ˜ - Á 12 + 32 ¥ 4 ˜ Ë ¯ Ë ¯ / . "" 6 ¥ 10 bh = "" 3 3 bh3 4 ¥ 83 ¥1 12 12 & / / & $ 3 G " H" !* " " "" ¥ 1 "" # ! ! J # 3 "" " !! ' "" # 9 ' ! ' & ! $ Ê Ix - I y ˆ ± Á ˜¯ + I xy Ë 2 677.33 + 165.33 Ê 677.33 - 165.33 ˆ 2 ± Á ˜¯ + 132 Ë 2 2 ± EE fi ! " & " #&&+6+10#.!51.8'&!':#/2.'5 'XAMPLE!clrk 5OLUTION 5 %! " # ! " %$ 3¯¶¾ɏ»¯¨»ɏ»¯¬ɏ´¶´¬µ»ɏ¶ɏ°µ¬¹»°¨ɏ¶ɏ¨ɏ»¯°µɏª°¹ª¼³¨¹ɏ¹°µ®ɏ¶ɏ´¨ººɏ-ɏ¨µ«ɏ´¬¨µɏ ¹¨«°¼ºɏ2ɏ¾°»¯ɏ¹¬º·¬ª»ɏ»¶ɏ°»ºɏ®¬¶´¬»¹°ªɏ¨¿°ºɏ°ºɏ-2 Ŷȍ ! 9 p , !* " # r " #" ! r q ! % * # ' ! # qKG # 9 $ " ! # ! " '! ! ' J 5 q¥ ! ! ! ¥r # " ! ! ! ŷŸżɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº y y Rdq R C ' , J # !q # $* " #! " ' J x C (a) Thin ring H ' qR dq x (b) Peripheral element in the ring ! # r q ! " " " " " " # ! # ! $ # " ! $* $ % J p 2p Ú dJ C = Ú0 J p R3Ar dq = R3Ar [q ]02p p r L9 r 'XAMPLE!clrcl $ I #¨³ª¼³¨»¬ɏ»¯¬ɏ¨¹¬¨ɏ´¶´¬µ»ɏ¶ɏ°µ¬¹»°¨ɏ¶ɏ»¯¬ɏº¯¨«¬«ɏ¨¹¬¨ɏº¯¶¾µɏ°µɏ&°®ȍɏŵŴȍŵź¨Ȏɏ ¨©¶¼»ɏ»¯¬ɏ¿Ɋ¨¿°ºȍɏ y y a a dy x = ky2 a a x O y x O (a) 5OLUTION J ! G " # ! ! ! 5 # ! # x = ky2 x (b) '1! 9 $ % ! # & ! ' ! # #! " ! ! ¥ & ¥ " " # ! # ! $ % #! " a % È y5 ˘ a ka 4 Ú dI x = Ú0 ky dy = k ÍÎ 5 ˙˚0 = & $ $* ! - " M ɏ ŷŸŽ -¶´¬µ»ɏ¶ɏ)µ¬¹»°¨ɏ 5 ! N fi ! 1 \ 5 ¥ = 5 'XAMPLE!clrcc 4 5 $¬¹°½¬ɏ¨µɏ¬¿·¹¬ºº°¶µɏ¶¹ɏ»¯¬ɏ´¶´¬µ»ɏ¶ɏ°µ¬¹»°¨ɏ¶ɏ¨ɏº¬´°Ɋª°¹ª¼³¨¹ɏ¨¹¬¨ɏ¨©¶¼»ɏ°»ºɏ «°¨´¬»¹°ªɏ©¨º¬ȍɏ 5OLUTION l dy ! "# " q $ ! q " ¥ " q& ¥ " %# X y q q %∵ " q q"# ' q q& # # C x R q 4R 3p X x q q ! # ¥ " "# # # q&# " # q¥ q¥% ) * " p /2 Ú dlx = Ú0 2 R 4 cos 2 q sin 2 q dq " %∵ " 1 4 p /2 4 sin 4q ˘ È ÍÎq - 4 ˙˚ 0 " 1 4 4 # 1 2 ( # # q %& q q +, % & q" p-#! cos 4q dq 2 q" #q % .# p /21 R4 0 Ú q # #q& " sin(0) ˆ o 4 Ê p sin 2p +0+ = 0.393 4 ÁË ˜= 8 2 4 4 ¯ 7 ) (q& -(8 �!0OTE '9: % + ; #& '9: 'XAMPLE!clrcd !ɏªÀ³°µ«¬¹ɏ¶ɏŹŴŴɊ´´ɏ«°¨´¬»¬¹ɏ¨µ«ɏŵŶŴŴɊ´´ɏ¯¬°®¯»ɏ¯¨ºɏ¨ɏ´¨ººɏ«¬µº°»Àɏ¶ɏżŴŴŴɏ ²®Ʉ´ŷȍɏ&°µ«ɏ»¯¬ɏ´¨ººɏ´¶´¬µ»ɏ¶ɏ°µ¬¹»°¨ɏ¶ɏ»¯¬ɏªÀ³°µ«¬¹ɏș¨Țɏ¨©¶¼»ɏ°»ºɏ¨¿°ºȎɏ¨µ«ɏ ș©Țɏ¨©¶¼»ɏ¨ɏ³°µ¬ɏ¾¯°ª¯ɏª¶°µª°«¬ºɏ¾°»¯ɏ¨µɏ¬µ«ɏ¨ª¬ɏ¶ɏ»¯¬ɏªÀ³°µ«¬¹ɏ¨µ«ɏ·¨ºº°µ®ɏ»¯¹¶¼®¯ɏ»¯¬ɏª¬µ»¹¬ɏ¶ɏ »¯°ºɏ¨ª¬ȍɏ 5OLUTION! < ' % & ) > "? * " ! ? " rp ! " # # "@ * MR 2 1884.9 ¥ 0.252 = " " 2 2 " # !r"@ - A ¥ p ¥ % ?-#&# ¥ # " @@( B ! ŷŹŴɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº %!& $ + (! * È R 4 h2 ˘ + ˙ " @@( B ¥ "MÍ 3˚ Î 4 È 0.252 1.22 ˘ + Í ˙" 3 ˚ Î 4 $¬»¬¹´°µ¬ɏ ´¨ººɏ ´¶´¬µ»ɏ ¶ɏ °µ¬¹»°¨ɏ ¶ɏ ¨ɏ º³¬µ«¬¹ɏ ¹¶«ɏ ¶ɏ ³¬µ®»¯ɏ ,ɏ ¨©¶¼»ɏ °»ºɏ ª¬µ»¹¶°«¨³ɏ¨¿°ºɏµ¶¹´¨³ɏ»¶ɏ»¯¬ɏ¹¶«ȍɏ 'XAMPLE!clrce y %! @ 5OLUTION *r x 9! L/2 9 ' ' ! x O dx L/2 "r # ¥ " ! ' " %r & # * " L/2 L/2 Ú- L / 2 dI m = Ú- L / 2 (r Adx) x 2 = r AÚ L/2 -L/ 2 x 2 dx L/2 È x3 ˘ È ( L/2)3 (- L/2)3 ˘ È L3 L3 ˘ r AL3 = rAÍ " pA Í ˙ ˙ = rAÍ + ˙ = 3 ˚ Î 24 24 ˚ 12 Î 3 ˚- L / 2 Î 3 " mL2 12 'XAMPLE!clrcf %∵ ' " r %& ! &°µ«ɏ »¯¬ɏ ´¨ººɏ ´¶´¬µ»ɏ ¶ɏ °µ¬¹»°¨ɏ ¶ɏ ¨ɏ ¹°®¯»ɏ ª°¹ª¼³¨¹ɏª¶µ¬ɏ¶ɏ©¨º¬ɏ¹¨«°¼ºɏ2ɏ¨µ«ɏ´¨ººɏ-ɏ¨©¶¼»ɏ 7 ) -# 8 y °»ºɏ¨¿°ºȍɏ dy 5OLUTION B x ! " R % . & h ' O ! " rp # ! * r ' 7) -##8! # " # y ! # " rp # ¥ # 1 rp # ( 1 { } rp R (h - y ) 2 h 4 R x ɏ ŷŹŵ -¶´¬µ»ɏ¶ɏ)µ¬¹»°¨ɏ ) * h Ú0 " Ú dl y " fi rp R 4 " " " ! h È ( h - y )5 ˘ Í ˙ Î -5 ˚0 4 rp Ï R rp R 4 ¸ Ì (h - y ) ˝ dy = 2 Óh 2h 4 ˛ rp R 4 h 10 3m p R2h ÈÎ- (h - h)5 + (h - 0)5 ˘˚ = 10h 4 1 3 p fi r r rp R 4 h p R4h 3m = ¥ 10 10 p R4h 3 10 ! "# "$ %& 'XAMPLE!clrcg !ɏ©¹¨ººɏª¶µ¬ɏ¾°»¯ɏŸŴŴɊ´´ɏ©¨º¬ɏ«°¨´¬»¬¹ɏ¨µ«ɏŶŶŹɊ´´ɏ¯¬°®¯»ɏ°ºɏ·³¨ª¬«ɏ¶µɏ¨ɏ ½¬¹»°ª¨³ɏ¨³¼´°µ°¼´ɏªÀ³°µ«¬¹ɏ¶ɏŷŴŴɊ´´ɏ¯¬°®¯»ɏ¨µ«ɏŸŴŴɊ´´ɏ«°¨´¬»¬¹ȍɏ$¬µº°»Àɏ¶ɏ ©¹¨ººɏǛɏżŹɏ².Ʉ´ŷɏ¨µ«ɏ«¬µº°»Àɏ¶ɏ¨³¼´°µ°¼´ɏǛɏŶźȍŵɏ².Ʉ´ŷȍɏ$¬»¬¹´°µ¬ɏ»¯¬ɏ´¨ººɏ´¶´¬µ»ɏ¶ɏ°µ¬¹»°¨ɏ¶ɏ »¯¬ɏª¶´·¶º°»¬ɏ©¶«Àɏ¨©¶¼»ɏ»¯¬ɏ½¬¹»°ª¨³ɏ®¬¶´¬»¹°ª¨³ɏ¨¿°ºȍɏ 5OLUTION! ' *+ , * " -##$ ## /. 67$ -##$ ## r % " 67$ r" + , ; ,* < 1 3 " < ; < , , ? , ; 3 10 " " A 3 10 1 2 8 p # . " 8 # / %9 ¥ "# 6 $ : # 8## 8 8 ¥p¥# 1 3 p r p¥# ! , ! ; ! , ,, ? ¥ /" 9" ¥ # A 1 2 #8 8 %" ¥ "# 6 $ " "r" . 8 . ¥ / %9 ¥ "#8 ¥# ¥#8¥ @ %" ¥ "# 8 ; , /" 9" 6 >/ - 6 ,; @ ¥ >/ - ¥ # , ,, ! "# "$"" "# "$ %& 'XAMPLE!clrch $¬»¬¹´°µ¬ɏ»¯¬ɏ´¨ººɏ´¶´¬µ»ɏ¶ɏ°µ¬¹»°¨ɏ¶ɏ¨ɏ»¯°µɏ¬¸¼°³¨»¬¹¨³ɏ»¹°¨µ®¼³¨¹ɏ·³¨»¬ɏ¶ɏ ´¨ººɏ´ɏ¨µ«ɏ»¯°ª²µ¬ººɏ»ɏ¨©¶¼»ɏ»¯¬ɏ¨¿°ºɏ·¬¹·¬µ«°ª¼³¨¹ɏ»¶ɏ»¯¬ɏ·³¨µ¬ɏ¨µ«ɏ·¨ºº°µ®ɏ »¯¹¶¼®¯ɏ»¯¬ɏª¬µ»¹¬ɏ¶ɏ´¨ººȍɏ"¨º¬ɏ¾°«»¯ɏǛɏ©Ȏɏ¨µ«ɏ¯¬°®¯»ɏ¶ɏ½¬¹»¬¿ɏ¨©¶½¬ɏ©¨º¬ɏǛɏ¯ȍɏ$¬µº°»Àɏ¶ɏ»¯¬ɏ ´¨»¬¹°¨³ɏ°ºɏ¾ȍɏ 5OLUTION ? @ ! J 6 + , ? "# # ! , 7 ? OC ¥ FG OA ,? C @ , C ! b/2 ¥ I h !; C ! ,B ,? , , , b h I @ , 6 , ! ŷŹŶɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº < '¥ ,? < ¥ ¥ , ,? ! b ' h ¥ < b h ' , @ wtb h ! ¥ I , I ,? 8 I ! h wtb wtb È hy 3 y 4 ˘ wtb È h 4 h 4 ˘ wtbh3 (hy 2 - y 3 ) dy = Í ˙ = Í - ˙= 4 ˚0 4˚ 12 h h Î 3 h Î3 h Ú dI x Ú0 dy y, Y y, Y A A R x y X G B O P¢ h Q P G x C Q¢ O x b B b , ' ¥ 12 ! ¥ ,? x dx C (b) Determining IY (a) Determining IX < h Xy fi ' m bht ; 3 2m tbh3 1 2 wtbh = ¥ = mh 12 12 6 bht K 2 , , L ? , ,? Wh 2 6g @ , ! ! , ,, , @?, 2$ !; ( @ ? , , , ;! , ' ( ( \ I ( & A , M M 1 6 I $8 M Ê ˆ ¥Á ˜ Ë 3¯ 2 ! 2 mh 18 ( @ @ 2 2 m(b 3 / 2) mb = 18 24 ∵ ! 8$ , ; * ) ! ! " # OA h !h Ê b ˆ ¥$ %& '# ¥ $!*! & ' # Á - x˜¯ OC b/2 b Ë! ( ) $ , , ɏ ŷŹŷ -¶´¬µ»ɏ¶ɏ)µ¬¹»°¨ɏ + #'¥ ¥ ,¢-¢ ¥ #'¥ !h Ê b ˆ Á - x˜¯ ¥ b Ë! ¥ # ! wth Ê b ˆ Á - x˜¯ dx b Ë! + 2 ˆ ! wth Ê b ˆ ! 2 wth Ê bx - x3 ˜ dx ÁË - x˜¯ dx ¥ # ÁË ¯ b ! b 2 $ ). ' 0 !*! 2 ˆ ˆ b / 2 2 wth Ê bx 4 wth b / 2 Ê bx 2 - x3 ˜ dx = - x3 ˜ dx # 2 Ú dI y = 2 ¥ Ú Á ÁË Ú Ë ¯ ¯ 0 0 b 2 b 2 # ) ! ¥ # # b/2 4 wth È bx3 x 4 ˘ 4 wth È b 4 b 4 ˘ wthb3 # - ˙ = Í Í - ˙= b Î 6 4 ˚0 b Î 48 64 ˚ 48 ' 70 $ ' , # ) 2m thb3 mb 2 ¥ = 24 bht 48 # ) 9 # * ( ; 'XAMPLE!clrci ) ) * ) # mb 2 mb 2 mb 2 + = 24 24 12 : < * >? $¬»¬¹´°µ¬ɏ»¯¬ɏ´¨ººɏ´¶´¬µ»ɏ¶ɏ°µ¬¹»°¨ɏ¶ɏ¨ɏ¹¬ª»¨µ®¼³¨¹ɏ·³¨»¬ɏ¶ɏº°Á¬ɏ¨ɏ¥ɏ©ɏ¨µ«ɏ »¯°ª²µ¬ººɏ»ɏ¨©¶¼»ɏ»¯¬ɏª¬µ»¹¶°«¨³ɏ¨¿¬ºȍɏ y ! 5OLUTION ¥!¥ @ : x dx t % @ r # r ¥ $! + : x O b ' + # ! # ! ¥ r ¥ $! ' # r! ! $' + a : # & *! : # a/2 Ú- a / 2 dlm = Ú a/2 - a/2 pbta Ma 2 = 12 12 # pbtb3 Mb 2 = 12 12 x3 3 a/2 È a3 a3 ˘ = rbt Í + ˙ Î 24 24 ˚ -a/2 3 9 # rbt x 2 dx = rbt ; *! $ #r ! # ' : 3 <@ * B? ŷŹŸɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº 'XAMPLE!clrcj !ɏ»¯°µɏ·³¨»¬ɏ¶ɏ´¨ººɏ´ɏ°ºɏª¼»ɏ°µɏ»¯¬ɏº¯¨·¬ɏ¶ɏ¨ɏ·¨¹¨³³¬³¶®¹¨´ɏ¶ɏ»¯°ª²µ¬ººɏ»ɏ¨ºɏ º¯¶¾µɏ°µɏ&°®ȍɏŵŴȍŶŶȍɏ$¬»¬¹´°µ¬ɏ»¯¬ɏ´¨ººɏ´¶´¬µ»ɏ¶ɏ°µ¬¹»°¨ɏ¶ɏ»¯¬ɏ·³¨»¬ɏ¨©¶¼»ɏ»¯¬ɏ ¿Ɋ¨¿°ºȍɏ 5OLUTION D ! 0 y !D !! + 0 Plate-2 t X & E + , , ' < 7 70 $ ' # & !E ∵ + , x #! (! # b ! ! 70 $ $' 7 G' 2 (m/2)h mb = 18 36 + $ ! b z 1 $ 2 2 b 3 * >? F G (m1 )h 2 mb 2 = 6 12 Plate-1 X b $ ; ! ' # (! # mb 2 m Ê 2 ˆ mb 2 2mb 2 9mb 2 mb 2 + ¥ Á b˜ = + = = 36 2 Ë3 ¯ 36 9 36 4 ∵ ! ! : 2 ! 3 # ! # : 2 # ; ! # mb 2 mb 2 4mb 2 mb 2 + = = 12 4 12 3 'XAMPLE!clrck !µɏ¨ºº¬´©³Àɏ¶ɏ»¯¹¬¬ɏº³¬µ«¬¹ɏ¹¶«ºɏ¬¨ª¯ɏ¶ɏ³¬µ®»¯ɏ³ɏ¨µ«ɏ´¨ººɏ´ɏ°ºɏ·°½¶»¬«ɏ¨»ɏ°»ºɏ ¨·¬¿ɏ/ɏ¨ºɏº¯¶¾µɏ°µɏ&°®ȍɏŵŴȍŶŷ¨ȍɏ$¬¹°½¬ɏ¨µɏ¬¿·¹¬ºº°¶µɏ¶¹ɏ´¨ººɏ´¶´¬µ»ɏ¶ɏ°µ¬¹»°¨ɏ ¶ɏ»¯¬ɏ¨ºº¬´©³Àɏ¨©¶¼»ɏ¨µɏ¨¿°ºɏ·¨ºº°µ®ɏ»¯¹¶¼®¯ɏ/ɏ¨µ«ɏ·¬¹·¬µ«°ª¼³¨¹ɏ»¶ɏ»¯¬ɏ·³¨µ¬ɏ¶ɏ»¯¬ɏ¨ºº¬´©³Àȍɏ #¶´·¼»¬ɏ»¯¬ɏ½¨³¼¬ɏ¶ɏ»¯¬ɏ´¶´¬µ»ɏ¶ɏ°µ¬¹»°¨ɏ°ɏ³ɏǛɏŶŴɏª´ɏ¨µ«ɏ´ɏǛɏŻŹɏ®ȍɏ O O d1 l l d2 60° l (a) Assembly of slender rods A C2 C1 C3 l (b) Centroids of rods B ɏ ŷŹŹ -¶´¬µ»ɏ¶ɏ)µ¬¹»°¨ɏ 5OLUTION ml 2 12 ! "#" $! % & ' % - &( ) &$ * ) 3 2 + , " . ( " " ) ) 2 È2 ¥ I rod + m ¥ d12 ˘ Î ˚ AO,OB + ÈÎ I rod + m ¥ d 2 ˘˚ AB ) 2 2 È ml 2 È Ê ml 2 Ê 3l ˆ ˘ Ê l ˆ ˆ˘ ˙ Í2 ¥ Á +Í +m¥Á + m ¥ Á ˜ ˜˙ Ë 2 ˜¯ ˚˙ AB Ë 2 ¯ ¯ ˙˚ Ë 12 ÍÎ 12 ÍÎ AO,OB 2 2 5 2 ml + ml 3 6 3ml 2 2 # 04 " ) 3 ¥ 04 ¥ 2 'XAMPLE!clrdl $¬»¬¹´°µ¬ɏ»¯¬ɏ´¶´¬µ»ɏ¶ɏ°µ¬¹»°¨ɏ¶ɏ»¯¬ɏ4Ɋº¬ª»°¶µɏº¯¶¾µɏ°µɏ&°®ȍɏŵŴȍŶŸ¨ɏ¨©¶¼»ɏ»¯¬ɏ ¯¶¹°Á¶µ»¨³ɏ¨µ«ɏ½¬¹»°ª¨³ɏ¨¿¬ºɏ·¨ºº°µ®ɏ»¯¹¶¼®¯ɏ»¯¬ɏ#'ɏ¶ɏ»¯¬ɏº¬ª»°¶µȍɏ y 10 cm 10 cm 10 cm C2 2 cm 2 cm A2 10 cm 10 cm 2 cm x C C C1 C2 10 cm C1 A1 (a) (c) ¥ " " #" " 9! 7 ¥ B 8 cm (b) 5OLUTION "7 8 & & & & ; A 2 cm 2 cm < * " 8 & : = " 9 ŷŹźɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº ! > % % ?@ 4 & & = " " % % - A1 y1 + A2 y2 A fi = 20 ¥ 5 + 20 ¥ 11 40 D CD4 $ D DC $ & & " C " = È I C,1 + A1 y12 ˘ + È I C,2 + A2 y22 ˘ Î ˚ Î ˚ È 2 ¥ 103 ˘ È10 ¥ 23 ˘ + 20 ¥ 32 ˙ + Í + 20 ¥ 32 ˙ Í Î 12 ˚ Î 12 ˚ . 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" #$ #%66' y 2 dm ( R 2 - x 2 ) ¥ rp ( R 2 - x 2 ) dx rp 9 = = : 2 2 2 " - . : 0 9 76 6 6 / R 3 rp 4 rp È 4 x5 4 2 2 2 x ˘ Ú dlx = Ú- R 2 ( R + x - 2 R x )dx = 2 ÍÎ R x + 5 - 2R 3 ˙˚- R R rp È Í 2 Î 5 5 + - 5 2 5 5 + 5 3 4 p r 3 <r + 5 ˘ ˙ 3 ˚ 5 4p R3 2 5 6 " % . #$ 6> " 2 6¥ 5 6¥ 5 8 qo 15 3M fi r 8rp R5 8p R5 3M = ¥ 15 15 4p R3 2 - 6 - . " 2 2 (2M h ) R 2 = 5 5 ! " #$ #%#?' 'XAMPLE!clrdj &¶¹ɏ¨ɏº¶³°«ɏ¹°®¯»ɏª°¹ª¼³¨¹ɏª¶µ¬ɏ¶ɏ©¨º¬ɏ¹¨«°¼ºɏ2Ȏɏ¯¬°®¯»ɏ¯ɏ¨µ«ɏ´¨ººɏ-Ȏɏ«¬¹°½¬ɏ¨µɏ ¬¿·¹¬ºº°¶µɏ¶¹ɏ°»ºɏ´¨ººɏ´¶´¬µ»ɏ¶ɏ°µ¬¹»°¨ɏ¾°»¯ɏ¹¬º·¬ª»ɏ»¶ɏș¨Țɏ¨ɏ«°¨´¬»¬¹ɏ¶ɏ°»ºɏ ©¨º¬Ȏɏș©Țɏª¬µ»¹¶°«¨³ɏ¨¿°ºɏ·¨¹¨³³¬³ɏ»¶ɏ°»ºɏ©¨º¬Ȏɏ¨µ«ɏșªȚɏ¨µɏ¨¿°ºɏ·¨¹¨³³¬³ɏ»¶ɏ°»ºɏ©¨º¬ɏ¨µ«ɏ·¨ºº°µ®ɏ»¯¹¶¼®¯ɏ°»ºɏ ½¬¹»¬¿ȍɏ / 5OLUTION - . " " / * . - ! ! / . / < r" @ + #$ 6 * * R h " #$ 6 / / - . / /, " / 7 / rp / " 2 dm x : 4 - . 2 ¥ 6 6 " "< 2 rp x dy ¥ x : rp 4 6 ¥ 6 - . 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( + , # " ) , ( - * -& ; ; # p ¥ 152 4 # ! $% A1 y1 + A2 y2 800 ¥ 20 - 176.71 ¥ 30 = 623.29 A fi S ( *,%( < ," bh3 20 ¥ 403 = 12 12 / -( D ( ; # J ) # - $ ; ;" #%! ," * C B ' !"!# $ 10 cm (b) 5OLUTION > A A ;A / I - ! ? - @! & x y (a) ! 30 cm ) / ) # ( - / p / - p ¥ (15/2) 4 4 / > - / /> / / / ?@ #+ A B E E , / ) > / , " ŷźżɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº C # " D ! $% ( ) / ( > ) * # # $% $% - & / ( / F > / G + ) @ # $ # + # , $% $% 'XAMPLE!clree !ɏº¸¼¨¹¬ɏ·¹°º´ɏ¶ɏª¹¶ººɏº¬ª»°¶µɏŶŴŴɏ´´ɏ¥ ŶŴŴɏ´´ɏ¨µ«ɏ¯¬°®¯»ɏŸŴŴɏ´´ɏº»¨µ«ºɏ ½¬¹»°ª¨³³Àɏ¨µ«ɏª¬µ»¹¨³³Àɏ¶½¬¹ɏ¨ɏªÀ³°µ«¬¹ɏ¶ɏŷŴŴɊ´´ɏ«°¨´¬»¬¹ɏ¨µ«ɏŹŴŴɊ´´ɏ¯¬°®¯»ȍɏ #¨³ª¼³¨»¬ɏ»¯¬ɏ´¨ººɏ´¶´¬µ»ɏ¶ɏ°µ¬¹»°¨ɏ¶ɏ»¯¬ɏª¶´·¶º°»¬ɏº¶³°«ɏ¨©¶¼»ɏ»¯¬ɏ½¬¹»°ª¨³ɏ¨¿°ºɏ¶ɏºÀ´´¬»¹Àɏ°ɏ»¯¬ɏ ´¨ººɏ«¬µº°»Àɏ¶ɏ»¯¬ɏ´¨»¬¹°¨³ɏ°ºɏŶŴŴŴɏ²®Ʉ´ŷȍɏ 5OLUTION! H D ' A * > > > J+ ¥p¥ rp * ¥ r * J+ # 2 m1 R 70.68 ¥ 0.15 = 2 2 * , + / / + # / > I! I! + , (@ #+ A ) + , (@ #+ A > I! D 2 ) 2 m2 (a + b ) 32 ¥ (0.2 + 0.2 ) = 12 12 J+ " , J J & 'XAMPLE!clref 2 ! ¥ > ¥ 2 # 2 K > I!A r I! >& #¨³ª¼³¨»¬ɏ»¯¬ɏ´¶´¬µ»ɏ¶ɏ°µ¬¹»°¨ɏ¶ɏ»¯¬ɏª¶´·¶º°»¬ɏ¨¹¬¨ɏº¯¶¾µɏ°µɏ&°®ȍɏŵŴȍŷŻ¨ɏ¨©¶¼»ɏ »¯¬ɏ»¾¶ɏª¬µ»¹¶°«¨³ɏ¨¿¬ºȎɏ¨µ«ɏ°»ºɏ·¶³¨¹ɏ´¶´¬µ»ɏ¶ɏ°µ¬¹»°¨ȍɏ ɏ ŷźŽ -¶´¬µ»ɏ¶ɏ)µ¬¹»°¨ɏ y y Y (x3, y3) C3 15 mm A3 A1 (x, y) C C1 (x1, y1) 40 mm 40 mm (0, 0) 30 mm 30 mm (a) Composite area L !+ 1 ¥ 2 ¥/ !+ p ¥ 152 2 ( ' - - - , D +J 40 3 @ / & - 4 ¥ 15 3p / , A1 y1 + A2 y2 + A3 y3 1200 ¥ 20 + 600 ¥ 13.33 + 353.42 ¥ 46.37 = A1 + A2 + A3 1200 + 600 + 353.42 C - - ( , / / ! - D +J / / ++ + , ( ! / / / ! > / D +J 40 2 P !)N + (M ! ¥/ - # - ! C x 30 mm (b) Component areas 5OLUTION $ D +J O - - C2 (x2, y2) A2 x 30 mm X ( , # Ê 30 ¥ 403 ˆ Ê 30 ¥ 403 ˆ + 1200 ¥ 2.47 2 ˜ +Á + 600 ¥ 9.142 ˜ ÁË ¯ rectangle Ë 36 ¯ triangle 12 &( ¥ >/ & > / ¥ ) + ? ! @ #+ B ŷŻŴɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº - ! C ) ' - A - , > & D +J A / - @ A > , A1 x1 + A2 x2 + A3 x3 1200 ¥ 15 + 600 ¥ 40 + 353.42 ¥ 15 = A1 + A2 + A3 1200 + 600 + 353.42 C - - )E , - ! ! - D +J > / > P ! ++ + , ) Ê 40 ¥ 303 ˆ Ê 40 ¥ 303 ˆ + 1200 ¥ 6.97 2 ˜ + Á + 600 ¥ 18.032 ˜ ÁË ¯ Ë 36 ¯ 12 Êp ˆ & Á ¥ 154 + 353.42 ¥ 6.97 2 ˜ Ë8 ¯ & - ) , # ! ( ! @ #+ ) ' * + !D @ , - ( & 'XAMPLE!clreg # / ) + , ( &/ + ) #J / $¬»¬¹´°µ¬ɏ»¯¬ɏ¨¹¬¨ɏ´¶´¬µ»ɏ¶ɏ°µ¬¹»°¨ɏ¶ɏ¨µɏ¬³³°·º¬ɏ¨©¶¼»ɏ°»ºɏ»¾¶ɏª¬µ»¹¶°«¨³ɏ¨¿¬ºȎɏ ¨µ«ɏ°»ºɏ·¶³¨¹ɏ´¶´¬µ»ɏ¶ɏ°µ¬¹»°¨ȍɏ y A A1 x C1 2b x C y y B x a2 x y 2a 5OLUTION! M ! , J + # ++ " ! + y2 b2 =1 y B C (a) Ellipse x2 dx A x (b) Quarter of ellipse ++ " Q - " + , *RS # , + ! J , , ++ + , " ++ # + ! K ! @ ++ *RS ɏ ŷŻŵ -¶´¬µ»ɏ¶ɏ)µ¬¹»°¨ɏ @ + $-% ++ D # ! ! x" + a" y" b" fi # # ◊ $ # ! 1- $ x x2 a2 2 a2 ◊ ! % " ¥ # # ! 1- &'( a % ! # fi ! q , # # 5 x2 ◊ a2 ¥ " #! " 1- x2 a2 ◊ ! # Ú dI y = Ú0 bx % # ! 1- 2 1- x2 a2 x # a ◊ dx )* q % #+ q q q# fi q # p." q# . # fi q# / )= #"> ' q ! 3 ! 3 4 * )*! # bÚ 0 p /2 (a cos q ) 2 1 - cos 2 q ◊ (- a sin q dq ) # - a 3b Ú 0 p /2 cos 2 q sin 2 q dq #- a 3b 0 sin 2 2q dq 4 Úp / 2 )∵ #- a 3b 0 Ê 1 - cos 4q ˆ Á ˜¯ dq 4 Úp /2 Ë 2 )∵ #- 0 p sin 4(p /2) ˘ p a 3b sin 4q ˘ a 3b È a 3b È q 0 = - + ˙˚ = 16 8 ÍÎ 4 ˙˚p / 2 8 ÍÎ 4 2 4 " # " # +" " * " * 0 \ &'( @ 3 &'( ! > ! #?¥ # > oa 3 b 4 # oab A # > B > # p ab3 p a 3b o ab(a 2 + b 2 ) + = 4 4 4 C% ! %! ."DE ŷŻŶɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº 'XAMPLE!clreh $¬¹°½¬ɏ¨µɏ¬¿·¹¬ºº°¶µɏ¶¹ɏ»¯¬ɏ´¶´¬µ»ɏ¶ɏ°µ¬¹»°¨ɏ¶ɏ»¯¬ɏª°¹ª¼³¨¹ɏº¬ª»¶¹ɏ¶ɏ¹¨«°¼ºɏ2ɏ º¯¶¾µɏ°µɏ&°®ȍɏŵŴȍŷŽ¨ɏ¨©¶¼»ɏș¨ȚɏÀɊ¨¿°ºȎɏș©Țɏ¿Ɋ¨¿°ºȎɏ¨µ«ɏșªȚɏª¬µ»¹¶°«¨³ɏ¨¿¬ºɏ·¨¹¨³³¬³ɏ»¶ɏ ¿Ɋɏ¨µ«ɏÀɊ¨¿¬ºȍɏ'°½¬µɏ»¯¨»ɏ¿Ɋ¨¿°ºɏ°ºɏ»¯¬ɏ¨¿°ºɏ¶ɏºÀ´´¬»¹Àɏ¶ɏ»¯¬ɏº¬ª»¶¹ȍɏ Y y y x R r a a O x O a ds dr q C (x, y) r dq y x, X R (b) Element of size ds ¥ dr (a) Circular sector 5OLUTION , ¥ F ' ' q! $ ) * # & ! ¥ ! G! , H # ! q! ! !3 ' q -I # q ! ¥ # " #) q q*" # *¥) & ! # " q q ! !3 q # Ja Ba % +a 3 a R cos 2 q dq dr = Ú r 3 ÈÍ2 Ú cos 2 q dq ˘˙ dr 0 Î 0 ˚ )∵ 3 ! R È a Ê 1 + cos 2q ˆ ˘ " q d dr )∵ "q # " q + * # Ú r 3 Í2Ú Á ˜¯ ˙ 0 2 Î 0Ë ˚ R a R 3È R 3Ê sin 2q ˘ sin 2a ˆ sin 2a ˆ È r 4 ˘ Ê = a = a # Ú r Íq + dr r + dr + ˜ ÁË ˜Í ˙ Ú0 ÁË 0 2 ˙˚ 0 2 ¯ 2 ¯ Î 4 ˚0 Î # Ú dI y = Ú R 0 Ú- a r * 4 fi ! - # sin 2` ˆ Ê ÁË ` + ˜ 4 2 ¯ ! q ¥ q ¥ q q q q " # R +a 3 Ú dI x = Ú0 Ú- a r a R cos 2 q dq dr = Ú r 3 ÈÍ2 Ú cos 2 q dq ˘˙ dr 0 Î 0 ˚ a !a ɏ ŷŻŷ -¶´¬µ»ɏ¶ɏ)µ¬¹»°¨ɏ È a Ê 1 - cos 2q ˆ ˘ ∵ q ˜¯ dq ˙ dr Í2 Ú0 ÁË 2 Î ˚ R a R 3È R 3Ê sin 2q ˘ sin 2a ˆ sin 2a ˆ È r 4 ˘ Ê Ú0 r ÍÎq - 2 ˙˚0 dr = Ú0 r ÁË a - 2 ˜¯ dr = ÁË a - 2 ˜¯ ÍÎ 4 ˙˚0 R 3 Ú0 r $% q 4 sin 2` ˆ Ê Á` ˜ 4 Ë 2 ¯ fi - ! ( & ' # 4 sin 2` ˆ Ê Á` ˜ 4 Ë 2 ¯ ( ) # ) fi ! 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" 1 " " ( ( ½! ! " v2 ds = Ú v dt t2 " # % ! ! ( " " " " " " " " ½ ! ) " / '0 % * $+ #$ % . " " ŸŷŶɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº 5OLUTION B " C ½ # $ 7 ( # 54 000 # $ 3600 ( D½ #, 7 ( # ' 72 000 # 3600 (D # " $ ( ( Velocity, v (m/s) " ½ ) $! " " E / & B 20 15 A 10 5 D C 0 Time, t (s) t = 20 s " ) " ) " ½ " ) #/ 6 v1 + v2 ¥ 2 'XAMPLE!cdri / ! % $ /&F4 # 15 + 20 ¥ 2 ! ,$ % ,$ / 7 8 5OLUTION Velocity, v (m/s) !" # $ ! 12 10 C 0 A " ½ % & D s2 5 s1 s F 3 E t2 t1 t3 B Time, t (s) " ' "! " ( ! !" F AC + BD ¥ CD 2 7 21 ! ,' 4 ) ( " # ) ( % $ / ɏ Ÿŷŷ -¶»°¶µɏ°µɏ¨ɏ3»¹¨°®¯»ɏ,°µ¬ɏ ' " ( " !" * * " ,! " # D,-. * ,! " # ( 12 ¥ 12 ¥ 1 ) * 4 ! ( ! ɏ ( ! ɏ 9 ) \ ( " ½ ( ½ " ' " $ !" ( * fi !" ( * fi # " ( ¥ * .1!cdrg 23/6! ( 12 ¥ 12 ¥ 3 ) * * 46 ( ! fi * ( ! fi * 47 6 .: 4 6 % * ! ¥ 6* -/&. * ,! " # D&/3 * * 8 ¥8 fi ; * ; *8 ; ) �! NPUJPO!VOEFS!HSBWJUZ < ") ( ! % %"!( ( ( ! " " " " !" ! ( = ! " # !" > "! ( % % " ( " ! " " " " !" ? ! ( ' ! ) ( ( ! " " " !" " ( bddfmfsbujpo! evf!up!hsbwjuz ( () ) , " # "! #! " " "! % " " " " �! 9ORTHY!OF!0OTE ) 9 " !" ( !" ! " " " ( " %" " ( % %"!( 4 6 : " 9 % "! ! ( # 4.: @ E6 "! " ") ( ! " ! H " ! !" > ( " ! () ) ( # " ! ) > " "! " %" ! " % " ! " ) # ! " # # " " % " Tjho! Dpowfoujpo < !") ( (! ( ! ! % h* %"!( " (" # % " " (! " ( (! 23/6/2! Npujpo!pg!b!Gsffmz!Gbmmjoh!Cpez! ( ! F ! "# ! ! ɏ ɏ ɏ ɏ ɏ ɏ ?F ! # ! " ( ( #! ½ ½ ½ ½ ½@ ½ ½ " * 47 * 47 * 47 * 47 * 47 * 47 * 47 # #") ( " % 7 7 " # 6 6¥ (7 6¥ (7 6¥ (7 6 ¥ @ ( 7@ 6¥ (7 6¥ (7 ( % "! ! #" #! < " " % G G G G G G * * * * * * * @ ! #" " " 47 6 ¥ 47 ¥ 47 ¥ 47 ¥ 47 ¥ 47 ¥ 47 ( "! % ! & ! " (( 6¥ 6¥ 6¥ 6¥@ 6¥ 6¥ (7 (7 ( 7@ ( 78 (7 (7 8 " # " ) ( ŸŷŸɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº ' " % " (( " " "! ( % %"!( #! "! I & ! E % # ) ( 4 I , 3 - / . " ( &6 " ( (" ( ( " !" (" ( # " ( , ) ( #" ( ! " ) ! ( F ( " !" ( " ( ! " % Myth Fact! Jo!b!gsff!gbmm-!uif! jojujbm!wfmpdjuz!pg! b!cpez!jt!bmxbzt! {fsp/ > " #! #" #") ( ! = ! &! #" " ! " " !" ) ( : " " ) " # s1 = 5 m t = 0s O t = 1s A 0 m/s 10 m/s S1st = 5 m S2nd = 15 m s2 = 20 m t = 2s B 20 m/s s3 = 45 m S3rd = 25 m t = 3s C 30 m/s s4 = 80 m S4th = 35 m t = 4s D 40 m/s s5 = 125 m S5th = 45 m t = 5s E 50 m/s s6 = 180 m S6th = 55 m t = 6s F 60 m/s + # Myth �! 9ORTHY!OF!0OTE J & E " ! ) " ) " ( % ( " " " !" ( .H" 8 23/6/3! Xifofwfs!b!cpez! jt!espqqfe!)tbz-! gspn!b!sjtjoh! cbmmppo*-!jut!jojujbm! wfmpdjuz!jt!bmxbzt! {fsp/ Fact! K < #! " ) !") ( ) ( # " !" "!" ! " ) ( Sbujp!pg!Ejtubodft!Dpwfsfe!jo!Fbdi!Tfdpoe L ( !" (6 ! " % ( " ! () " # ! " !" ( ) ( : " ! " 4" ɏ & ! #! ' #" ) ( % " 7 7 ! ! ( 7 7 7 @ 7 ! !( !@ 7 8 7@ @ 7 @ 78 ! 7 8 7 ! @ !" ! 23/6/4! 9 % ( ! !( !@ ! M 23/6/5! & " @ ! $ , ( ! " %"!( ( ( # E & M , , ,+,), - Myth B!Cpez!Uispxo!Wfsujdbmmz!Vqxbset " " ) ( # " ! ( ! " ) = ! " > "! ( % %"!( " ( ! " % > ( % ! $ % " ( # ' " ( #") ( ! $ % " (( " " " " > %"!( # ) ( "! ) %"!( G > ) " (( " F % ! !" %"!( ! ( % %"!( " ( % %"!( G ! " ! J % ŸŷŹ -¶»°¶µɏ°µɏ¨ɏ3»¹¨°®¯»ɏ,°µ¬ɏ ( %"!( %"!( % Fact! Uif!ejtubodft!usbwfmmfe! jo!frvbm!joufswbmt! pg!ujnf!cz!b!cpez! npwjoh!xjui!b!dpotubou! bddfmfsbujpo!bsf!bmxbzt! jo!uif!sbujp!pg! , , , - " (( " !" ( % %"!( "! ( % %"!( G ! " = " % ! !" " %" " ' ! �! 9ORTHY!OF!0OTE < %"!( " " ) ( ! % " ) = ! !" ( % %"!( Hsbqijdbm!Sfqsftfoubujpo!pg!Npujpo!voefs!Hsbwjuz ½ " ( E # 4" !" # 6 & ! #") ( ! % % " " %"!( #! ! (" ! ( # & & E , ) ( ( " ! " " #" ! " ( ) : % ! ) ( ! " I "# ! ( ' ( " "! ( ! " ) ( ! ( "# ! " ! ( ' !" # ! # ) ( % & 8 > " * , ) ( ½ ( ! " " " " !" # ! ( " (L " ! ( = ! ! " I "# ! ( ' ) ( "! $ ! #! ! ' " ( # ! " " " !" ! " ! (% "# ! " ! ( ' ½ !" # ! # ) ( % & 8 ' ! %"!( " ( ( % %"!( # ) ( " !" ! " % " " ( ' !" # ! # ) ( % & 8 Ÿŷźɏ Displacement (m) %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº 200 180 150 100 50 0 1 3 2 5 4 6 7 8 10 11 12 Time (s) 9 10 11 12 Time (s) 9 10 11 12 Time (s) 9 (a) Displacement-time graph Velocity (m/s) 60 0 1 3 2 5 4 6 7 8 –60 Acceleration (m/s2) (b) Velocity-time graph 0 1 3 2 4 5 6 7 8 –10 (c) Acceleration-time graph ) 'XAMPLE!cdrj ! 8 8 8 '' (©) & & ® 9 ,+ % $ 8 (¨) 5OLUTION # * ac (2n - 1) fi ! 2 $ %& %' + %' * - 10 (2 - 1) 2 fi " * ut - 12 gt 2 = 0 - 0.5 ¥ 10 ¥ 62 = | - 180| Bmufsobujwf!Tpmvujpo, , * , - + / 4 " 3 ɏ ŸŷŻ -¶»°¶µɏ°µɏ¨ɏ3»¹¨°®¯»ɏ,°µ¬ɏ ! $'5 : 'XAMPLE!cdrk / % $ $+ . (¨) % (©) 8 5OLUTION 7 * , ;4+ ® 9 ,+ (ª) 9 : ; 4 $ 4 4 < * P + h1 > % : 4 ? @ * ɏ u = 20 m/s ; & * ; + ½ ɏ fi 202 20 * fi + C ¥ ! D; ht ¥ 4 < @ & + 4 %& ; / / & % fi ½ ½ 3 O A Q < # 4 *4 ' ɏ <@ B & * + + * ¥ ! ; 4 ; fi + + *4 ' 4 @ < ; ¥ 4 ) D; ± 400 ½ F + + + ½ . + 4 * + ; C ¥ ! ! ! % * & * 4 G # D; * fi ! 1 2 3 ! D; # /(!01, H 2 3!0, # ;4+ + ; D; / % * + * 4 * $ * Bmufsobujwf!Tpmvujpo , 4 4 < J # + 3 ! 3 D; % + D; J< + + & * J < I & ; + + ; 4 ; ! + ; &% + ; , * ? ; 4 * G & J ,; * 4 K * 4 J & J ! + *4 + ;4+ * < ; 4 J * * # K ½ 4 . + K Ÿŷżɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº 'XAMPLE!cdrcl ; < 0+5 = ! 8 $+ % ! ® 9 ,+ % $ 6 / L 5OLUTION 4 D; A , : ; : + * M 1 2 fi ! L ! # N 1 2 fi OD + ¥ ¥ Bmufsobujwf!Tpmvujpo 3 N + ;4+ N % % 4 J , 3 M 44 + B 4 < Q + * + + * 23/7! ;4+ 1 2 C fi ! L ! fi ! ! ! - P ? & # ; + ¥ & * # 4 4 * % * % 3 L + N �! NPUJPO!VOEFS!WBSJBCMF!BDDFMFSBUJPO ! * + 1 2 ¥ ! ! 4 fi L + .1!cdrh ; * # OD $ + 4 L $ - + D; &$ 44 % & & / OD 4 % ** + & Q ; ; Q + ; R % ; & & ; OD & # + + % 'XAMPLE!cdrcc ½ 9 + '>($» ? 2» $) % »92 @ »9+ º9+ 5OLUTION OD : 3 + + $ \ * ? % & * ½ ½ 3 Q 4 *; ds + dt ds dt %& 4; $ fi R 4 4 R Ú ds Ú 0.5(2t + 3t 2 ) dt % + * fi $ R R » $ % 4 * ɏ ŸŷŽ -¶»°¶µɏ°µɏ¨ɏ3»¹¨°®¯»ɏ,°µ¬ɏ H R + % dv dt H RR R d Ê ds ˆ d d 2 2 ÁË ˜¯ = {0.5(2t + 3t )} = {t + 1.5t } dt dt dt dt R R¥R R 'XAMPLE!cdrcd R ¨ 9 »2 A 2» $ ? ' / $ % ¨ / » »9, »9$ 4 2+ R 5OLUTION !R dv dt 3 ½ > Ú fi ½ " 1 $' % "9 $$% " : SR Ú ¥ $ ½ $ !" # 4 1 4 !" \ ¥ $ fi $ ½ $ % $ 4 ɏ ½ & 2 4 ¥ '*" Ï 4 Ú ½ dt Ú ÌÓ 4 +"+ ," \ +"+ +"+ - +" +" +" 5 20 4 - 4 2 +5¥ $ 2 $ ," + fi $ " $ $ +"+ ¥ .1!cdri ¸ + 5 + 2˝ ˛ " $ " $ 3 +" ¥ " ¥ ¥ 23/8! �! BCTPMVUF!BOE!SFMBUJWF!WFMPDJUJFT! - / /6 : " < > $ ?+ : # bctpmvuf!wfmpdjujft 78 $ / ; +: # $% - " " $$ $ " + " ŸŸŴɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº > $ $ %" % / $ 4 /" + $$ > < sfmbujwf!wfmpdjuz < / $ ?+ : # / > /% > <" 4 @ 4 / ", 4/ $ " - " " + $ $ $ / % " " / $ "/ ?+ $ + +: # > $ / : " " - " " + +: # > > < " "/ "/ + ?+ Va = 70 km/h Vba = Vb – Va + - " " " / $ "/ " $ "/ % "/ " (c) Determining Vba (b) Determining Vab (a) Vb –Va –Vb Vab = Va – Vb Train B : - " " $ Va Vb = 50 km/h $ $% , " " ?+ $ + < % $ Va = 70 km/h < > +: # > + > < ?+ +: # Va - " B $ Vb –Vb " –Va Train A Vba = Vb – Va Vab = Va – Vb Vb = 50 km/h Train B (c) Determining Vba (b) Determining Vab (a) + , Usbjot!Svoojoh!po!Tusbjhiu!Usbdlt!Jodmjofe!bu!bo!Bohmf ; - " $ <" $ Train A A 8 $$ $ $ / > < ! A @ % < " < " "> > : $$ $ / $ q $ $ $ / $ / $ ɏ "C - ŸŸŵ -¶»°¶µɏ°µɏ¨ɏ3»¹¨°®¯»ɏ,°µ¬ɏ " $$% " " $ $ % /% D " Va –Vb Va = 70 km/h Train A Vb q Va Vba = Vb – Va Vb = 50 km/h q Vb q Vab = Va – Vb –Va Train B (a) (b) (c) ) @ Bmufsobujwfmz "! $$ $ 8 $ / $ " , '8 $ ! $++5 'XAMPLE!cdrce '0 8 $ " E" #$ % ,'+5 B % 6 + 5OLUTION F $ I $ G" % ++ "/ : # 72 000 + # H "/ " ? : # 3600 / /% % %/ ++ + + $ % " / " "/ + # : $ $% : 350 d ?+ 5 Vab ? : # ! 'XAMPLE!cdrcf +$ % . ,'+ %/ + " 54 000 : # 3600 $ $% : # $'+ ! C $ . $$ % & ,+ + 5OLUTION! F C $$ $% G" + /$ % DC $ # /% +" %/ + $$ $ $ >" + " # % + $ $ ++ $% %/ + $ % "/ + " 8 > < < ŸŸŶɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº < $ $ %" $ " $ /+ +" # + * ++ > $ $ " ¥ + +" ¥ +" ¥ + fi " $ % < " + "/ fi "/ " + " /+ > " /+ ! /+ E /+ E # > + " #&&+6+10#.!51.8'&!':#/2.'5 'XAMPLE!cdrcg ! *( ) #' % ; 2# * . 8 ® 9 E 4, D " Ɋ $ % 5OLUTION ! + . )- % )/ * " #$ %& % ' () )* %,) , %& )- % . " % % ) % ( * *% ( . (' 8 9 ' ( :'% - ' *% ( Elevator v=0 B / ) 1 7.5 m/s - h1 A Shaft ; v fi fi < ¥ < $ ; ) + ( v- ( :'% $¥ < ) - =' , - vv- < 'XAMPLE!cdrch $ h2 ' ( % , - / < ¥ < $ ; ! ! ! 5OLUTION! $¥ + )+ # C ! < $ $5 3 3 05% ,' 8 ,+4 6 # $ $ F F h0 = 37m %) ')% ( ,1 ' fi v; ¥ < $¥ < + )+ $5 3 3 05% )( * *% ( ( :'% . v; fi v- < % *% Bmufsobujwfmz !* % ' v; fi fi $ . & 108000 3600 % B ,+ % $ ɏ ŸŸŷ -¶»°¶µɏ°µɏ¨ɏ3»¹¨°®¯»ɏ,°µ¬ɏ @ , %& ,1 ) ; ( % B % % )) ( ,1 ) ; $; % ) fi ; fi % %) ')% C < %& ( ' B % ; $ $ = % 1 2 1 2 $; ¥ ¥ ; -A % % %( % % )) ( ,1 - 9 - - $ ; * 'XAMPLE!cdrci ! ; ¼ & ¨ ,' * $' - ¼ ¨ ɏ ! 5OLUTION C D: ! $ * % ! % ( E', % D: ; ! ; < " D: ¥ c 2 $ * $< 10 c 2 9 c 2 19 c ; 2 ¥ < $ c 2 ; < $ $ $ fi $ * ; 9¥2 2 'XAMPLE!cdrcj fi ! $+ 0+ . 2 G 5OLUTION @ " F H G ( :'% , %) ( :'% ; 12 ) ( # % - * ; E ) %, ( ( 1% ( fi ) % ¥ % fi I 1 2 * :'% %) ( % , % * % ¥ ; - ¥ % ¥ ¥ ; %) - ) F G fi ( * ( - fi # ( ( ; % ¥ * $ %) ( % ; # -I I ( % ' " $ ( %) ( % ', :' ( ŸŸŸɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº ' < C ( :'% ¥ J - ; ¥ < ( % * fi ; v 'XAMPLE!cdrck #$ ¥ (, ¥ < $ ; K<I <I % fi #$ + ) )+ + ! 5 $+ % 2+ % . ½ 5OLUTION @ , ' & % ) % C ɏ ½ + % ɏ ½ & & ( :'% % @ - , 8 * fi ; % ¥ ; (&) ½ @ % ( , (&) ' G :'% ɏ ½ % ( ½ ) fi ; * ½ (' % %& % ( % , * * ½ ; ½ ; # fi ,1 1 - ¥ ¥ ¥ ; G fi ½ + fi fi , * G # I ,+ ,'+ & 0+ ½ 5OLUTION # 1 I % G ) F * % % % ) % ( % ' * ' )1 ( ) % % # + ɏ % / / ; " ; ½ ½ ; ; -" % )) $ # % ( (" F ( % 8 fi 1 2 ; fi fi . . " = ' 1 % "F ( % % H ¥ % ; 1++ ¥ Speed, v (m/s) 8 * % ( )1 ! 'XAMPLE!cdrdl . % &) ,1 % % )) ( % (% ( (&) ; % % & 1 ¥ ; ) < $¥# fi $ fi D s2 0 A ¥ s1 E t1 fi ¥ ; ; < C V I F t2 s3 t3 B Time, t (s) ɏ ŸŸŹ -¶»°¶µɏ°µɏ¨ɏ3»¹¨°®¯»ɏ,°µ¬ɏ %) ( % % )) ( ; ; %) ( % I :'%) %) ; 'XAMPLE!cdrdc ; ; ! # ' 8 & », »$ ¯ 8 ¯ 9 21 >®», » $ ! 5OLUTION! . ' ' *% ( ( ; E', % 1 2 < $ < $; ) fi ;# % % - !B"D " 410 410 I " 30 ; I ; I fi " # \ % 1 - = \ ; 'XAMPLE!cdrdd , % ( 1 2 < $ ; < + ) % 1 2 * 1 2 < $ 1 2 % < $ 1 h $u 2 $ ( % ; $ ; 1 2 ) 6u % % 1 hu 2 < $ u ! », & »$ / »1»2 5OLUTION! . ' * % ( , < < L') :'% )1 - G %) ; ; $ F! %& < % * 1 2 * % ) 1 , < $ % ( < ,1 % ( 1 2 - 'XAMPLE!cdrde fi fi ,1 * * % ( %(( @ * * , - % * 1 2 u1 u 2 & º $ (¨) » * (©) (ª) 5OLUTION % % 1 2 ! º 9 ,1» ? 0» $ J 2» 2 K % ( ' *% ( ( 1 ; 2 < $ ( :'% ; $ - -%)) ¥ ; 12 < $ H % K v 0 ! "## $% ŸŸźɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº & ' ( ½ v $ % , ds d $ dt dt d½ d $ dt dt ) ! " # #% ) * " + !% ) ¥! 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" ! fi 8 " 7 8 , $% $ % ¥ 8 "!¥ 8# " ! ¥ 8! ) # " + # ! ! ¥ 8 " ¥ 8 "! ¥ 8 " )# * * "* " 8* " $)#% + $% $ % $ % 8 ɏ ŸŸŻ -¶»°¶µɏ°µɏ¨ɏ3»¹¨°®¯»ɏ,°µ¬ɏ 'XAMPLE!cdrdg ! 1+ ,' +' / % & = > v1 - v2 60 - 15 = Rate 1 = > < !7 ! / + / \ # = > > > ! > + ½! ! 8¥+ ½! ½ v1 - v2 60 - 15 = + Rate 0.5 ! < 7= > 60 000 8 > 3600 15 000 8 > 7= > 3600 ½1 + ½2 16.67 + 4.17 ½ = 2 2 ½ !¥ 7 *+ / / - = > ½! ! % 7 B ½ / E # = = #87# 7= > > ! > ½# !¥+ +#8 * ) !) # * + ) #87# ) +#8 * ) !) # 7)+ )+ #7 E = > $ 7 7+ 8 8 >% = 7 7+ 8 v1 \ / 'XAMPLE!cdrdh $+ = 5159.7 16.17 " ! » (©) # + #7 " # + 8 ¨ ¨ 9 $ A 2» ,+ % / 4' % % ' = ɏ , + $' 1+ ; < & % % 5OLUTION # % B & (¨) * * * K ŸŸżɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº 5OLUTION G < v7 ! > d½ dt fi : \ 5 ds dt v !"# fi ½ t Úu d ½ = Ú0 adt fi v Ê 3 2ˆ ) Á2 ˜ Ë 2 ¯0 v : *7 fi t Ú0 (2 - 3t ) dt v" Ê 3 2ˆ ) Á2 ˜ Ë 2 ¯ t tÏ Ê 3t 2 ˆ ¸ Ú0 ÌÓu + ÁË 2t - 2 ˜¯ ˝˛ dt Ú fi v $% fi " Ï t3 ¸ Ìut + t 2 - ˝ 2 ˛0 Ó 3 \ / ) ) ! " $ % 2 $ % , : v7 ! -. $ % Ê 3 ) Á2 ˜ Ë 2 ¯ 2ˆ v : !"#¥ 7 !"# > Ê 3 ¥ 52 ˆ ) Á2 ¥ 5 ˜ Ë 2 ¯ ! *7 fi + -. $ % 3 ) ) ! " ) 87¥ *7 2 $ % ! ) 6 103 fi 2 -. $ % " 2 ) 2 - v 3 2 87)! " 7 ! fi ## "7 : ## -. $ % 3 ) 'XAMPLE!cdrdi ) ! " ## ) $ ##%! " ) 87¥ 2 º 9 $» 2 A E»$ ? ,$» A ,+ / (6.33)3 2 º . » - K ) (¨) (©) K 5OLUTION! / 5 v , $ % 9 !#"+!) ! " ds d $! # " + ! ) ! " dt dt dv d $ ! " * ) !% dt dt -. $ % $% ! % ! " * 6 ! " * ) ! fi ! " * ) ! $ % $ % ɏ ŸŸŽ -¶»°¶µɏ°µɏ¨ɏ3»¹¨°®¯»ɏ,°µ¬ɏ 9 9 $ % 9 ! !¥!" * !¥ " * ! " * ! " * ! " -. $ % 6 ! " * 9 fi 7 7 !#"+!) ! " !¥ 7# " + ¥ ! #"+ !) ! " !¥ "+¥ 7! ) ! ¥ 7" " J E " 'XAMPLE!cdrdj ) !¥ " " 7 "7 7 " $" % ! E+ % 5 ! 21 5 5 3 <" 5OLUTION G % " + = > $ 4 $ 4 " 4% 9 6 # = > $ % q % L + 7K 4 ! 7 North –Vt West Vt a Vct f East 45° Vc S-E South " : 4 4 4 "4 f " 4 ) $"4 % * K"q L4 * K " 7K 902 + 362 + 2 ¥ 90 ¥ 36 ¥ (cos 135∞) = a " Ê "c sin f ˆ ÁË " + " cos f ˜¯ t c L4 #7K "t2 + "c2 + 2"t"c cosf a ' 4 L4 $ " Ê 36 ¥ sin 135∞ ˆ ÁË 90 + 36 ¥ cos 135∞ ˜¯ % : ) ŸŹŴɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº ' Bmufsobujwfmz ( ( 4 4 \ a ( + – K 4 "4 4 # –" 7K # –" 7K " + – K ) " * K " 7* 7K : 'XAMPLE!cdrdk = ( ! ) –9 : 3 #$ ,4 % % ( % ) + 5OLUTION G <" " " *= > 72 000 ! 3600 9 8! = > 8! = > + K q >M " *= > ! L4 $NB% 18 000 3600 4 $N,% 7 4 $ON% 4 L4 ' 4 > DONB B Vst a –Vt C Vs 90° Vt O A 1 $ " OC2 + BO 2 a " Bmufsobujwfmz \ ! OB% "t2 + "s2 = 202 + 52 Ê BO ˆ ÁË ˜ OC ¯ ' ( " Ê 5ˆ ÁË ˜¯ 20 : ( ( 4 4 ! – K 4 "4 4 7–+ K 7–+ K " ! – K a * K" K = ( –9 : : 57//#4; .1!cdrc Defines dynamics, and distinguish between kinematics and kinetics. Ljofnbujdt ljofujdt = 1 (0 ( 5!0 ; QS 233 ' !' P: 5 4 , 3 * 3567 ɏ ŸŹŵ -¶»°¶µɏ°µɏ¨ɏ3»¹¨°®¯»ɏ,°µ¬ɏ .1!cdrd Define position, displacement, velocity and acceleration for a body in motion. usbkfdupsz 9 tdbmbst $ .1!cdrf . T bwfsbhf!tqffe % T bmhfcsbjd! Represent graphically the motion of a body, by drawing s-t, v-t, and a-t graphs. $v % .1!cdrg List the equations of motion of a body under gravity. 9 $ % . .1!cdri <#<7<8<+< Differentiate between absolute and relative velocities of a body. , O O , ! +/2146#06!(14/7.#' � � # = > � � � 8 lim v Dt Æ0 Ds Dt > lim D Æ0 v v ds dt D½ D dv dt 9 v = u + act (1) 1 2 act 2 (3) v 2 = u 2 + 2ac ( s - s0 ) a (4) Snth = u + c (2n - 1) 2 s = s0 + ut + (2) i Valid only for constant acceleration � Displacement, s Integrate Differentiate Velocity, v Differentiate Integrate Acceleration, a � 4 4 4 + ŸŹŶɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº ! 5*146 #059'4!37'56+105 < 5!0 3 A & & A ' 5 ' ! 5 > '! ! @ B 3 0! 05 @ D C! = 0 5 =0 5 5 ==!0! !? 51 ! >>0 ? !5 ! = > 0 ' !@ '! 0 C! !5@ > ! = 0! 05 ! E " A t.u 0 > F 3 # H( ! 0! $ A ! 51 0! !> 3 ' ! C! ' 1 = ! 51 G!0 @ # !5 =0 5 ! 0! ( 5!0 C! C 0 % ! ''! !0 ! b.u '(0C! 5 ½.u '(0C! 5(0 !0C = !@ @ % -';!6'4/5 .1!cdrc .1!cdrd .1!cdrg & .1!cdri ' " ! ' ! 4'8+'9!37'56+105 * , & , * , * - *. *, * + & + / 6 / 6 / 6 ɏ ŸŹŷ -¶»°¶µɏ°µɏ¨ɏ3»¹¨°®¯»ɏ,°µ¬ɏ 78 9 / 6 / 6 ' / 6v ) & ! 1$,'%6+8'!37'56+105 #!/ULTIPLE %HOICE!3UESTIONS $ " ' : ;< 9 > / 6 @ ' / E< > @6 / 6 D; ' ? " / 6 @< ' / 6 D ' : > @6 / 6 E< > / 6 H D @ E + ' D " / 6 h @g $ ' : ?< 9 > / 6 A : $ : 9 ' / 6 ; " / 6 D @; / 6 F< ! / 6 G@ / 6 EA; @< > / 6 H< " " / 6 ; / 6 E@< / 6 A<< % 9 / 6 H I @< / 6 E< " / 6 E; E? - > / 6 @< h Hg > / 6 @; ! - " / 6 G; ; @; E / 6 Hhg / 6 @G F A E / 6 " @hg " / 6 v( ) ' / 6 @ E< > % / 6 G ; A E . / 6 / > / 6 v( 0 < (A / 6 A > / 6 v( 0 @ / 6 v( 0 v 1 2 @ E@ EG " / 6 D > / 6 ? > ŸŹŸɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº 2 Velocity, v (m/s) Acceleration, a (m/s2) 1 +3 4 0 1 2 Time (s) 3 –3 2 6 4 8 10 Time (s) –1 –2 + " E@ EH " E< / 6 D / 6 ? / 6 H / 6 E< $!(ILL $LANKS!3UESTIONS # , " JJJJJJJJJJ ' JJJJJJJJ vojgpsn!npujpo ' JJJJJJJJJ " % JJJJJJJ ' JJJJJJJ A< - E E< JJJJJJJ %!/ATCH %OLUMNS!3UESTIONS 9 & ( E /6 / 6 9 / 6 ! & ( /K6 /L6 / 6 / 6 9 /.6 & ( /6 / 6 / 6 / 6 E & ( /K6 /L6 / 6 /.6 EE ! EE 9 @ ɏ ŸŹŹ -¶»°¶µɏ°µɏ¨ɏ3»¹¨°®¯»ɏ,°µ¬ɏ &!5TATEMENT %ONFIRMATION!3UESTIONS $ /6 M / 6 M /6 / 6 / 6 / 6 ' /6 / 6 /6 / 6 @< 9 > ?< ? ?< > / 6 / 6 #NSWERS!TO!1BJECTIVE!3UESTIONS $2% I( > ! & '! /(! + $J% J ) /(! @; $&% I ' & ( /(! / 6 /L6 / 6 /.6 / 6 /K6 / 6 / 6 $K% " ! ! .& L0 / 6 / 6 / 6 / 6 /.6 / 6 /K6 / 6 /L6 /(! /6 ! 24#%6+%'!241$.'/5 ' E; A > ' 8 > N2 ' 9 A;< ' & D > N2 A? 9 > G< G; P G@ 9 > 9 * > @P @ @<< , ; N2 < G; > ' @ EA AA EA AG; P E< - ' N2 A< ;< A ;P ŸŹźɏ %µ®°µ¬¬¹°µ®ɏ-¬ª¯¨µ°ªº ' : ' ?< 9 > G< 9 > @< 9 > @ @;9 ' N2 * * +A " % '* A @R / 6 E< 9 P Speed (km/h) 60 40 20 0 0.25 0.50 0.75 1.0 2.0 Time (h) ) " " D< D > @ N2 ) ' EF ? ? / FH - > ' @< @<< E > @6 N2 ' ; ;H H P A 9 N2 ' @< D P I : / E< > @6 N2 E; > P A< ' */ ! A< > ' E< > @6 N2 ' E< @ D; - E 9 S N2 ' E< > / FH < GE P */ N2 ' ' @< T 9 F HE P > @ > @6 D<< ' "9 @; > E< F< > / ; E< > @6 6P 9 8 N2 DD ED; P ɏ ŸŹŻ -¶»°¶µɏ°µɏ¨ɏ3»¹¨°®¯»ɏ,°µ¬ɏ * 8 ' 8 /*6 @ ; U?@ * /6 * N2 + " 9 ' : $ E<< 9

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