A bicycle odometer (which measures distance traveled) is attached *jen-h1/ vsvili3r m6sj;oy, near the wheel hub and is designed for i 1yn *,h/e6mslv3ijsr j;-ov 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at conk4x+mkpo;wz - stant angular velocity. Does a point on the rim have radia-;p 4w xkmzko+l 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 either component of linear acceleration change?

Could a nonrigid body be described by a single value of the angular vb40dms(3z vi telocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Explainauxf -chnt 40ldg:1y 87 augd,fs3u:h.

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 k ;u22*gwjk5*c9uit;*gpn9j hyiexuhands behind your head than when your arms are stretched out in front of you? A di5hupji;gk g*t2 k *unxy*w99uce i2j;agram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the peda:u 4nzu+jb .xk*f ol;zl cranks. In which gear is it harder to pedal, a small rear sprj;.znb xfku +*l:uo4zocket 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 witur(4 +i9dmcsv -/ r6q41amytds0uekrh flesh and muscle concentrated high, close to the body (Fig. 8–34). On t yaqmiscer9dkr-r 1tu44+/d6u0v( smhe basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fic 7 tgi/h;4qt lmo,tl4g. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero? If tya 79mxj7 3gnxd)p6ubhe net torque on a system is zero, is the net force zeroy)7u 7n9px xb36 adjgm?

Two inclines have the same height but make different angles wijs+2 ugqq-s+s th the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? +g-+ss jsqu2 qExplain.

Two solid spheres simultaneously start roly mt-m1/ve/rrcmn80,hfa, if ling (from rest) down an incline. One sphere has twice the radiusrh/0t1vm /y-fimr, ca, nem8f and twice the 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 mass. They jsettc/i( d lsuf8c16zkaxnyg(b z .42: *6yfstart from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? (ycz*41(ylsig saftdk 2f6:nz/ bu x8jtce. 6Which has the greater rotational KE?

We claim that momentum and angular momentum are conservejm-o/w juyybr)/ms nm.+x o-2d. Yet most moving or rotating objects event mr// mb+)-yysx -jwj2.oomn uually slow down and stop. Explain.

If there were a great migration of peop( btd0hmk wi1i (/hisg :m,.nif:zo +mle toward the Earth’s equator, how would this affect the length of the d m(:m+dfishm(nb/i. i:, o0gziwkth1ay?

Can the diver of Fig. 8–29 vsb,w)3.cehqg3w n uid9dd:vz;f29ndo a somersault without having any initial rotation when she leavesdz ,: 2e3n s)d9wv.vd3; iqfcuhb9gwn the board?

The moment of inertia of a rotating solid disk ab9 gnxidb6nfe7p 6/ .keout an axis through its center of mass is 6knb f7.dpe96g / xeni$\frac{1}{2}WR^2$ (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 hol5s lpta4v2t7k5vny2i ding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay thevai4 t5 syvnk 7t5p22l same? Explain.

Two spheres look identical and have the same mass. However,).vwi:. myq slsp67e2 rd-v cp one is hollow and the other is solid. Describe an experiment to determi2yrqp 7s) . m -:pscvldw6ive.ne which is which.

The angular velocity of a wheel rota3gq-l6i6;u9l in4dzmsjds3 y 4hoo* dting on a horizontal 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 poinyzd 4*s6lmo3ol d6 gs ni -3;4uiq9djht at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the e7f1lp xbnxu v.8 gb:lw33omw.dge of a large freely rotating turntable. What happens if you:w3vfln1ml3xpb.8 gu 7.ox bw walk toward the center?

A shortstop may leap into the qdv,pbpg55e tem 1ap:lxk q*+(5t.qo 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 hips and legs rotate in the opposit*o xe5p k.e+mv( 5:pad q5,qqtgpb lt1e direction (Fig. 8–36). Explain.

On the basis of the law of conservatiiwvv.j 5eb5eqpa a22:on of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways j5.qv 2wpba 2a:5eevithe second propeller can operate to keep the helicopter stable.

  Express the following angles in radij 5;:spu /5i hoc3uzy,odhu y-1a 2sifans: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincidence. Calculate, usigr)kg 1f/5h6bcupz*+e vd0j fng the information inside the Front Cover, the angular diameters (in radians) of the *ub01 krgg6c)dz/j fhvp+e 5f Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Mo,ra*v fq2mybbmn* /:t on, 380,000 km from Earth. The beam diverges at an angle *ytbbq* mnra2v/f:m,$\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. When the m-45ffm h ngug.ls-bg.w6alm; otor is turned off during ope;gw4..ng 5gah-m - 6sb fllmfuration, the 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. If the b/1 kto aap-3amall makes 15.3mptkoaa-/a 10 revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutions do te ul0 1vowc.9vh9 .1vceluow0ve wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates a/9dvv y v883yl8mzbvele ,n5et 2500 rpm. Calculate its angular lm8 8l e vyvey3n5d89bze, vv/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. 8–38).bchx6g +hp-h; (a) What ihgb -6;xhhp+ cs 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 the Earth (a) in its orbit around the Suw )3un 9yvctrz+vlv) +lw rh8qq8w k3 v1aw(3mn    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a point.f evn76x/twyo,j2fd )1 n ,eotzym8 l
(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 centri,p o(gno6arx0 fuge rotate if a particle 7.0 cm from the axis of rotation is to experipnorgx06,a ( oence an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheat8d by15 s37.ubl -iubl9ue,at fa+yel accelerates uniformly about its center from 130 rpm to 280 rpm in 4.0 s. D9be57-3l. tuubf,+ail d1a 8 a sbyutyetermine
(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 pl)zzcpefi8t2 c97y ,$R_1$ 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, astron 1f3y 6qb zwpia8;iil+auts aboard the Apollo 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 pi3 zf18qybiail;6 w+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. Througc6w08 u0ocrl rozdva 3u2k1o :bn4-t rh how many revoluno2 8w-c uz crvotbr461aru: kld00o 3tions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calcula+x gp81898 uz(+k2ukjsggyd+rcg ckcte
(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 hi54 dde 2jrdf+jghspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolut d4r5+ jdefj2dions 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 unifvbaisu2/wp 0hd3,0s formly from 240 rpm to 360 rpm in 6.5 suw /ps3h s idb0a,0fv2. How far 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 850rev/min It turns 1500 revolh9eo krks/-3 ctg;cc3utions befe;htkgc9 3 rs3kocc-/ ore 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 car redu;4 q si:i;:hgp um17nnewz-y dces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.7z4:pdi1 mwqn;hs-;gi: ye nu 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 revolutio:/ ghtb; nkkwahz xlr -7ujq18 78py1ins as the car reduces its speed uniformly from 95km/h to 45km/h The t: inlkk7/1xzh jbu q8p-;rga8t7w yh1ires 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 climbi8 myz( siuadfo2szjb0ux;;3 *ng a hill. The pedals rotate in s m;i zx(dy2fz;*sjbu uo0a38a 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 i 5omhbs7m vi: j7 qq nv:*qmse(*qql/8s the magnitude of the torque if the force ilh/j b7en:qs qq:v7m*is5mv* om qq 8(s 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 n9o7 hus7k 9qveo5de,b qz8 6tFig. 8–39. Assume that a frik87uo t6, qe9eno9hd 75z qbsvction 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 .8yqqv o xn/-rwhich pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then relea v8x qo/-nr.qysed. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening to ax)ogok1e g7*eeqx zho56rk* ) torque of 38 kgze * ox)*ohrg)e15kqe o76x$m \cdot N$ 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 radfyz e83nd c-8j6iz .r*rd7 ugdius 0.648 m when the axis of rotation is through itzrg3ez-*j cd. unr87ydd8 fi6 s center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm e8n)( tmw whm38l m8mbin diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignorww8nh8 eb3 )mm8(m tmled (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in w4rp4q j2r f4 cs9nhn5rjpo:2a horizontal circle of rorh4rsp2q 4n9pjjwrcn 42f: 5 adius 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 cp )z1;t 5ei .ffsaz9kh7mdef4, m 6dovonstant angular speed (Fig. 8–42). The frictitz)sk 1 m e;7fodadpi95.,4fezmv6f hon force between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point objects shown 8,en9nlxd 4g) 5vbv3*avczpcin Fig. 8–43 abcp)lcbvv d4na8 gx53,*ze9 nv out
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen atoms whose thkv) )q 2yw9gjotal 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 cylindrij8le . z9wzo6l,gnx5gpn7.slcal satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 36p,5w .7xj s gl.gnl69 no8ezzl00 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylifpjph1g 6x ) wrc qa4uv k6r62:..usadnder with a radius of 8.50 cm and a mass of 0.580 kg. Calculatexud6).w:psjkr havc 1 pa 2g6rq6.uf4
(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 bam8jvj8k-;us gofr9i/v zpn/ 6 t, 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 hand-driven me/0t sbsr6vzg( br73 .xy6tv9y -br raurry-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 mxrb0rt y s/g9(-vz36vt6 byubs7r. raass 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 rotating at 10,300 rpm is 4no v-14kvd igshut off and is eventually brought uniformly to rest by a friction-v dik4 ogvn14al 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 a 3.6-kg ball at3iku6:iy ez sl /zq1h9 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 :k p 47bu5u+gz m sg;jsafh;d-thrown solely by the action of the forearm, which rotates about the elboszadhk j;+: b-;4gpsmgufu 5 7w joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, as shown in Fig. i-fdu -1uy se+p8yv;j8–41y pdyf+i-j8 -sue uv;6.
(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 masses, u jtl(n52v9b3mjnl/-xnx1xb $m_1$ 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 hamszwsvje. 5bx; am 07-dmer from rest within four full turns (revolutions) and r; xmv0ssdb57.e w jza-eleases 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 iners f/u7 jdek+/ntia 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 deverf-0:,pgcmztxlp.un q jn: -)lops 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 9.0x+v*qp . tm;;8ta x,ujhmlz/tuw:ed- cm rolls without slipping down a lan hp u8 ,- lm.tdw:m;eu+jtazv /x;t*qxe at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of thg8vtfm5 n8*c c b,v0xze Earth with respect to the Sun as the sum of two tex,*gm5c nv bt0 zc88fvrms,
(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 muchz2 gh8l1sg9 rkjgyczm-- ay7 )cdw09z net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a-s c g19lkmj cay09g -2dw)zyg7hz8rz solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls wk,qd-+gb sn b4ithout slipping down a 30.+ d,g4qkbn-bs 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 it0so*di d5kld/s tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end ofd si5*ok/l0dd the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum o;x .x+ii(96)cfcvou s x7fmdhf a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an angular speedx;6uchdfvo ifc m.sx(x 9)7i+ of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grinding wh l6,i* vzdae/bbann,c6a -za) eel of radius 18 cm when rotating at 1v zb )e,6,* ac-n6anilad /zba500 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 g+ sfrl,;sy9b his side, on a platform 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 decreases tl+ gsr;bsy9, fo 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) czi7m*xj )6g1w9ry dnqs) 7(uc gvjux;an reduce her moment of inertia by a factor of about 3.5 whencg s9*wmixyu ) x6v7(g u1j)nz; rjq7d 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 izu h+ ;prx t3bma66woqr,(7lincrease her spin rotation rate from an initial rate of 1.0 rev every 2.0 s +r (t,m6urxi;6azpw7qh3l ob to 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 rotating around a vertic ,c21nebfr9q9,3cpav /fxmk t t y4+dwal 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,fkpr9fa,dm+13xwc2ttqcy /be4 n9v 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 skater spinnii h,t+3q n.rhuevb)2t ng 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 the Ez:c h9,.thndm*/ hzvoarth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is dropped onto 0y cf ko22nd:aan identical diskn:oa0dyfk22 c 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 2.4p 0)c/wxtm x2d $rev/s$ 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 theos0bt f .we(0m center of a rotating merry-go-round platforse0b(m0f.ot w m of radius 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 anfrfk0+szj6 -jty c)y e5k oj *+4baw,hgular velocity of 0.8 6jjyjazrhc)b*of 0+ skyf ke-,wt+54 $rad/s$ 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 white dwarf, losing abw 0bnks;37q 40* fh,hd+ilqqnw;c lvpout half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momenqnf*bls dn,4hh w ;3 +0v w;7q0lpkiqctum, 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 involvedemtk 3(ph)f) 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 m9dbm d47r gec)c4l xmje* o*($ 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, initiallyagi qm)*lghp dkga g, 4t2-2.h at rest, that can rotate freely without friction. Td4g2qpgaih. ggl h m)a- 2k,t*he moment of inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of + qvq*jm0s:;v cuz.hia 6.5-m diameter merry-go-round turntable . uz0sv;hqv+c :j mi*qthat is mounted 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 ) lt6c*s *7ac)ih e7vrcyuiw.*l0mueof the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spescy)waclhr06lii umt 7 .c **e7*vu)ool? How far does the spool’s center of mass move?

  The Moon orbits the Earqr (e qnc-u ; am7zfh77tckr(-th 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, trqar7zc eum -hrtnc7-7;fk(q (eat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate ozc*fo2tu- * 7+yfz lpqf 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 pjfgkz1+jygo(b : 3*eof radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interv*: og31kyj bpzfjge(+al at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cascx(anep w9+n4 c1l -ylindrical 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 diameterc+e1 9 c4x as-(annwpl 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, hoi-q7s5wdjl// qce cfb359 ugx4c- qdomz l3h4w is the angular speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times as great, wkoq(t.u . n-y93xfuh4wym 2kbm (imo , ,mmsk1ere rotating at a speed of 1.0 revolution every 12 days. If it were to .nu4ximoomm k- , ktyk. , 9m(sh31bm2yuqf(wundergo 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 energyhv y,mxj;u 35o stored in a huv3xhymjo ,;5eavy 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 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 illustrateq7gq6n0gj(qj s an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rollinguocq36h+ g3ef 4l1tbf 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 freelk :en3ty5 jpgm y-/qqkh /07kly (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 acceleratikqn-p/ t 0hj kkmey7y/ql:g35on 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 radius R. It is5b ttybfdw;-8p uc0 d:: kt.jv standing vertically on the floor, and we want to exert a horizontal force F at its a8;dj - w:pty0fd u.v:k b5ctbtxle 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 on a flat road is 3tel 8fr/9hjxmxv gz 3j2f /)zmaking a turn with a r/xe8)zzj9 l2jmt3x v3fhgf/ radius The forces acting on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg 6m+qdwvn ;7ulzot ;3 mt6dt8drock into a sling of length 1.5 m and begins whirling the rock in a nearly horizontal 7ud w8om;m z+dt ;tqdv 636nltcircle 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 royl r6osb+h s 17wrx 9.amm+:ibp11wuz ds, making reasonable estimates for the dimensions. Then calculate the ratio of b :bowrhlu sax1wi.1+6ms 1p7zm9yr+ 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 consists of two cylindrica5xt r t/ds*d9f:sryd4q mh45rl plates, of mass /r5f:ssmttrdd*dh 4 rq y4x95$M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 rs s. /h7xrpk7oim4u0c4tx 8cfolls 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 of the ls7.fckr/mp 7i c8 hxs0o44uxtoop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not ah/w-s0vex j33f zd1 massume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed unif j lglro+x.i;pmw4fr6)y5m )n ormly from ll6.i+n )pymr of5rmwg x 4;j)90km/h to 60km/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