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Ch 7 Homework
Name:
Homework problems are from the Serway & Vuille
LOth
edition. Follow the instructions and show your
work clearly.
1. (Problem 7)
A machine part rotates at an angular speed of 0.06 rad/s; its speed is then increased to 2.2 rad/s at and
angular acceleration of 0.70 rad/sz.
(a)
Label physical quantities in this problem using letters you choose.
in)tial' otn6'vl*n vsteci{V
anso lar velpcitT
r'^ol /,
$; n^t
.
teler o"t io n
0 ' 4 CI nrd lsz' Ar'6 n tar c^t
(.\/o
rt'dfg
=0,o$
il { * }.L
Nz
'= argly +La ParL Yl*ates
49
(b)
Find the angle through which the part rotates before reaching this finalspeed.(First, write
down an equation you will use and substitute numerical values in the equation)
Lt{
'
::-
d" 1-l $
t L
Lt),' +
N,
iLr-z.sY
"'!'-it"'-"
z&,
-.
-ri)
..1
,FJ;/*l
l:
.
L:_::-: **t
(c)
ln general, if both the initial and final angular speeds are doubled at the same angular
acceleration, by what factor
is
the angular displacement changed? Why? (Hint: Look at the
forrn of equation 7.9)
fror^ +ha ek} ,
tAe anputar
I
isptaten,
,{
nt'
tt^z ah6^r^r 5peeds ^tve
it
'
9r!)':-.!'*'t'
>oL
sloaulcul
.%::"y!:"y
= + tfr=Jil
; l+0
'#
t
2.(Problem 12)
A 45.0-cm diameter disk rotates with a constant angular acceleration of 2.50 rad/sz.lt starts from rest at
t
= 0, and a line drawn from the center of the disk to a point P on the rim of the disk makes and angle of
57.3' with the positive x-axis at this time. At t= 2.30 s, find (a) the angular speed of the wheel, (b) the
linear velocity and tangential acceleration of P, and (c)the position of P(in degree, with respect to the
positive x-axis)
(a) Complete the table below.
Phvsicalouantitv
Variable
r
Angular acceleration
0
lnitial angle
go
lnitial angular speed
uJs
Final angle
01
?
Final angular speed
(r)1
?
Linear velocity
Ta
(b)
Numericalvalue
Radius of the disk
D,LLSln
2,5o totl/rt
I ,oo
rod'
O ,oo rad/,
?
ngential acceleration
a
?
Using variables defined above, find the angular speed of the wheel and substitute numbers
to the equation.
fu+= tN, *^1,
l= \r;""4
---.....-
(c)
_---
Find the liner velocity and tangential acceleration of P in terms of the variables above and
substitute numbers to the variable.
.17.=
rNf
a
At=rd.-=
(d)
l@
Find the position of P..
N'L
0;*0,:
+
jot'
0+ = q,:4!-?\rl= 1,ct r^)
+*-)r
I
*
/t
1' /
O
3. (Problem 16)
It has been suggested that rotating cylinders about L0 miles long and 5.0 miles in diameter be placed in
space and used as colonies. What angular speed must such a cylinder have so that the centripetal
acceleration at its surface equals the free-fall acceleration on Earth?
(a) Draw a diagram and label physical quantities
using variables you choose.
-/
l= lo *,1a5
r> a.gerle"f
height
ri,J;as
o+ tru
of tk rylinler
a)> c^uf,ular t perl
ff,:
lr r+l
Cen
cy';dlt
tt i peta L arre lrr rx trox
(b) Convert distances from mile to
m.
).= (o2,)les t
r:
(c)
2^" $h;ky
1,6 o
q3 nl
d#
l6 01A4r1
I *i lc
I ,6oc4 ,3*'t
1ry.) | e
Find the angular acceleration in terms of radius and angular speed.
C^,= l/N
Find the angular speed of the colony when the centripetal acceleration at its surface is g =9.3
m/s'.
rN
CA
t-
o
N=
L
E
r/
t---u.
0
,O+1 + rat/,
-"-------
4. (Problem 19)
One end of a cord is fixed and a small 0.500-kg object is attached to the other end, where it swings in a
section of a vertical circle of radius 2.00 m as shown in the figure below. When 0 = 20.0", the speed of
the object is 8.00 m/s.
9\
v
(a) Draw forces on the diagram above and label physical quantities using letters you choose.
tn = o .(oo Vy fil^iS
*l'e ohera"f
f
o'ln
= ),o
p-
-
LO,O"
ait= $,ao*t/j
1,r.d;$
s r{
rafi^*t'o,r
x"ngle .t +k
fongcnl.i,-t
T
tension
w=^3
w e ;6.h1
cxy
t,*yr,r{';*e-
6c
Cr,,
tr i pr+af
oloeci.
spnl
of +he oSjert
af(eic r^4
o{t.vlw
t'o nt
^1{an
(b) Write down the tangential and radial components of forces.
Rad
ial
d
irection(y-direction)
Ta
Tension in radial
direction
tangential
direction
Weight in
tangential
direction.
T
Weight in radial
direction
ngentia I d irection (x-direction
Tension in
- %q
a
co50
Fnety
Fnetx
0ciony
(c)
/)
-?nf S;n0
fui *+
Find the tension in the spring.
Tf
(d)
)
--
ofi
-F
r
"tr\
+
tr19
G
hd$
Find the tangential and radial components of acceleration.
(1,-^Y)' the c?^+e /
th< rotntion
-
l,,----l
6L =-%9;40
7-:,r, ^rr')r
"r )
rtockwise otiwca;o')
(e)
Find the magnitude and direction of the total acceleration.
J*r"t 6i
^=
-
)?tL
")
1 /)././J
d = torl (X)
=
ls your answer changed if the object is swinging down
S,q
l"
o' Love -'-^.Lis'
toward its lowest point instead of
swinging up?
M
(g)
o
chans€
Explain your answer to part (f)
The
o,tcelero'l
o{ f}.s
Sffiel
ion olegnts o n
, so +t/'* $irtc'f;oot
tle
e "f +le
n)**s not w*tbr,
fgurr
r*"1,ai<nAe
5. (Problem 27)
An air puck of mass mr = 0.25 kg is tied to a string and allowed to revolve in a circle of radius R = 1.0 m
on a frictionless horizontal table. The other end of the string passes through a hole in the center of the
table, and a mass of m, = 1.0 kg is tied to it. (See the figure below.) The suspended mass remains in
equilibrium while the puck on the tabletop revolves.
FIGURE P7.27
(a)
Label all physical quantities using variables you choose.
1r
spe"cl "f
It= l,o t^
ro,rJ
irts
o
f
ftl
t
the r ot*t iort
o,L 4kat
0ll, -- 1,,ol.q
r
'lvruss "f h
,t^a.ss o+ ''n >
T
Trn
6.c
QnLr;peta{
,r1,
"
(b) Draw a free-body diagram ofthe
1
sio
n
oxcrclera.'tion
puck.
ryf1tO\c
TF
+1,
-J
(c) What
rt
is
the tension in the string?
t@
,fi
=
q,8 M
J* ,ul
(d) What
is
the horizontal force acting on the puck?
T:=- q,tM
(e) What
is
the speed of the puck?
{ =h,6.
=
,
g
1f = .lIIE
*11
V
=
(,\ h/s
1*r
5. (Problem 34)
A satellite has a mass of 100 kg and is located at
2.00 x 105m above the surface of Earth.
(a) Draw a diagram and label all physical quantities
y hatt ol tA€ Sa{etlit4-.
l": zpo*106n altilv^lc of *setetli'le.
.fti> loot
4/t = looPcj-
{hflI
t =
in this problem using letters you choose.
2-,ooulob
ke:
Rr
l,r1
J
(b) What
is
the potential energy associated with the satellite at this location?
v1--
(c) What
is
g
q Meh
fE, + r)
the magnitude of the gravitational force on the satellite?
r
G Aerq
(t2e + h)
*
;'641;*l
lz1r.!5
"f
a{ *t* &'rfA,
fk
Ear4l1
7. (Problem 37)
Objects with masses of 200 kg and 500 kg are separated by 0.400 m
(a)
Find the net force exerted by these objects on a 50.0 kg object placed midway between them.
(Zoo t cfi L$o
,-^
L(jl
tuL
,o bo1)
(Soovy} L Eo.o kg)
1- c=l
( o ,>oo 1'
(o.lro?n)-
'
t,4'trtisN
+,l4.tf,sU
)I
ao,^1,.rol
tksool< 3- o\; er't
(b) nt what position (other than infinitely remote ones) can the 50.0-kg object be placed so as to
experience a net force of zero?
a.
Find the net force exerted by the 200 kg and 500 kg objects on the 50.0-kg object placed
distance d m away from the 200-kg object and10.tt00
-d) m away from the 500-kg
object.(see the diagram below)
ft -
@-'
2OO
r
lzeL=
L-
0.4OO m
0.400 -d
rC
kg mr-50
q rt rfrs
d,'
htr=So}
+h
-
fi t,dt
c#
(r..-dl*
kE
b.
Find the distance d at which the net force on the 50.0-kg object is zero.
f4:,# { FmL
Fxer=o =
4lt
futt
W
e
=
d2.
llz
4vt,
@"
friy
,
lv,,
Ol
(-r-"
c
t+
V
,H"ll
fr,
=
lmt
rtff
$h*
t_
dL
c t* ,H.)
8. (Problem 41)
A satellite is in a circular orbit around the Earth at an altitude of 2.80
(a)
x
105 m.
Find the period of the orbit.
qffir @
, Mefr
/1f
6Me
=
/},
LTt(PE+'"- ))'
{*.6+L)
T
f=
-l =
2
[*
7,8o
x /oe
*t
1f6tae
Fu+L'
Qn)'trsrA )9
6las
=/
B'7
los
L^ +3
l-,
(b)
Find the speed of the satellite
l/-
2
f* ( jae +-J' i
=
l= G,sqr^/s
le4Ee-a
I
l*****dr-Ft<*
(c)
Find the acceleration of the satellite.(Hint: Modify Equation 7.23 so it is suitable for objects
orbiting the Earth than the Sun
ft:
g
f
E_ru
9. (Problem 71)
A 4.00-kg object is attached to a vertical rod by two strings as shown in Figure below. The object rotates
in a horizontal circle at constant speed 6.00 m/s. Find the tension in (a) the upper string and (b) Iower
string.
T
(
_+-
\l
s.oo
\l
t+
FIGURE P7.71
(a) Draw a free body diagram and define a coordinate system. Then, label all physical quantities
using letters you choose.
1r> 6 ,ootn/s s1'tenl "
rlyl
f
i
-Ti'*
$
4rY
-l
c-
I
,!
a
r
6
)
lti
rl'e o bd ec{
: +.oo|*os| o{ oLsrc{
I = ).ootn
A
I
len|+a
"f
e-ch s*ring
= 3.'o^
I
g
) 1a"2le
f :
f
l2e{ween
+k tu'o r+fih,t
d
o'h.^
r- /rcs9
I 'f
ll'r
=
ft'- t*t
ro fo'f ia
o- eiitt*)= +t, 6
vt
I jz cr1
(b) Write down horizontal and vertical components of the forces.
y-direction
x-direction
Value in terms of other
variables
Variable
T,
Tr
(.oS
0
T'7
T,,
t
Tut"s
T.
E
O
l^/
% 6 = ')ny:
I netx
Value in terms of other variables
Variable
{-
T, Sin&
- T>5in0
"rt 3
Fnety
/-1
r
(c)
L-/
Find the tension in the upper string.
*
- A;rectloLt
cos7 -7 T,t
0,*T.)
=
^yf
y -ol i rec'l io Yl
ft,- T)t;n0 -^t
f,{r
(0
*
:7
eqa
t|
Ir-'>
(d)
=O
-
-4t
/r
L
+:nlfa
(togg
-F-
foi6L
r'
'ffi7
- string.
Find the tension in the lower
eho Tt
"fi
r mlf\
L rto|g
"4 *
s.
rt
0 )
Tz-
T,-7,
.lh1fz
rroSO
fial
=
5
in0
-o