A bicycle odometer (which measures distq9n hqqu s3ev3*-jnr; ance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-is ev39* rqhq-3nuqn;j nch wheels?

Suppose a disk rotates at constant fzru.tcrq(x8 z i3b ,,angular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/orx8,zqbu(cr 3 i,ztr .f tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single vaw+ vw 7qyatx5u 6qw37mlue of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a lam 8n4xbch(1w arger force? Explain.

If a force $\vec{F}$ acts on an object such that its lever arm is zero, does it have any effect on the object’s motion? Explain.

Why is it more difficult to do a sit-up with your hands behind your head thx+ hjnj+qd3lx5* xs6w;wf gy/an when your arms are stretched out in w+njwx/xyd3jlq sg *5+ h; x6ffront of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets ajgsh0+n ;1 (s. oump7o*bj)8costkplr s 7l7ct the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a larcks0c;b+ u8gp)ps (j msh slo7no*71o.tj7lr ge rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to p(koq xs0kvt 5re8.ask* 5i.wrun fast have slender lower legs with flesh and muscle concentrated high, vt 0exs o.rks55 qa (*wk8.ikpclose to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a l7 i2t.bzpk,o bong, narrow beam?

If the net force on a system is zero, is the n pr+:mbzvjk,,bfk v ,12mr e/jet torque also zero? If the net torque on a system fb1ekb/k,,rpj, zjm:rv 2vm +is zero, is the net force zero?

Two inclines have the same height but make different angles with the horizonb/:ruor1d. h/ gc c8jetal. The same steel ball is rolled down each incline. On which incline will the spo:/d 8gj.he1r c b/rcueed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from r zl2g( hyn5,/srxfkv7 est) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the 5z 2vy/(r7 ,fkgslnxh greater total kinetic energy at the bottom?

A sphere and a cylinder hav -.;i.xbrsgw2 ax upo/w on(z:d:b) jfod s.5be the same radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic en. 5sd ow d2xbx.gbf j-r(iwooa:) p.z/un:b; sergy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum kkbkb w9b .w/ *6ydkb;are conserved. Yet most moving or rotating . wd/bwy9bkk6kb*b ;kobjects eventually slow down and stop. Explain.

If there were a great migration of people toward the Earth’m j5)z +9my06hv+igxpv flym+s equator, how would this affect the lf0 g ++y9v)iym65zjlhpx+mmvength of the day?

Can the diver of Fig. 8–)o w0zj d/mih329 do a somersault without having any initial rotation whenido0z j3/w)h m she leaves the board?

The moment of inertia of a rotating solid disk about an axisid),lf +q wc k* yuezqw/di45u x/x-t7 through its center of kqzdqtuly-i5/+u)w xw,df *ixc/e4 7mass is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on vwlzj *d 20-pq40 wan3:bugcja rotating stool holding a 2-kg mass in each outstretched hand. I2wp- b awd0zj:*j vgc0u n4q3lf you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identicn:gb o n g .hinld)+c4ki4:ke6al and have the same mass. However, one is hollow and the other is4 kbn nceg):i 6n: +gk.o4lhid solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal t.(x1if orb:d.er 8po axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the . bxti8:1f( opro.der wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on therhn7rt6uvy j y d2/ a4pgz 9o.;*2kosi edge of a large freely rotating turntable. What happent*6;y2u4 yps 2z.aoiov h/g drk9 rj7ns if you walk toward the center?

A shortstop may leap into the air to catch a ball and throw it quicklt2au- g5dgsw+ ;5xamly. As he throws the ball, the upper part of his body rotates. If you l2;5d x w-g5atlsaug +mook quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law ofyak;r7e6tn dfx 9twb,k d,z4t ,j c*cyk4zz;6 conservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propee7wtnk, , zz4 bka*6;;ct4 cy ,yfj 6txddz9rkller can operate to keep the helicopter stable.

  Express the following angl sw rpi:8u+4 cak0hwv(es in radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincidence. Calcu6 pb*whyk 0,ak6(jvgn late, using the information insn*gyb(a 0 p6,jwh6 vkkide the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. Thef8urv xb(op7q(mva.9 beam diverges at an anf (vux(pr9. mob8va7qgle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. When the motor is turj8 yo; 4in3(moeythd0ned off during operation, the blades slow to dit(4o;h 0yoe 38jn myrest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the ball makes gu z-z5fily+ *15.0 revolutions, what is its5iz u-* lyz+fg diameter?    m

  A bicycle with tires 68 cm in diameter th5se7mk27 pwcravels 8.0 km. How many revolutions do the h57e2k p7csmwwheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate its a5 .edbvoj)9q engular veloce ).9 jdveq5obity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.0 s (Fig. 8–38). (a:cpc,sywo;;ylf7:3 0v b gadx ) What is the linear speed of a child seated 1.2 m from the cenbxl3s 7cvy:ayo pg;; f0,wdc :ter?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its ogubpvgh- bq3x p8-0yf: )jrg6rbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of as2 7rzc-- orfu point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.0 cm from the ax,3zzg ac2a i(lc p3.ecis ofpcci( lg3 32 z.c,zaae rotation is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rhufg6.+o s(hd k or.9ppm to 280 rpm in 4.0 sh pf krh6uso .g9.do+(. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radiuf; h-;rsb nnpoh0y5h8 s $R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboard the Apollo spacecraft put themselvyi7eritybv6 3n1h3 + ks.wpw(es into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylindeh(s .bwv t13yi+36nyier7kw pr with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpm j-t ,b g0iw-fyq.(,ow idu yu4in 220 s. Through how many revolutions dit(f,.qu 0,i wyo j-d4-bwyu igd it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 120 m/it/uo,rqrbzh0 45 6tz)gcduk..t r0 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flyingey uo-l g3i4+f highspeed jets in a whirling “human centrifuge,” whig-elyf 4u+3 oich takes 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from )bj*, djown9tlw sy:+240 rpm to 360 rpm in 6.5 s. How far will a point on the edge ofj,wljbt *s9 o:y+)nwd the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turu;naob-fb1j/t 0rr, hns 1500 revolutions before it comes /bubr,j;a1 thnf-or0 to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions asl;3vd9wgf1r z the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diam3dl; zfvw1r9g eter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the lht9 w5j x8x7rpc5hh* po*3cd car reduces its speed uniformly from 95kh3 5 d*wpjch8 xr9t7lpxo c*h5m/h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each pedal when cli4qs:a ga ue9 fsg-r6pv)m9)m ambing a hill. The pedals rotate in a circlesvs4murpemf g6aaag9: ))9-q of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cmicwq5 ko 12as, wide. What is the magnitude of the torque if the fo12aowk5i,c s qrce is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fig. 8–39. Assume r j7 d8ufabiix 58ee;xa77 gscb.u-) nthat cb7)e erg a8nu7.df8x-;7ub j5i isxaa friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a massless rod whi ah,cmn gmw9og. fn2*)ch pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net t*hg mocnf ).9a wnm,2gorque on this system.

  The bolts on the cylinder head of an )b ca3c7gj d 68rw0jcmengine require tightening to a torque of 386bd0jmc8wj3cc)7arg $m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of radij c+ qq-osl7c:us 0.648 m when the axis of ql7 sq-c+o:jcrotation is through its center.    $kg \cdot m^2$

  Calculate the moment of io-.x d*p mcks lc8mw)1nertia of a bicycle wheel 66.7 cm in diameter. The rim and tire hav w ml8)c sxk-om.*1pcde a combined mass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a t 70c94i -cuwkdfkmr r)b 7ccj(hin, light rod is rotated in a horizontal circle of radius 1m udjbk 0r 4cf(ck- c7w)7ir9c.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wut( m4tfl+-ytheel rotating at constant angular speed (Fig. 8–42). The frictil(ftyt u t4-+mon force between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point objects shown in Fig. 8–y.t p *z.u l)d;zrv9fj43 pl.9r)z; * yduftvzj .about
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen atoms whose total mass i5s2/ *p9iblw,tywqa *rp5x 7.bto uats $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the correct r4nqe y04,xb ieate, engineers fire four tangential rocketib04 e 4nq,yexs as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform ct ha5 ,muf6(brv, sbe5 (u7a:qckh tm,ylinder with a radius of 8.50 cm and a mass of 0.580 kg,ub( 5cht ,hamum5tv:7e6kb(rs qf, a . Calculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, acceleratiwa1;wdw 5rz+4cv0 wzw ng it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven merry-go-r.5k0c6k fz+kb e)ivo(ge2 cuv ound and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge5+oekkvvegik2 6 ufb0(c. z )c. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and/ kxpz8w(2n ql-trvyi b ,8x-k is eventually brought uniformly to rest by a frictional torqvy lr2w,b8ixk /t (zqpk-xn -8ue of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a 3.6-kg ball at 7fxa hrrhl27 *a 0i/,iz $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the actio*v 3pyon52*if cbt +htn of the forearm, which rotates about the elbow joint under p yhvtc o *bf*t2in3+5the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considerieol; q 5ov/p cgcy(*,92teived a long thin rod, as shown in Fig. 8–46v*o;ty oqpcc(gli,5 /v92eie .
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine consists of tw0d 9ay rl hn )zvebq+upi/b/4;o masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the,b1n x4n/ -h almma9sq hammer from rest within four full turns (revoluti-qsmb na 1,h4a x9/nmlons) and releases it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a momen6r1lvui zp.vq t/ 68ipt of inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torqueyz9z9* .qvhmg7)ygv,kuqf 8 x of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls without slsdyv 1.ojde,.h l8bdoi1;-p nipping down a lanb 1di;lj8p o1es,d.-.hvny doe at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Ei 81r3ipb9 n:twfe;0pr spd r0arth with respect to the Sun as the sum of two te 0ri f;tepbp83d: 9s1n0rwriprms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius of 7.50 m. How much netr. +y2u02eaoa ni:rd qydg/+j work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder. 0yaagjqu iy.dn/ o+2 err:d+2    J

  A sphere of radius 20.0 cm and mass 1.5 mnm-m +r245jsmya /.sqasq t:hadc )g4+skw80 kg starts from rest and rolls without slipping down a aw g 2 +:y kdmmn5cqa)/5ma-4s.jtsrq4+sm hs30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

  Two masses, $m_1$ = 18 kg and $m_2$ = 26.5 kg are connected by a rope that hangs over a pulley (as in Fig. 8–47). The pulley is a uniform cylinder of radius 0.260 m and mass 7.50 kg. Initially, is on the ground and $m_2$ rests 3.00 m above the ground. If the system is now released, use conservation of energy to determine the speed of $m_2$ just before it strikes the ground. Assume the pulley is frictionless.    $m/s$

  A 2.30-m-long pole is balanced vertically on its tip. It sta *jxxt).fgxnf-m 2o 6arts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of en xo-6fgmjn*2 fa.x)tx ergy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the en * xwyx,ya;jc*d of a thin stw*a; *xcyy ,xjring in a circle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg ;*c)golr-5x8 jipv on uniform cylindrical grinding wheel of radius 18 cm when rotating at -8 5;v* ixpjloog) nrc1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rozk) (zc b*o+1 xldi*zmtating at a rate of 1.3rev/s If he*zc +xzbldik1(o )*mz raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the oh.5kuf hk u-n.ne shown in Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her anh5khuu-k .n f.gular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation glg+g4hvqe e ev-; k,1rate from an initial rate of 1.0 rev every 2.0 s to a final rae;klg4gg q e-+,1 hevvte of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotati 7,orx 2ahsqm(ng around a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. Wh72 m, a(qorhxsat is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinning at 3. +,7b:a mit+ hbcalyj*wg:owe; j;s6 w5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum of7x (e* yd dhvlg.8l) fo6vrkj, the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is 7y b5,a8akpy-nxcq/he/ url: dropped onto an identical e7:yyl5/n cr u bk/-a8, pahxqdisk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4wjbg e 0qxz(06z/ao d2 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotating merry-go-roul5txa + d16xhnnc1.bqnd platform of radius 3.0 m and momendt1h5lxb+1xn aqc. n6 t of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with aafy86nfzmicq;dm*0 0(5r1(mrm f nibn angular velocity of 0.8c0 (*difry8f1 ( r6mm m0nmqnzabf5 ;i $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventua :xrh5* b4i s(gpfl1pglly collapses into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nea5 : lx*bgghrp(si1fp4 rest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in excessfaevqb1j 9c/bz.n +1 b of 120 $km/h$ at the outer edge. Make a crude estimate of
(a) the energy,    $ \times10^{16 }$ J
(b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 $kg \cdot m^2$) of radius 100 km and height 4.0 km.    $ \times10^{20 }$ $kg \cdot m^2$

  An asteroid of mass em0d9 (htmjy+$ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, thateydf,:n7ioz,q f i34t can rotate freely without friction. The mom ezi7n, 3ff:qyt4,odient of inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands 0vhqjiflf4r,7cig pn g (83a1i *ck1pat the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bea,gfi08hckqg4n3l pv1ja7i pf i*(c r1rings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground .-t4q7i,y+s3 3lsi3d j6 iypotas z jdwith the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behjqiitdpi3z,+o34 -3 ssj tyadl7s 6y.ind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces the Earta63f0 -lklp wqh. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter caqf3p lal 6-0kwse, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rat f+*e 4yzdklaa//t-qhe of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylin(a*+nwjtl 9hrxh8i0 euk.go6der of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the tx gwo0 nu hhkr9eti*j+l 6a8.(orque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical dp7ks*qpzt3oxd+ko 5on1 , a8d isks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindro1* +n,dt ko7 pkapsdx5z83q oical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is thecgno q.g2q te*06e w6o angular speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but wira3h ar (.g/xwth mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the sta x3ra.hr(agw /r is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for o1vl 5hg/pw1hq al75p it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be ablep5g 1h qw plvha7l/5o1 to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates rtxh47yaw:1t c3ehht7l v.y 7 an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal a-hy .m.ta v; jp5kt-usurface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., we ignore fri+z8 v8*tl vl0sc,ekb zction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts avt8,0 *clzk8ebzlv+s t the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is sthvxs . f+j(;,d+pofj/)lj ynx5 uf7ttanding vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where hd ,j(ht5f po/fjn+.xj+s yxu7 fv;)tl

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a turn witha 8cck+9wx oxq14 b4n:yjr ,bh a radius The forces acting on thecbahx nr:4cw b+ 9,xqjy o1k84 cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sliyih kxb(crr* u;h6 a;s zcjs/e+(;mf ,ng of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0b;*csjzya+s/hhcrf r , mk ; u((x;ei6 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylin*z/tx6-a yfhwz7 tnw8der and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightl*nyfz/ z tx7wwh at-86y against her body.   

  You are designing a clutch assembly which consists of two cylindricls9-ls3sl:z 97w zohr al plates, of mas -rslz 3 9wlhs9zos7l:s $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the looped rough track of F+8tmkadd. v q,tmxr/- ig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point+ t-qdm8 av/drktxm ., of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do v5 80bf)(vskg wucpk0not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its sp k5jhtett,9 t0 :y1ujpeed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each tire? htu5ty:0jp9 kte1j,t $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s