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1.
Driving
inyour
carwith^9-9"9999',*
abump
intheroad
that
hasa circularcross-section,
asindicated.If theradiusof curvatureof thebumpis 35 m, find
the apparentweightof a 67 kg personin your car asyou passoverthe top of thebump.
V--\f*'lS
I-:35rn
pn= 6'lK3
j l-
'd
*F.'
>i h -- txr - -13
-F-L=Fr..r-F3
-\--
I- c- thS- -
- m !t
r-
ru
r
* t-l.l -- rp
' -(S*i l =Fs
61l<3l@*t.Bntsl = Fr.r
3U I N: Fu
2. Refeningto thefirst problem,at whatspeedmustyou go overthebumpif peoplein your car
areto feel"weightless."
ZF = Fs -F l
- f_- f
tq--
_r -
f-c -thg=
f*=flN
\ - N - \- S
l
\- N
-F n
\-c_ +ts5-u
F c - =r5
'yi3-=f3
r
v=m
! =Fs,.clm).
v:- \8 .5 9 s
To test the effectsof high accelerationon the humanbody, the National Aeronauticsand
SpaceAdministration (NASA) has constructeda large centrifugeat the Manned Spacecraft
Centerin Houston. In this device,astronautsare placedin a capsulethat movesin a circular
path with a radiusof 15 m. If the astronautsin this centrifugeexperiencea centripetal
p: \Srn
acceleration9.0 times that of gravity,what is the linear speedof the capsule?
G" = qo\
l+
\l
vf
f,-'\,
frr
C. =
c .r
It
-?
- \
e
\1
r
Ya
3 G .\* J s :\r
4. As part of a circus act, a persondrives a motorcyclewith constantspeedv aroundthe inside
of a vertical track of radiusr, as indicated. If the combinedmassof the motorcycleand rider
is m, find the normal force exertedon the motorcycleby the track at the points
a.A
b.B
c.C
l-r
TT
{_.)
tu o a a
T
r .-
-\\
Frc
rN-
lr
wl- c
*_t'
rIr-J -
s-] r- _
-Fr
r
-. t\1
t-
- fF J
,f
Fc = Fr.t
ilnt' = Fpl
FNf f
\1
|-
iu
r
r.-r
Z \- = \-''r-13
l -c: F N- 15
tq -1-\-,3-' tr g1
m'tD
_l, lJ_
rJ
lrnq
-
\- NJ
parksis illustratedbelow. In this ride,peoplesit in a swinglhaJ
A popularride at amusement
is suspended
from a rotatingarm. Ridersareat a distanceof 12m from the axisof rotation
and move with a speedof 25 milhn ->l\.E'is
a. Find the centripetalaccelerationof the riders,
Vr? -
Q=
|
r C | .y .\
-
6 t.a*ts)-- \ O.q5'/s"
\E r'.
b. Find the angle0 the supportingwiresmadewith the
vertical.
=r \- _
Zlrr_-\x
ZFr= Ty Ft
F3 =Ty -
F g = -T a
- Oc-- Ot,x
-
\O,\5tr/54=g1r.,X
*onO=fr
o=ton-'(*)
!
r=l\b.8"
=
T
o fo.-,tft#i)
If you observea ride like that, you will notice that all the swings areat the sameangle 0
to the vertical,regardlessof the weight of the rider. Explain.
Y\uSS C"rosSeSourt
f-{ o. c\-i'\c\,o.dul*,Cr emF\
r
the
o n3\e wi\\
Swinq
r
-l
rno\res tn O Circ\e-
bj_ tho s 0 n'\s, Sor cn 5 v e-n SPe,ed .
6 . Venusorbitsthe sunat a radiusof 1.08x 10" m with an orbitalperiodof 228days. Find the
massof the sunin ks.
Y-=-\.OBXlO" .n
F. -- F5
\? - T
-
r
-xr
o.r r r '
{
- f}tdqVt
= lq)6 qq,iocs
(T "
\ c.,rrs
\-r-
\*
\-
\f.t"{q
C"ft=- \frara
r
\t
I--
-
Sflr
\\> :
\ff'
\ff a(r,oz*ro'J3
13
G T*
(
T
Y\ > -- \.q" * lb'" K3
A ball on the end of a string is swung in a vertical circle. The massm of theball is 0.25 kg
and the radiusof the circle R: 0.75 m. The position of the ball is markedevery quarterof a
revolution in the diagrambelow.
a. When the ball is at position IV, the tensionforce in the string is twice as greatas the
weight of the ball. Determinethe speedof the ball at position IV.
T
F,- Fc-.
o-9
rrI
?t3= {*l'
\=d }3 r
rv
'
V -- da&gr"vrq(o,"rs-\
]t
\ -- 3'1-/:
b. The speedof the ball at position III is 5 m/s. Determinethe tension
the ball passesthroughposition IIL
Fr = o'a-:t(s^rs)* -r o.f,sr3(q,zr*a)
ZFy=Ft-F.3
O"9 S rn
Fc=Fr-FS
r.-r q + r3
h
\7
rl-
-
r - +*3 =
ft= \o .E N
11
lh
c. Suppose
thestringbreaksjust astheball is at positionIV.
i. Describethesubsequent
motionof theball.
Straighf
tap
5too
AccL-\crale-S
(dc<ctor:iks\
ii. Use the speedyou calculatedin part (a) to determinethe malimum height the ball
ofi"::,"t *)gay
reachesafter the string breaks. ;nrl'=p^gh
h=
= o . T 3 rn
=h
6
*
o,"i,'. --l'.
8. At what height abovethe surfaceof Mars must a satellitebe placedto be in a synchronous
orbit? The radiusof Mars is 6.79 x 106m, its massis 6.40 r 10t' kg and its period of rotation
is 1 .03d. T= ?8,qqas
r
tfi\r
F.=
It- r 3tr:
'+=H-
drl = Gren
,=fGF
\ - { -F- \r_
Yt -
Jftr
T
T
r
\\
T
r..jre*=a
f J \ Gn l ]
rfv
,r,f1.-
J= -\
{
n: _
r-=
,-\ Ac
n
f . O 5 x lO ' rn
cv.vr,\rvr\
h= O.O5xroq--c.'lqxo6a
l f\_,."
9 . An airplaneis flying in a horizontalcircleof radius1.0km. Whatmustbe the speedofiffi
plane if the pilot is to experiencea centripetalaccelerationthreetimes that of gravity?
O.=
7^
J3-
Ve
r
VA
r
.jrg" = v
\ = \'1?*'la
10. If the pilot in problem 4 has a massof 75 kg, what centripetalforce actson him?
F.:
rrr Qc-
F. = rn3j
F. -- ts K3(3\(q.Bl^ls\
Fc = Sboi X
I 1.A trick motorcyclistmovesin a horizontalcircleinsidealargeverticaldrumwhichhasa
radiusof 10.0m. If thecoefficientof staticfrictionbetweenthetiresandthedrumis 0.60,
whatis the smallestpossiblespeedthatthemotorcycleandridercanhave?
X Ft = Fq-F5=O
=
ft rt
!F .= - 3
Fnr=$
1
Fr.,= Fc
=S
F,..r
-/
\
{.'
S\=ry
v
l0* (9.8t'nA\
\ = \E,l "t/5
12.A 1.5kg toy motorcycleis moving on the insideof a verticalcirculartrack(r : 1.0m). It
arrivesat the top of the track with a speedof 5.0 m/s. What forcedoesthe track exerton the
motorcycle?
Z Fv- -F1-Fxr
- Fq -- -fg*,) ' = * qo
(
F'uJ
Fn)
F
rn.Ja
r N. -- ?
- -rn g
= \,src,rFP
= l).
F,'r
- q,xn^A
13.A 40.0kg childtakesa ride on a Ferriswheelthatrotatesfour timeseachminuteandhasa
diameterof 18.0m.
a. What is the centripetalaccelerationof the child?
G. = \-[da
raorl)?
o. = 9, (o.u3\
0 .=
\e
l 'G m/5a
f-
(d = lrff-S
c-r : P{f (0" o{ar Hz*)
( ! = C,\Et r q."r lS
-- 0.()6'
\r/
b. Whatforce(magnitude
anddirection)doestheseatexerton the childat the lowestpoint*of theride?
F. = tr*- Ft
F. +tru= Fnu
FnGc--+ )-vrt -- fg
\ 013(t . to*r's\ + \ 0\co:(R.gro,1*1
=- Fnt
c. What forcedoesthe seatexerton the child at thehighestpoint of the ride?
- Fc: Fnr-tr1
Fc -- F3- tr"r
Fxi=trS-FcFr.f = I'rt3-
rnClq-
Fr* = \ 0r1 (q.grnls'- l. G%\
v
Fr u= 3 t8 .+ N
d. What force does the seatexert on the child when the child is halfivay betweenthe top and
bottom?
'l Fxl
t
l--"--1
t\
Jr=^
r3 ---Fru
mB=Fru
lbq(e.8r*ls.)=F^r
Sq;N = Fl..J
'v