A bicycle odometer (which measures distance traveled) is attachd.lbf 9u3niy no/z;) ned near the wheel hub andnl )/n; d.yub3fon9zi is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant anguld4; wk k.unf.ear 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/or tangential acceleration? For which cases would the magnitude of eku 4dfnekw.;.ither component of linear acceleration change?

Could a nonrigid body be described by a single value of the ang *ix i7*/tgjayular velocity $\omega$ Explain.

Can a small force ever exert a aw4;d2u*mek6/h auya; l c4/cej i.fbgreater torque than a larger 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 headoa2,(tlnb 13mhjx j f) than when your arms are stretched out in front of you? A diagram may help you to3jmo1 f( ,x2bjth)nal answer this.

A 21-speed bicycle has sev.m kcmj-w- um* vj-xxt qf,h2swu;03ken sprockets at the rear wheel and three at the pedal cranks. In which gear is it hardec*-vxxwt k2j qk ,m 3mu.-whu j-0smf;r to pedal, a small rear sprocket or a large 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 run fast have slender low f wj1o wpqb3,h-r)w0jer legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamih1r3wboq f jj)0ww, p-cs, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, na/f0ig8rfol3 f.w v9 lv.v8kq mrrow beam?

If the net force on a system is zero, is the netb+t-r+g m mlw k2*;tzh torque also zero? If the net torque on a system is zero, is the net t*;mg-+2t +zrlhwk bm force zero?

Two inclines have the same height but make different angles with ooesl.8cu7;evw 6 nifhc d5(*the horizontal. The same steel ball is rolled down each incline. On which incline will the 5fc(i7o ;v.6su*l8dnc oe h wespeed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously sta* p 9z-apjq,li5( iajiv5z)arrt rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which ra zai)ivpiprzj* (aq5 j,95l-eaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same toebj*f :o8 d88fa x 7e*76bckei77y* ctkxqemass. They start from rest at the top of an incline. Which reac7e7afei*ye * ockttj8x:* 7kqe 86 do7bx cbf8hes the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet m(n lyhhmgy (4+ost moving or rotating objects eventually slow down annm(4(hyh yl+gd stop. Explain.

If there were a great migration of3vnb4y : 0f1-q aogg8lecw t-r43k gin people toward the Earth’s equator, how would this3 fr a1:n-eg48k c4o3v bt-wniyq 0glg affect the length of the day?

Can the diver of Fig. 8–29 do a somersault without having any iyj:l k /w3a1xinitial rotation when she leaves the board1lxj ay:w/i k3?

The moment of inertia of a rotating solid diyisdms*+ca4b*,y o 15y m7lo:a2 ykpam3ii p:sk about an axis through its center of y4o l:a 3ibcs*m*17y,p:m asad+5iom2ypk iymass 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 a rotating stool holding a 2-kg mass in eaceul1/bf95d5 e pu2 zohh outstretched hand. If you suddenly drop the masses, will 2op5ebhe5d 1f/luu z9 your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identw)f;snci6o iy+so( x* ical and have the same mass. However, one is hollow and the other is solid. Describe an expn xi sis)ow6 f*yo+(;ceriment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axlzj dh0i0 8wxwc 9xrm33/dr+pfe 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 ofj x8 wxm93/ d+pf0wdz 3rrihc0 the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotatii a c4)g;h-u fw,x3tz-t6.rce otr*nfng turntable. What happens if yo*h)c g-w,;rttfaxri -znoe3 f.tc u46 u walk toward the center?

A shortstop may leap d6ad447 uul5ty w/bgyinto the air to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hipsta7 d45y /g6ubduy lw4 and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discuss why a h)jl c rm;;zeofk87kfgly.,09q *h opn sj7 1rselicopter *z g; 9nmhf;. ekjlrjsyq8c77or) o0ksf1, pl must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the followin4x(zszaye2) sx () midg angles 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 becausx7a nxdj6e*x9+ gesi dmak5g+7 d3rz *r7t /iae of an amazing coincidence. Calculate, using the informx+j nr x63k+da7i/*a 57sd9damrxi g7geet*zation inside 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 ji j:fs . i:oouuo;/(kMoon, 380,000 km from Earth. The beam diverges at an angi;(ouo fs:.u :j /ojikle $\theta$ (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 qzcug px lp6r0 ;c7+ojsx: g*/the motor is turned off during operation, the blades slow to rest ixp:g/;cu7js+* oz0p 6g xl cqrn 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 a5rrh 7k y 2gxee03lw.wnother child. If the ball makes 15.0 revolutek7r gww520lxe r3h.yions, what is its diameter?    m

  A bicycle with tires wo/d2vq93 i yw68 cm in diameter travels 8.0 km. How many revolutions do the w9qo/ 3vid ww2yheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotatf5:pi* g)cay ogcc )jbw0vq-7 es at 2500 rpm. Calculate its angular 0c)gfy*i5c- 7pwjvgo :abq) c velocity 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 oub7o*7apg 2 mh7vtw*+3cnpc f ne complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child se btfcn2o h+77a*c7pv *u3pm wgated 1.2 m from the center?    $m/s$
(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 orbit around the S9 j*6b zjf--z-ebiwxf un    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear spmch,jv3h6nx883 8bik c+lbn qeed of a 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 pazztg-a jbw sk:6guhg:)k +6aovf19 (zrticle 7.0 cm from the axis of rotation is to experience gbaaj gh )g-66k zvo9 w:zft1(su+k z:an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter whe0,t.uvjd2: d6y nifrnel accelerates uniformly about its center from 130 rpm to 280 rpm in 4.0 s..d vjtr 6n:,02finuyd 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 radius v0 i vx;8)4-y,heu-8fus-vycrwg xz u)xjm (d$R_1$ 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 mcyqp(x;r 3(z9 uvx8 +p tv +y(c7hanaaboard the Apollo spacecraft put themselves into a slow rotation to distribute the7p( axr9;n+yxv z(mhpay38uvt q +cc ( 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 cylinder 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 in 220 s. Through1 z qddz( .ko7pcat7(q how many revolutions did it turn in this ti (pkctzq( 71 7ao.dzqdme?    $rev$

  An automobile engine slows down from 4500 rpm tosown-g(x 9ovp 09nt:e 1200 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 flying highspeed jegz sa5mjddyr9sfpi+g/l5na5;4:ul5jb ; d( gts in a whirling “human centrifuge,” which takes 1.0 minyz ;l s5m/gu(d:j 45 ;idj9d+p bnafsg5gar5l 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 240 rpm t-pg62na v4 cuuc6x edp0;f6 dyi,v5 mlo 360 rpm in 6.5 s. How vp5a- i6,u;ed0 y6pcmcf24uxv6nd glfar will a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850reqw 7z-ym x:co:v/min It turns 1500 revolutions before it comes to a stop.yxcw- ::mzq7o
(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 6rixzfw4s10rdda gd94 b2( y,d 5 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The dy,rzi09wxbafs d d4g(2d1r4tires 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

  The tires of a car make 65 revolutions as the car reduces its speed uniformly szei)( g:+ 2l.uyxta cfrom 95km/h to 45km/h The ta:z2.u +e(sg)i clyx 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 climbing a3v rp2p- exqo8 hill. The pedals rotate in a circle of radius or 3ppvx8q2-e 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 endolihc k3gaazx3n 27u- 7 ;vm,l of a door 74 cm wide. What is the magnitude of the torque if the force is exeag7 ia;nkho -3l xcu,mvl3 27zrted
(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 torq4/ zz z9ncj epxi qr-wn92e4w9ue about the axle of the wheel shown in Fig. 8–39. Assume that a izw qc9ez nrez-x/9j9n 44 w2pfriction 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 which pc1 for7co tr2bm7+mh8 ivots as shown in Fig. 8–40. Inirmrf821cb+ ct7mh oo7tially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinduxjj x2 ujg(m+/)9niaer head of an engine require tightening to a torque of 38) uu2j(mg/+x ajjxni9 $m \cdot N$ 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 radius 0.648 m when the 8eg6n9 re2lk8i ;vheu axis of rotation ish8l8 2u6gri enk;ve e9 through its center.    $kg \cdot m^2$

  Calculate the moment of inertia oymy ,js 5zjrq; opd3y;82h- fbf a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 2o3 hpsy zb rqyfm,j-y5j;8d;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 thin, light rod is rotated in a hori4kdc64ad;-5cv iba wrzontal circle bc k54 d;avd-i w4cra6of radius 1.2 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 wheel rotating at cons (qr* d5kwyic7n 3qmn)tant angular speed (Fig. 8–42). The friction force between)nr37kq5d ny*(c i wqm 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 obj(-x1ro4 -s nxx yiy9sdects shown in Fig. 8–43 abou9- 1(y-xoxi xnyr4 sdst
(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 at /u85v;olzg ieoms whose total mass is $5.3 \times10^{ -26}$ 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 8/v25bkvw34uv c uxvorate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, whatx3kc8buu vw o4 v2/v5v 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 unifor)) bhsu2iz rll4 vem86m cylinder with a radius of 8.50 cm and a mass of 0.580 kg. Calculathmuslz 6 8b)eli rv42)e
(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 swing:bn0dlh21 nlhs a bat, accelerating 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 hit;,ngsakn.b,i lk7 * and-driven merry-go-round and is able to accelerate sinn.;kab,i ,kl 7 *gtit 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 edge. 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 rotatiq /)cm*o3oai cng at 10,300 rpm is shut off and is eventually brought unifor /) oc3*icoaqmmly to rest by a frictional torque 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 a5klakh5v , zj0hud(w;ccelerates a 3.6-kg ball at 7 $m/s^2$ 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 action of the fowb3lis 9ksi 2sw:n 7v)rearm, which rotates about the elbow joint under the action of the triceps mus2lis:i79sbvw s3w kn )cle, 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 cr 3ty( kuct43h7os3juiu5/aponsidered a long thin rod, as shown in Fig. 8–46.c4rukis/ ao(tjpy3 u733t u h5
(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 two j p*kj1x9 ;ya)8grm lejhuk4 (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 hammer from rest within four fulhkhl- -tk tv,;l turns (revolutions) and relet-h,; vh lt-kkases 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 moment of inertia jqow ,- 48ra10l mjbwpof $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine deveq.g., ,k,sza;zqbkpo2i kf13+q zi wmlops a torque 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 r pvt4f(( l1/q fhgilo0x5.y/)eal n mholls without slipping down a lane at )./l0 (p(ohtl n54q1vxyief fh/agml 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sunkud4 .wbq0--cgv+-o:jz j hw k as the sum of two terms0: bozv. kw-w-cg4kd-uhq+ jj ,
(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 bk5 wwf c.tswd. z4i98 ;+hqiwa radius of 7.50 m. How much net work is required to accelerat .dk.4ws c9zwqiw;+iht wb8f5e it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls with kz4t -/p(gftylb2-po out slipping ktf4zp-gol/bp yt2-( down a 30.0 $^{\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 balance**k kg- h5up1a -li-tiztam (td vertically on its tip. It starts to fall and its lower end does not slip. What will be the speed oktk*i1(ita l zt-*m -pu g5ah-f the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of 4q39sc fefu)g zj6a c9a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an angulqc 6fsuc43e99f a)zgjar speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momenshk0npgg1cz6v;ae k4-wj-7 rtum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotating at 1500 r;ea4h0 p c wk g61zv-7skng-jrpm?    $kg \cdot m^2$
(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 platfojw6+0xp sc.+i7w sw vmrm that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decm7jc.s 06sivp+wxw+w reases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29)4sl*ay1 ljb,d can reduce her moment of inertia by a factor of about 3.5 when changil,b* jd4ysa l1ng 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 angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial rate of 1r)q,i-apt z-g3c) qy(n2px o 6dsjit).0 rev every 2.0 s tog-t-r3),)nio(62tc y) sxzpqjia q pd a final rate 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 fh0e.tkl/ 0 ferotating 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. What is the frequency of thf. fklte0 0eh/e wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skatxc qj-78h re(/g uyp.xer spinning at 3.5 $rev/s$ 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 o+/ oo9)lnsvh hshp;j( f 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 or 4qbi,x ,uss5f inertia I is dropped onto an identical disk rotauqsri x5,,bs 4ting at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns a* a l87tb()u ykalpqzu b137yut 2.4 $rev/s$ 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 standslr njt**s, +8 ,xivpozbju9(f at the center of a rotating merry-go-round platform of radius 3.0 m an s *(9l+ ut8v,*zjprj,obixfnd moment 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 rot+n 3l i amrj76+auwutpx +a01rating freely with an angular velocity of 0+ nu l6tmrxw a30ij++7 1aupar.8 $rad/s$ 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 eventually collapskxtsuukd5rp sc6c h57r m/,.0 0)ndm res 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 nots0 c/ ) ckk5mdpr.m,hxud 0sr5ru67n angular momentum, what would the Sun’s new rotation rate be?(round to the nearest 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 i0c)fp gs ae9yzje+m 71qca -g:n excess 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 liq 08lx-juk+lw 5 p:f$ 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, that cf59pduaf;q 8:xvmo5aan rotate freely without friction. The moment of inertia of the person plus tm9 8 fax;q5a:f vod5puhe 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 at the edgut,cdqq)3 e0b 2b*wmpn nb.:we of a 6.5-m diameter merry-go-round turntable that is mounc *b b0puwbw.2nq:,3qdnet m)ted on frictionless bearings 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 with the end of thc(tqc y(8dgx19q (zj he 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 behind thcg(q dx hy8(c tj(1zq9e 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 Earth. Determ k9*q+axc d +ax0pk pl2oxoyx7x fa*7,ine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting theox7 yx xqkl,2++xp9op0a af k*cad x*7 Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at ag(zn8zi vx;*c 5n: zlmu5lqy9 rate 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 uniformt 8v7osxs tsg,9u d2g* cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant 7gsdgs9v t *2,8 otsxuangular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 kg9 7xtrj,)ud go and diameter 0.075 m, joined by a (concentric) thin solid cylindrical 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-loo u d)txr9jg7,ng string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speji/+ u ezp*v++wtyy8s ed 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 with mass 8.0 times g6hb( ou(q mjc9a u:vk+iz-r /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 star is a uniform sphere at all times, and that the lost ma v u- +uq(rkh/:(b6cajo zm9igss carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a lot zqsb3,rt2.7c geg h gqin(v6.i:ry5w-pollution automobile is for 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 able to t.e t5(in y it b, 6c7zhg3:rqs.vgr2gqravel 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 1ny*kbq8(nigx9 xi 1san $H_2O$ 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 su. v )ec6o lvzv2g;1b(c,7bv)xup dopjrface 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 igjg 60guvu)6xpha5 jf +nore friction) 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 at the center of mass of the rod, as shown. [50a6j fgh+xguuj)6 pvHint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on thlll1fsqlk/x5.s( c- tpj n. y9e floor, and we want to exert a horizontal force F at its axlnts jl( sf.p9clyxl5k-.lq /1 e so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s ugb 0 iy9 869qtqmjpqvd9;n9q on a flat road is making a turn with a radius The forces acting on the cyclist and cycle are the 9nqviy98 jqu9 gt;qbp9m0q6 d 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 sling of6mh8k 5+ffczagnc* a1 length 1.5 m and begins whirling th n+fazmchc58 a6g*1fke rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 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 cylinder and her arms as thin ro-d*hybmo(qpm;+ zo1 ylh/5guds, making reasonable estimates for the dimensions. Then cal 5/bmu(dqmg; *+h y o-lh1zyopculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consistsbsptykam -7 0s4b5+k n of two cylindrical plates, of mask0yt7bkp5+ 4 s-nambss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 Fig.tn.js ; q8pj+q 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 of the loop without len;qsj p+jqt. 8aving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assry;5 5y1wxc )bw nib3kume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as tuumovp.94j u 3 e 5nn mjsgs2v/)/kj2phe car reduces its speed uniformly from 90km/h to 6u2s 5 )vm/o2p 3vsj/ kugpm unn.jej940km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(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