A bicycle odometer (which measures distance traveled) is attached uu 46kkndjfrc oz*yfz2 4(q/l(ax*4h near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inchzqr cu44f*o* kyxz24 nuklj6ad(/(hf wheels?

Suppose a disk rotates at constant angular velocity. Does a+tmstdn.0 xff yu128m point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, 0 t8x+mn df f2t.1umsydoes 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 descr-apmj82j4s szzmu0 ;yibed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater tord *5j0 7t 2xvbhxlip9j8qbb7 lque 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 diffim aviq56fyof j4t whd48t14l(cult to do a sit-up with your hands behind your head than when your arm44(loyffv6 aw1 dt4mh8 i5 tqjs are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rg 6;sq8idwn+v+psnsen n5, h4 ear wheel and three at the pedal cranks. In which gear is id ws+sipn+esq4;gvh n8, nn6 5t 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 fast have slender lower legs vhs5rmf efl80 5k: 2b kt 2jn;xpssp;)with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explf: kpl;fr; 5ssbn xk2)ejv5mp08 h s2tain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carrtp9p ;xf8 7hz:nrht)xi ja +c9y a long, narrow beam?

If the net force on aq0ikhicf*/tgj ;m x(3 system is zero, is the net torque also zero? If the net torque on a system is zero, is the net forgfhq*i/kx tmij;3(c0 ce zero?

Two inclines have the erdx-l.x 8)1q x(cxrvb(-russame height but make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the rsrv ( xx( )lqx18duc .--xberball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest)i6*b tzak7 :6(lfouvh 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 i:7(6zaufvo6kl *h bthas the greater total kinetic energy at the bottom?

A sphere and a cylinsr0e. (4whaf(atr5w o n uw(y7der have 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 n7tw.f(4e(ray wao (0w5h s urbottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum ardxv/v-olf p,: e conserved. Yet most moving or rotating objects,o: /pvfldvx- eventually slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equator, hvam 7hct *j:l2ow would thialm 2tj h7cv:*s affect the length of the day?

Can the diver of Fig. 8–29 do a somersault without having any t 9 b3w,6fxmrl: .d1eftqr-pdinitial rotation whf:r pl .wrqed1bdftt m6 3x-,9en she leaves the board?

The moment of inertia of a rotating solid disk about an axis through its l sr1ftph)m1*v euvx * ef)w;9center of 1v*;ehs*x e1ftvur w)ml) 9fpmass 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 rotakmak/-/7 v jhrting stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.a j7 /r/k-mvhk

Two spheres look identical and have the same mass. However, onebqx-0+. /h8a7 -xb t /(h ekqphfmoora is hollow and the other is solid. Describe an experimehr p-bx f-oehkt(b/ q7 08ox+mq/aa .hnt to determine which is which.

The angular velocity of oogjk+ z7q/ /ma wheel rotating on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angul o/7kmz+o qj/gar acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a fftwvr-:9+ykt9n )q *sttg2c large freely rotating turntable. What happens if you walk-+tvf9cnt t*t:gr2 y9 )fqk sw toward the center?

A shortstop may leap into the air to catch a ball and throw it quiqc+s8oi2t pg; m8uiek s,bnw81+ .hqjckly. As he throws the ball, the upper part of his o+ ,8iwqi e1 2s+t uc;snb. pgmjk8q8hbody 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 conservation of angular momentum, diszzh3f(av(v: rm 7kzs5cuss why a helicopter must have more than one rotor (ovzk(hv3mz5rs(: z7 far propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles ina hu :dz7(.nsz 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 Eart jo0uh: we( r+34mirmih because of an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Eart+(i3 u4jimhr em0:wroh.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Mrr :1 d,tv*xptoon, 380,000 km from Earth. The beam diverges at1pt: rdt x*,vr an angle $\theta$ (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. tm6s4 i hdeecs g-.00yavhi*84 kj:ohWhen the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow dowtsy egkcij.v 6ai dms* h0 o048h4he:-n?    $rad/s^2$

  A child rolls a ball on a lr.coojgh +w:; w4/mcp evel floor 3.5 m to another child. If the ball makes 15.0 revolutions, what is itwomgo:cr. 4h+jc pw/;s diameter?    m

  A bicycle with tires 68 cm in di3twkf + f+a5ifameter travels 8.0 km. How many revolutions do the wheels ma3+ 5it fwfka+fke?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calcaf 2bdnf l3rvt10;yz/ulate its angular velocif123b da ;f 0tlnzv/ryty 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 comple:3gxj-un anp w 28hv;)lfbi s/te revolution in 4.0 s (Fig. 8–38). (a) W/nl h-x)gasnj: f83 vup;2b wihat is the 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 velocity of th 9e+.mmwjt ifyf-64g(b s7gi qe Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear spe6j, kkz irj*8ped 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 particle 7.0 cm from the ax8+7 w3 fylxrfzis of rotation is to experience an acceleration z3w7 y8f lx+rfof 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130vx r-kyh2:h5znl3g w 6o 5ep+v rpm to 280 rpm in 4.0vn : eo356yhkg vhzl-r+2p 5xw s. 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 radiu b-qo/4 u2h rw9encq9ks $R_1$ 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 pub chq ;ocj5v* )9bb;yzt 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 5c);c byzqobh 9bv ;*jcan 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 unuyr2t5-cd xpl8. d2vfwf))u fiformly from rest to 15,000 rpm in 220 s. Through how mtw 85) )frvypc 2udffdu2-lx .any revolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 -zz 4 54)cpbr kvr;fy9kdbj1v 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 tested for the stresses of flying highspeed jets in a whirling /k1 p kfi/xoyzn0,+ qw“human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before rqzywkn,0+/ 1/okipx f eaching 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 2.t+ ln5(8,qfv41j yuq gda vwb67joha 40 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have trj 8y j 6uq.5(vqv7gfnloa wh1dt+,ab4aveled in this time?    m

  A cooling fan is turned off w;aw,:j6ewigrx83 lk7 f;r zswhen it is running at 850rev/min It turns 1500 revolutions before it comes to awrl:i38rkefjsz; 6w,x7 gw;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 mr4v;.1da v 8o fg7b sa h,+xw.0dscxpkake 65 revolutions as the car reduces its speed uniformly froa h. ;wovasd1,7xb0r+. g f4 dkcps8xvm 95km/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

  The tires of a car make 65 revolutions as the car rz;/-4rq srq peeduces its speed uniformly from 95km/h to 45km/h The tires have a diamese4-qr q/pzr;ter 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 aliq9z+ozm eyofrx- ),5l her weight on each pedal when climbing a hill. The pedals rotate in a czqx yir,zf9)o e+5-moircle 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 w.ex0gv5a m(o a door 74 cm wide. What is the magnitude of the torque if the force i wxv.ea 5om0(gs 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 o2 ) aapv5adk:of the wheel shown in Fig. 8–39. Assume that a frictaakpdv :oa52) ion torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are w xrc6aut 2ev947:ehy attached to the ends of a massless rod which 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 torque on th 4uyhxwt arc9ve72 6e:is system.

  The bolts on the cylinder head of an engine require tightening to g1o-1-l6 4 d irsklgmwa torque of 38 wg-d64moi r11ll-s g k$m \cdot N$ 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 of6./io cziz-vinx)1x e inertia of a 10.8-kg sphere of radius 0.648 m when the axis of rotation is through in-xiv6i xi z zeo)c./1ts center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. The9ztj(bh8bkf :)f p*ag rim and af*k)(8 b:bg9ht pjzftire have 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 thin, ligy2ehiy 2 wjq8gux,m8 9ht rod is rotated in a horizontal circle yu ,8g 2ym w89xhj2eiqof 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 wgd oro ua;z 4:018 eva-;wkq 9e4muvtat constant angular speed (Fig. 8–42uavkoe m;w8rov :wq-z194u4 0dte;g a ). The friction 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 ofqyr6 a.ue4t3rbvr05 h point objects shown in Fig. 8–43 r64r.r h35a0y vt bqeuabout
(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 whosz ,qw0all,s3 ;4fmfn r)z 0 /cp-bfnxle 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 ay 7,gqw ysywj7b v/d5+,) ivput 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 o5j7 ui7+ d/wvgqy s)w v,bpy,yf 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 cylinder with a radius of 8.50 cm anddxx60)nbaqta 9a 3zy* a mass of 0.580 kg9d0)tn3 xaxa b*6qyaz. 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, accelerating5li;avtu,t i3 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-round anruhjbg g)-stk.-* 2 mkgpj,t 2np zd-8d 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 edge. Calculate the toskg2dp -8 - kgb*j un,tz t2h)pj-mgr.rque 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 5z6 r0*atvxig6.vj fqshut off and is eventually brought uniformly to rerg aiz*qf tv6.065jvx st 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 accelerates arce3m2 t;m-xez.3tth 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 forearm, x +63e; usvnp 46sld+eup*hbowhich rotates about the elbow joint under the action of the triceps mel +b d6;xp+se36 4 hv*uuponsuscle, 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 considered a long thin rod, as shown in F*mvb66o. jsr eig. 8–46..eoj66v mb*r s
(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 :oi 7 w-g2exopof 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 from2kt:d1 js o-b1xgfia 5 rest within four full turns (revolutions) and releases it1: 1b2dk fijosx- a5gt 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 ier 2hs4rk +ry)nertia 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 torquzi 2witbyyk :+:fu 6,ke 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 slipping down jy6sf 2jlw3c-q , *klva lane at 3y2sv6 jq *wc-ll,kj3f.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic -)u:acnl6,z 5paig )ss pao)ol7r e(penergy of the Earth with respect to the Sun as the sum of two terms,a),n saocu r5pl:s) -76zlo pgei)pa(
(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. ix60ypoge ofk;;n65j How much net work is required to accelerate it from rest to a rotation rat e56xn; ;jf 0o6gkpioye 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+::lub wedgzz sw0.bbs( o:z; g+-vah.80 kg starts from rest and rolls without slippin:gbbz l+ov:+ss ; ab. hd:-wwg eu(0zzg 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:*b *yy(x pulo balanced vertically on its tip. It starts to fall and its lower end doe**ub(xyoy l:ps not slip. What will be the 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 -rf4y/ it ls2fof a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an angular spe /y fr2f4itsl-ed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindricc h u1evho /88+biq2caal grinding wheel of radius 18 cm when rotating aue8 8q+icahboh 1v c2/t 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 2ttuxm bm:u ( 8cuzs+2that is rotating at a rate of 1.3rev/s If he raises his arms to a horizoncz8 sm(u ux+m2ut2b:ttal 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) can redcw1yyj, 7 y-hxxl)lr ,uce 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 , l7x,h1)yjcxryyw -lin 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 ratewdkh) 4mw(tb3 of 1.0 rev every 2.0 s to ak4(d w)w hbtm3 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 rotating ar2 (ku4no;wt) mh z0ufhound 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 cu (o)24t kh0fnm;hu wzlay, approximately 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 p aa;(s0 /j(t rl5-jo2a wbmr ;gn0fpra figure skater 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 angular7e-lt u: fxs5y momentum of 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 ismy. 8 dkzqcu*( dropped onto an identic z8dcmu yqk*.(al disk 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 2.5mee2l yi1 b,j4 $rev/s$ 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 ce1 h ,jwuad61egnter of a rotating merry-go-round platform of radius 3.0 m and moment of inertia e16w1 hjg a,ud920 $kg \cdot m^2$ 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 witd(b); mbmhv(xh an angular velocity of 0.8(b h( xdb;)mvm $rad/s$ 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 collapses into a ift e3i4m+-xtwhite dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assum 3+tm-i xe4itfing 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 winds in qt8)ti c77thjexcess 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 cq)9+vhcz) tb$ 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 ahe(b/,t(ensb platform, initially at rest, that can rotate freely without friction. The moment of inertia of the persb bee( h/n,t(son 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 ofptj;ass5t pc (-vh/p6 a 6.5-m diameter merry-go-round turntable that is mounted on frictiontpc(ha; t/ 5vsj-6sp pless 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 ropev szvm--) hdi8-1 buoyv7 6:wk)nooa u rolls on the ground with the end of the rope lying on the top edge of the spool. A person grabs--o vhdw v )7)uo1vkmszi: oabn86u -y 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 far does the spool’s center of mass move?

  The Moon orbits the t9dce5ccpb v1ndih ,(hr6 )+jEarth such that the same side always 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 ctcc+6e,h(ddbn1 j5prv 9)ih a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates froady:okh s8cl. h(q:x+m rest at a 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 uniform cylinder of radius 0.2qg2(k y:(c9wfc(-k)-vwq r4kztwb2 kd tf0m q0 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque d2-4mk c)2ftw :fgvw kqbtcykz 9 dq(w0r q-k((elivered by the motor.    $m \cdot N$

  (a) A yo-yo is made -;3iaqabk o5: i3rv du moex,:of two solid cylindrical disks, each of mass 0.050 kg 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 th;- miq3b ku5 viox: eord3,aa:e 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 speed of the rqa. q tx;1 .wie0t7jiyl)ff5 zear 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 Sunm:1lr/nrh-e5 8dyy v7qhn +ba, but with 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 star is a uniform sphere at all times, and that the lost mass carries off no angular momentum. rl m/ e-y1q85hn advb+hry:7 n   $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for itv6bvyx(fmsg q+v s/ ,nfl(m9 ;v,4jyy 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 n vy ,bx (j mf+yvyq6sm/4(,;gl9svfvto 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*(5 frvfkoc .k 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 (ho)guq 6oa63imp op) is rolling on a horizontal surface 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.,4 n9dysnt jce 9c;gv43.kyp0u we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod i4 . nnuvj99p0cysyg3ct ;dek4 s 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. [Hint: See Fig. 8–21g.]

A wheel of mass M has /qoc zy pltr5f9vr1;. radius R. It is standing 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 lyoz1/; c9p.5ft vr rqh, where h

  A bicyclist traveling with speed x82 at2dj9/y wulwtp+m+ 6myjv=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist and cycl6t a2ywmxw2y/l++8 m9 pt dujje 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 lei30+)w 0ya(nj gnqysnngth 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 5.0 s. What is the torque requir30jn a)(i+sn wy0y qgned to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solis w9htz5g,zk e/p3b3aa; wv8 rd cylinder 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 tightlywaspkzah ;rvw/893gz,53 e tb against her body.   

  You are designing a clutch assembly which consists of two cylindrical platesxzu2hm b98/jw , ofh 29zwubm8 /xj mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 ra .l9oh)re mlbm njb6/gc0i82bdius 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 it is to reach the highest point o9oe2 nb8clbbmj0mi h.6r/g )l f the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not a gft0+bc0xbd4z .,e lhssume $r\ll R$ h =    (R-r)

  The tires of a car makeu/-+vn q z viht.(lo 3k)uin3uq0b1 ak 85 revolutions as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What in /3qb u.3 kk-a) viu +qt(lvznh01uowas 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