A bicycle odometer (which measures distance travelead ro+jk.h25x d) is attached near the wheel hub and is designed for 27-inch wheels. Whaa5dh2 j+ ko.rxt happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular ve t5q814tdq;)as0 vh(lumdm f glocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly,4u0q vgh5)tt a81 (l;msmfddq does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be describef6 rl3 b(,slnsd by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger f-n4*t+el)e4tj bmz e14 jwdvzorce? 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 difficu1v* g:hlm(9dhy s/ef ult to do a sit-up with your hands behind your head than when your arms are stretched out in front of you? A diagram may he1*sd( hhmgeufl v9:y/lp you to answer this.

A 21-speed bicycle has seven sprockets a +ubz uku7 sh32-rz,qat the rear wheel and three at the pedal cranks. I+ qs -z,b 7h32uakzurun which gear is it harder 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 f.zd+5vm br/eewvtw5+k9v ,u y mx9)mkast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this di9w,z+m.vvkv t/w m)+m xeu9 k5ybd5er stribution of mass is advantageous.

Why do tightrope walkertp4buu7 gu.+qs b7bzzkf (( v+s (Fig. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the net torque ,p lod owr/8rm78 tr)-z6t jlgalso zero? If the net torque on a system is zero, is the net fo ,opd)ljr tm/w zlg8t68 ro-r7rce zero?

Two inclines have the same height but make different angles with* dzcbt 9opy tzmtu8(yf5 e,2+ the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be grea* +t m92(ozd teu8bytp5,cy zfter? Explain.

Two solid spheres simultaneously start rolling (from rest) d4 uqgb0ia)hz t8 5we(mown an incline. One sphere has twice the radius and twice t8igqzbha )0 umet( w45he mass of the other. Which reaches 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 7/7l(sbgeb -fm* 2j-gk xq usimass. 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 totak-2b l7bqisjsgm-7e u/f* (gx l kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular moment4ee 3 p,wlshe2rcv:0 num are conserved. Yet most moving or rotating objects eventually slow down and stop h4p ,0w23eersn :vlec. Explain.

If there were a great migration of fpandg 0iqbv3*5/ n7dpeople toward the Earth’s equator, how would thig5 ndv37i*/d panbq0fs affect the length of the day?

Can the diver of Fig. 8–29 do a somersault without having anyt 6s,t6f1 cnctdcwh+; initial rotation when she leaves the b6+, hc6tc t c;dswtfn1oard?

The moment of inerti0xtbzf7 fcneo4p t5 y0 2w.9fca of a rotating solid disk about an axis through its center o7btf ofefz.4wy59tccnx0 20 pf mass 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 holdt+bgr7sc/o;focf(* 6hmh0d of e)y8yj u 8sj: ing a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity h;f78(0*je/h: 8f)ycrt d m+f osgsyc 6objuoincrease, decrease, or stay the same? Explain.

Two spheres look identical and , e pzb(u-qq5gdh yy)3have the same mass. However, one is hollow and the other is solid. Describe an experiment to determine which is whz (3,)5upgbyeq- y dqhich.

The angular velocity of a wheel rotating on a horizont7xdzs wgfi+xgenit b03( :; -ual 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 to xzt3ubg0ew7 xi i-( g;+sd:nfp of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on th-;:5tirze2z e /*avseapj+nwe edge of a large freely rotating turntable. What happens if you walk toward the ce r/ a+tia-evjpe5 ewnz*s;z2:nter?

A shortstop may leap into the air to calrn-o8. *oxx u ijy,2ozi-s)etch a ball and throw it quickly. As he throws thel-- ieo.yxrnoux8i*soj ),2z ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conserukez )- osy5wqk,w;4t vation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more w,kq-yt ;we kz)4uo5ws ays the second propeller can operate to keep the helicopter stable.

  Express the following angl,w(eh l4,ngi .y+ipures 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-kp+;fu4 risy u6u3gq5)rzpm of an amazing coincidence. Calculate, using the information inside the Front Cover, the angus)r34murup 6-pqk+5zfg yiu;lar 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, 380xo7-ppqzq)il 8.a bdj a38 k1qyf0dw4,000 km from Earth. The beam diverges at an ang8d.f ykpbq13ja0d x 7l8i4 q w-zqaop)le $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blende 0udio+f v1k5tr rotate at a rate of 6500 rpm. When the motor is turned off during operation, tok5df+ 0t1v uihe blades slow to rest 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. q+.tvva z 4o1 h+jwt kvr+y*k2If the ball makes 15.0 revola o k.q+4 vh+vk2w v1ryzjt+*tutions, what is its diameter?    m

  A bicycle with tires 68 cm i.hio8kljfod; 7+7sk;woiw . rn diameter travels 8.0 km. How many revolutions do the ;w idjfs87k; r7i holk .w.oo+wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate its k* nvq2(uq7l cangull 7u(nqk2qv* car 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 one complete revolution in 4.0 s (Fig. ;:,g3e:wzhboh )t gv9) y(u 4z wmqlxv8–38). (a) What is w: )( 3h9:;tzggmzlohe4)xy wbvq,u vthe linear speed of a child seated 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 ve)7 z cqzkp)c/klocity of the Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed h gp+m ;8geb(92f n acc3k)clmof 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 c3pbs:kd ymm h:h8* 5oparticle 7.0 cm from the axis of rotakcdom p53m:sy 8*hh: btion is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uz c:)co- pgyokwb xe ,;f:l/,eniformly about its center from 130 rpm to 280 rpm in 4.0 s. De c-; cyeglwzofb,p,k: /eox) :termine
(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 p1ex; pa jf fq 4a6f1ct12ul;9zvm2tc$R_1$ 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 Apql qgepqm9 g3 m7h77o3z+v q6jollo spacecraft put themselves 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 cylinder with a diameter of 8.5 m. D3q oq+jemq7 pqvl zg7mh39g76etermine
(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 uniforvbw79n/9 + 6gmqxbns dmly from rest to 15,000 rpm in 220 s. Through how many revolutiom99/ 7bvsx+gqw6nbdn ns did it turn in this time?    $rev$

  An automobile engine slows down from 45b9sdzx r-d 0msa3n/1 cq29 t+i9 cbrkz00 rpm to 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 testeda/ysk4 9iqb+f for the stresses of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final 4+y/akbf9 iqs 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 accp/5aobs luko67bh3hp,d 88 vnelerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the opv6pa3h57bb8n u8kdh, ols/wheel have traveled in this time?    m

  A cooling fan is turned off when it is running a +*od* p1tpcsxt 850rev/min It turns 1500 revolutp*d1x tpos+* cions before it comes 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 as the cphqbifs/x w4/6 2 i 9w dj:67;wubjnjzar reduces its speed uniformly from 95km/h to6 bjbq6 xu;/jnhw 9w sfw 2/iizj:7d4p 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

  The tires of a car make 65 revolutions as the car reduces its spe:lbp bq* lb)/ued uniformly from 95km/h t q*lpbblbu:/ )o 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 wmg1 hi3a8x7lu eight on each pedal when climbing a ha h7xl8i3gum1 ill. The pedals rotate in a circle 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 cm wide. What is the mef cp4xcw34d7g p (,gvagnitude of4x 4 gp3pe(vg fc7,dcw the torque if the force 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 F3* m.--p zkpniz,4c-fz i kpb)x4lmfwig. 8–39. Assume t- .lxmz-bw *-mp4 )pkkifnzi4, 3zcfphat a 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 attaches0 (x,kpdbykah)j z:) d to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held x( jzbsad0 k p,:)h)ykin the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an enn4 t ma;8ehvch1v/m t0gine require tightening to a torque of 38 ;hvam n4/mhte vct018$m \cdot N$ 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 inerti z+gp(4jn7d yma of a 10.8-kg sphere of radius 0.648 m when the axis of rotation is ty4+n(m jpdz7ghrough its center.    $kg \cdot m^2$

  Calculate the moment of inertip9 mx;.wca z2syt3-w2tva2x h a of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of w a29zs xc th2-.3vmwpy2txa;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 hrwlp1 a+,v sb:orizontal circle of radius 1.2 m. Calculas 1+pw arvbl:,te
(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 w7k8giy-hn u/ 8bfs pf3heel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and the 7p3/k8hbsgf 8 nuyfi- 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 jrn96;vcmj br (.nb9+zzz uv3 of the array of point objects shown in Fig. 8–43zv(9nbzzmr n3b+r . 6jj9 v;cu 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 consistsd7ve;oh d/x mklw,t8 hs1c6a3 of two oxygen atoms 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 cylindry f2ai 7v3s bk qzc+dd a0q(vobcmi20j--l)h(ical satellite spinning at the correct rate, 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, what is the required steady force of each rocket if the satelliteq2o03dybi qa vmc2l s(i-z (haj+07c d-bkf )v is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinnj kzs ( ia,tmsm(m)/0der with a radius of 8.50 cm and a mass of 0.580 kj(/,sa mms(z 0ktm)i ng. 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 wv, h:f5:p bmz8 wtmb1bat, 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 tangenti-bxtsw4 wju:g t3e:+ kally on a small hand-driven merry-go-round 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, -wbet: :wu4 tkg+3jsxand 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 rotating at 10,300 rpm is shut off and inns *sp)o26 nulwc7c /s eventually brought uniformly to rest by a frip unl *n6/s cc)wo7n2sctional 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–d9s8g,y )b m;.ooske v45 accelerates 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 foro-0r(rpmf pji4o3wi+:u ow3u earm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10o wuu4w r om0or:3 (ipipf-j3+ $m/s$ 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 co6al 7h x s g)xm+,/ni5aqqs01qs(*yrqwmh ; jrnsidered a long thin rod, as shown in Fig. 8–46 q lh xn/(rsy r1qa,50jwqgmq sx+6m;*a)ih7s.
(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 cons v/amr*m 1d1mikn3+xdists of two 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 frcd( cm8x/o id1/k5ef our full turns (revolutions) ac51efd /x i/md (kro8cnd 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 haidsv . poa.-p;s a moment 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 ;3 dgk 1e/ky iv,e8s.z hace4ha 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 ,(qfd0 8g0op locrg6m7/ vd4 ovtufeh xj4 +7jradius 9.0 cm rolls without slipping down a lane at 3.jefgt ohv( 4c7ugm6 oq v0dx4d/fo j+l87p,r 03 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the sum of/z *tcc x4( tpawo4xqlybe 631 two termsb6 (/xwtqxtc4 ap yceol341*z,
(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 net worot3 -qxtf 86k1,lcwm dk is required to accelerate it from rest to a rotation rate of 1.00 revolutik1f ,-38d mwt6l qtocxon per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.8h73ebx:q: k rp0 kg starts from rest and rolls without slippingh:xkr: p73b qe 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 balanced vertically on itscm+ni md4 oo6lh(u4- g id iqfe3il:76 tip. It starts to fall and its lower end does not slip. What will be iq43mg7e-o c:u6i hii nl l6 4m+ddo(fthe speed of 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 a 0.210-kg ball rotating on th oy 2hgw8a vmww-(b:l.e end of a thin string in a circle of radius 1.10 m at an angulb l(ghowv.w:mw8-ay 2 ar speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular m;,(bgxia/kh /+z9or:elhuoxc *; tzkomentum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotua l(r ihzt,kbzc9/gox; *:kx ;/ho+eating at 1500 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 rotating at a ratexxo8)l s*mv/3 yqq /zblr9f 2f of 1.3rev/s If he raises his arms to a b3rx v9/q2* /s)fylq8xomfz lhorizontal 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 one shown in Fig. 8–29)0dsnaw m38, u1cmx0 lzl;zzhr1*ddj; can reduce her moment of inertia by a factxdu0l zlh 08am,s1 dz3*jc1dn;zm;wr or 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 angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin :hye p1i.(ea ea1d/i zrotation rate from an initial rate of 1.0 rev every 2.0 s to a final rate of ph.e 1y: e za(ii/1ead3 $rev/s$ 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 rotating around a vertical axis through its p zxn+n-9ou 4zyt/ fw8aq9 q6lcenter 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, approxiw/n9u zozt f6 ypl9nq +48a-xqmately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure ydxh4np:* *fu: srk6zskater 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 of (v.uep 5nw iqj 2q73o(.rvrpithe 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 0 jcyk wvsb4cx;j f *c*+d7q(mof inertia I is dropped onto an identical disk rotating at angular speecksybx*0+( mv4qc*d;7 wcj fjd $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2pvw5ti.in9mp) w d764r ttve,.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 st: ala-n6en ,dmilcoif )d+ss+ fm (9,rands at the center of a rotating merry-go-round platform of radia6 9,rc+n fi-d: nd+l eo) amss,l(fmius 3.0 m and 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 rotating freely with an angular veekynqw5.9oi6 g*) faelocity of 0.8 i9fn5.eakoq* )we g6y $rad/s$ 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 cori ,m.hi* w+hxz*xw h(llapses into a white dwarf, losing about half its m(h r*xz,wi. wi hm+xh*ass 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 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 )u4xlqk+wno .5 v ahs-winds in 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 an3a,i22ojraq( l ek0u/wr*e$ 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, initiallyxjxm mvnj;wz8pu75 ;2 at rest, that can rotate freely without friction. The moment of inertia of the persw;mn2;7 pjjvxx5 8zmuon 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 at the edge of af04m xl:( p ubbw04rbzck g1o2 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia of 170 pkbc:0ouxzr2 wfbg44(m bl010 $kg \cdot m^2$ 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 the rope lying on z zp23f *pdvw(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 the person without slipping. What length of rope unwinds from the spool? How farp *wv(pdzfz23 does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side alwa)mypxdjsv qws )j i)1t/sq4(6 ys faces the Earth. Determine 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 y1jds) /qi))pst4vjmw( qx6 s a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a w(:m- b utqn g+icwc*rxs2h) 9rate 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 ky1s48o xv*w 7;qey lpshape of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval 7qos ylp wveky;81x4* at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of twoe ezx)lrm/429y-miez t 79nkl solid cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrm/e 4kx 979)iz-m2zerl nlye tical 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 the angular- + rjjj1*juisdy;t ;g 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 with mass 8.0 times as great, wer5aezr03ign y 0e rotating at a speed of 1.0 revolution every 12 3 0g0ayi5er zndays. 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 mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use ene40, f-f4xruu +-lgnayc6bc uvrgy stored in a heavy rotating flywheel. Sl-an0ffr +g4- 4ucyv6xu, ubcuppose 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 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 a6yet7z-gbp )5 x7f m ty2vpa4en $H_2O$ 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 surface at speesy:5(rbjtp2 in*6 a hfd 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 pivotx )zl f/fvyql6g )09cf freely (i.e., we ignore 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 oqyg f)cz f6f9/l0xl)v f the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on t8 :z5b brxktuhfh4g l zk397k2hsl2r 1he 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). bxhfklg2:zs5 k k2b71ul h43r9ztrh8The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat rke h6z(yf9/7u/of fewoad is making a turn with a radius The forces acting on the cycweye/ (f9/of7 k6zfuhlist 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 sling of length 1.5 m and be3aasd* 9xubt ;gins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest t3;ax*sbu9dt ao 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 c n8 mpe(/fi wv-dhl+/l4*cew 05bkyh wylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinniiw y/v/mwb0lc8*fle-kd nhh+ep4 5 (wng skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindrical plates,3vik3-*tmm.p vmr9va ) re3oi of m- vm9e3v 3i)ima r3*.tkomvprass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 radi ;n2v*rtwhl ,dj6bo t7us r rolls along the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if *ln27od;twtj v, r h6bit is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do )f1;x,so)immz9hhb x do20u wnot assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the cahm(cx uuhb2/ns8b. i6 b,hko2r reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular a/.hbknm(6hhb82 2bo,u ixucscceleration 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