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Physics 218 Honors Final Exam; Secs. 201,202,203; Fal l-li Name: Section. No: NOTE: Points are noted on each problem 1 .( 12 pts) A certain quaternary star system consists of three stars each of mass m moving in the same circular orbit of radius R about a central star of mass M. The stars orbit in the same sense and are positioned one-third of a revolution apart from one another. a) Compute the net centripetal force on one of the stars of mass m. (HINT: do not forget the gravitational force exerted on each mass m due to the othe two r identical masses in addition to the large mass M) , b) Compute the period of the three stars in terms of G, M, m, and R j — .— .) lz_ C€3c2z (ZL b Fr YY’R” ; _....2_ / —1 ..—2 - vi) ‘a-3 0 •1- z ‘1 1’ hs’ Zb 3K 2.(10 pts) How many cubic meters of helium, are required to lift a balloon with a 400 kg payload to a height of 8,000 m? (Take PHC 0.180 kg/m .) Assume the balloon maintains a constant volume and the density of air 3 decreases with the altitude according to the expression Pair = Po , 000 where z is in meters and ’ 8 e = 1.25 3 kg/rn is the density of Po air at sea level ;::M s1- L’f. . 44 1k), + Vg I’ ‘8 C p - ) •h 3 J2s;F-r/ ; _ J 0 9t 3.(lO pts) A flexible cable of weight W hangs between the two pegs located at the same height. At each peg, the tangent to the cable makes an angle 0 with the horizonta l. a) Find the magnitude of the force that each peg exerts on the cable. Draw a free body diagram for the cable b) Find the tension in the chain at its midpoint. Draw a free body diagram for the half-cable Express your answers in terms of W and 0. 7- a) to 4.(12 pts) A spool of thread consists of a cylinder of radiu sR 1 with end caps of radius R 2 as shown below. The mass of the spool, including the thread, is m,and its moment of inertia about an axis through its center is I. The spool is placed on a rough horiz ontal surface so that it rolls without slipping when a force T acting to the right is applied to the free end of the thread. a) Find the direction of the friction force. Explain. b) Find the magnitude of the friction force exerted by the surface on the spool in terms of T, R ,R 1 , m, 2 and 1. a -r S 1J4e- A79J Y +he- v’y’t 0 / d / hi / çi-r ; €le (i Ci) iZ1;? -TT 1’- jr7C- a: TCT-i 5.( 10 pts) A 0.500-kg sphere moving with a velocity given by (2.001—3.003+1.00 k) mIs strikes another sphere of mass 1.5 kg that is moving with a velocity of (—1.001 + 2.003 3.00 k) rn/s. The velocity of the 0.500 kg sphere after the collision is given by (—1.001+3.003 8.00k) mIs. a) Find the final velocity of the 1.50 kg sphere b) Is the collision elastic, inelastic, or perfectly inela stic? Justify your answer — — 0 D’s — / ml 9)g -3j * -if 7- ‘1:— 1% h r A - ic 2f1 = (_ C 6 J i -/.S, ?? & 1 b L )c z —4 be-to, L. ki;E i’fter z -1--2 ‘. ‘h,zE_e_ I c I(,LS3 6. (12 pts) A 5.00 g bullet moving with an initial speed of 400 mIs is fired into and passes through a 1.00 kg block as shown below. The block, initially at rest on a frictionless horizontal surface, is connected to a spring with force constant 900 N/rn. The bloc k moves 5.00 cm to the right after impact before stopping. a) Find the speed at which the bullet emerges from the bloc k b) Find the mechanical energy converted into internal ener gy in the collision. C n4OO - L7 y 0 i’C yntt2,r 0 Y LL “II lo C-4- /4 7tet b41i--/ i .c Ccv’ M8Joc1e, 0) - ro ‘y ).; pi.s e ?2 , 04 I1 I1 vp4 , i, , POb4(f) “-3 —3 I “V h) Erry 1 ’ 9 d’’ 3 S1 ’ O 2- —;P6K i’ ; (JoO*.A) ,z.. 2 ),Izcj-i EJ7 >‘ -1ird 12 t212-J ei y 7. (12 pts) A toy cannon uses a spring to project a 5.30-g soft rubber ball. The spring is originally compressed by 5.00 cm and has a force constant of 8.00 N/rn. When the cannon is fired, the ball moves 15.0 cm through the horizontal barrel of the cannon and the barrel exerts a constant force of 0.0320 N on the ball. a) With what speed does the ball leave the barrel of the cannon b) At what point along the barrel does the ball have maxim um speed? c) What is this maximum speed? 15.i w,r-ene-rpY Th,c 4-L 1 pr’ I,, -ca z 232O ± 1 L -- — be 0 -t -I c. VSiiyC2) I•s / 1 5 i ” )’ Pri21 I fDrY t,e.’ 2 j52)) mojf c’cCt)t t6t c_it 4 I : 2 c .j -? I ( S3cXJ ;:• p(e J 8.(12 pts) A string on a musical instrument is held under tension T and extends from the point x 0 to the point x = L. The string is overwound with wire in such a way that its mass per unit length jL(x) increases uniformly from jio at x =0 to IlL at x = L. a) Find an expression for ji(x) as a function of x over the range 0 x L. b) Find the time interval required for a transverse pulse to travel the length of the string. Express your answer in terms of T, L, IlL, and i7L1L/’ ?‘;Qt/: c! ,Ao) z/-4, z:_14 2_ V.’ t’2Z /44;,? ‘5 ff 4$.,? 5 r cI’ 32 _; L+-P k)k4 j ____ _____ ____ __ _ 9. (10 pts) Two wires are welded together end-to-end. The wires are made of the same materi al, but the diameter of one is twice that of the other. They are subjected to a tension of 4.60 N. The thin wire has a length of 40.0 cm and a linear density of 2.00 g/m. The combination is fixed at both ends and vibrated in such a way that two antinodes are present, with the node between them being precise ly at the weld. a) What is the frequency of vibration b) What is the length of the thick wire? Tz4Loi’ — L 75 /L4.2 — 3 S ,.t’—IO L%j II(1 r: - v 2 _ rF -r V4 2 1 - ,A_3 S;n’ t2P1 r1 W VLI A,