A bicycle odometer (which measures distance traveled) la.n hq* vjmfn18:p:ais attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-iap j.na1q8ln:f m* :vhnch wheels?

Suppose a disk rotates at constant8 ,q8g 5jwlfk ,axd,ny angular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angul,8 x,8gny l daf5jwqk,ar 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 angulafu:vx2+ u8pvd r velocity $\omega$ Explain.

Can a small force ever exertk0 damwt 3m-7 x :(r9faeioou( a greater torque than a larger force? Explain.

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

Why is it more difficult t3ds v.-mxma7m42vdxqc fze+ l qi h77:o do a sit-up with your hands behind your head than when your arms are stretched out in front of you? A diagram may h7qezc47+xxdfs-:aqm m.i d v23vlm7 help you to answer this.

A 21-speed bicycle has seven )bpgo0kl ;-mfo 04il ssprockets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprock0oo mlp4 )k gb;-0slifet 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 lozq,; o d2cw9 yxk5ejt n9//jss/f9n:jiobz3 awer legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution oo;9fbdzn s//nojtj q2i: swz, /3cxe9y5ka j 9f mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carryncjiw9 )(e.ds a long, narrow beam?

If the net force on a system is zero, is the net torque also zero? Iok2t;1f iv xj3f the net torque on a system is zero, is the net force zerktv ;fojix32 1o?

Two inclines have the same height but make different angles 05a-0)oqbj mw ett+ibwith 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? Explain. i0+tbt)m a-bq 0e5jow

Two solid spheres sim 6l.lk7 hcqxx9ultaneously start rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the ot 6hl9q. xk7xclher. 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 masz lx nkqq*;g)yod(epw,r58b9 s. They start from rest at th*;g5qywk rl(,deb8zpn )x9 oqe 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? Which has the greater rotational KE?

We claim that momentum and angular momentum are cons1;b -lei2o muserved. Yet most moving or rotating objects eventually slow dow-1i ;l oemus2bn and stop. Explain.

If there were a great migration of people toward the 9wf*d6w,k -zm:fiqu) w ha7wa Earth’s equator, how would thiwa h)m*,u6i9fwzww7f k- d:qa s affect the length of the day?

Can the diver of Fig. 8–29 do a somersaulttd)0o 0o euper2q. u r;2kgu8d without having any initial rotation whe.otuue2 rrk duo; q)eg200 dp8n she leaves the board?

The moment of inertia of a rotati;ctn- 2hdffb -nwhjf3 y7 .7rrng solid disk about an axis through its center oftr7-d7 .cb;f2hf hnyf3jw-rn 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 holding a 2-kg mass in 3);m f)sy* j9f65jouwrykd xpeach outstretched hand. If you suddenly drkm fx*j)9; dfypr6u w5)so3jyop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is hol:tnj,q ,+(dmbrtf7txe 8vs,elow and the other is solid. Describe an experiment to determine which is w( d ftqt+ 7ee:t8,,vnxmbr,sj hich.

The angular velocity gi3bo)cjml4rg:2-is 53 ssvvof a 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 angular acceleration points east, describe the tangential linear acceleration of this poi )rbvj54so3 sg2ii3 :svg lcm-nt at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on-s*4 c -pycra3o pnlj) the edge of a large freely rotating turntable. What happens if you walk toward the center? n-clp*js-p 3a4 co)yr

A shortstop may leap into the air to catch a ball and throw it quickly.pm 3* 3p3 m;gahcxq5pc As he throws the ball, the upper part of his body ro53;a3g m3mh cq*pppxctates. 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, discuss why 0mgztk9b41 +v7rbur ja helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep7vbr k9z+140gujtrb m the helicopter stable.

  Express the following v6wi.pr 4zfo; z-6obq angles in radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth bex9cku iq+n- 3kcause of an amazing coincidence. Calculate, using the information inside the Front Cover, thikkc-q+ n3x9ue angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beam div:od19v+o qj+ to8. .jgmkhgsterges at an j k.81d+ moqgvghj.tt os+9o: 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 rotxyp6 6mseeuhv w7l :7*lp 0b+ yj(o+zs v04qtrate at a rate of 6500 rpm. When the motor is turned off during operat: 6 evq0 lz(vjs7botwrl y+ u0*xheys7mpp +64ion, 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 b9z s;hu)wqcc w15 lrk2ec( 4ulall makes 15.0 revolutions, what is its )cwl1zec 9uw4;q5 ru(2ck lshdiameter?    m

  A bicycle with tires 68 cm in diameter travels 8gv.s. j zl;b7 2uy-/wzj ale6p.0 km. How many revolutions do t.w gz.yl ujl7 - ;jp2s/beazv6he wheels make?    $rev$

  (a) A grinding wheel 0.35 m in l-ch h2)+0kskfb+ n,c(i rhpidiameter rotates at 2500 rpm. Calculate its angular velocity in i r,lc0s)+h pfhkk-h+ nb c(2i$rad/s$ $\omega$ =    $rad/sec$
(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/kgk. 5bt89l 6lmhjdyqn1m r8 in 4.0 s (Fig. 8–38). (a) What is the l6k hy1 5gd8mb/k9 nrqm.8tljlinear 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 i(bslgks *ig rc(e p/sj- p5vfgr439i7 ts orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speedzvz bu*c4r 8:c 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 centrifugegot3m+:fi 2y5cd2j 4wza 2fen rotate if a particle 7.0 cm from the axis of a3 2 dn:mi4w+je2f yzctof 5g2rotation is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates unifc 4 02nz6k9hb-pe *9u(fxcrgoiz q(rn ormly about its center from 130 rpm to 280 rpm i4zrp-f0irk9hnnco2((xuz*qg c 6 9ebn 4.0 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 radius tagkl)bkninf ) o 91,kih(:k5$R_1$ 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 th3 th7 r+bkauoy35 s4tz +nal8te Apollo spacecraft put themselves into a slow rotation to distribute thea r3to7ulk4b85 t+n+ syh3azt Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates unifon6xoc93c 4l)9d scz5 e m+stnlrmly from rest to 15,000 rpm in 220 s. Through how many revs6e9l)3z9om+cns5 t4nd c xclolutions did it turn in this time?    $rev$

  An automobile engine slows down from 453;rmg:1:ln l,m9n9mqbaian p-qz b u*00 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 highbce.hq 8e(f 9w*jw3ls kpp00espeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final spe .lfpj 830scw eh*ee qp09kw(bed.
(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 accah/k2deg euk 7ct1 9a-elerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled in this time? t1 c kg97/ued2e kh-aa    m

  A cooling fan is turned off when it is running at 850rev/min It turns 1500jrc )pb bpiexlul 1t,).m2q,1 revolutions before it.1up 2lmj),cixqbt )r lb,p1e 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 reduces its s l72in y-0fv+beyk-azg d6ao/mob65a e5cg) apeed uniformly from 95km/h to 45km/h The tiresml ad z5ce/ bn+v62aoa-gy g e5oib)67f0kay - 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 reducex :r93tg8vobhc *w2l 2u0vqwas its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.82arc8 3 9bv0 t*xuvo2wgq:hwl 0 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 we:8 +3joozfm(dx5q)yx2wii i eight on each pedal when climbing a hill. The pedals rotate in qmiix5:)8 d32 ofwiy(x+je oza 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. Wh6 uxpojw,zy jve::k u15m,:fo qmdn) 3at is the magnitude of the torque if the forcv) w:f:oj,xz kjp,um1n6d u3y:5m eqoe is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fig. 8–39so*/ (3pqlerf . Assume that a friction torque *e 3sl(qr/opfof 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 ofuo/(dn6hh2/e6)btre/ nj w6tf2we6fn-q cw n a massless rod which pivots as shown in Fig. 8–40. Initially theweu 2(feqhwct/d j 2n/o6f) ebnh /6-rnn6 6tw rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of anevxw8hi g0+t 3 engine require tightening to a torque of 38 x ht+vg3we8i0$m \cdot N$ 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 ofo1y *sg;j+sioca m ,1j inertia of a 10.8-kg sphere of radius 0.648 m when the axis of rotation is ,smo;+oyi1 jaj *s1gc through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle whe febs9c9 hde-3-e+ +nxhvi7 mzel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The masshc9+f ee zen7s- mv+3bdi-9xh of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light qhe 9s8f1d)39aofw0i6 ml * thgmd:dg rod is rotated in a horizontal circl*d1d9 6ml:9)e3 d t m8aghgqfso0h iwfe of radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating at 1;7*zw17(5 rjfw42tgszgb 0l a orqyuxtti 8pconstant angular speed (Fig. 8–42). The friction bgwrt4 z7ofr7;l5(2pat*01zyws qiux j18t g 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 6x dv,-30i z 65jyj 5gih yyqln+i/ehithe array of point objects shown in Fig. 8–43 abouyei+h6iqi 3/l,jxy 56d-5 iz0gnv yhjt
(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 u+k)2qog3ly v 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 cylindrical satellite spinning at the correct rate, engvy*3z d-tnv-uineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of*nu3vzvdty -- 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 cylindfl y/g069. n+6iq z lsgk;fybber with a radius of 8.50 cm and a mass of 0.580+ll ;b / nffkby.6szygg6i9q0 kg. 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, accelerating it fro 1eqego1g h0 m-:fp/ug);d+mh4m(o* bs mok dgm 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 smajgt18l 2y-ic*)zvix s ll hand-driven merry-go-round and is able to accelerate it fv-i)c8yl ijsg1zx t2 *rom rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpvgju .v,c(oe 0vo*2hvee;yjcb1) (le m is shut off and is eventually brought uniformly to rest bb )eco *yc;g ,2jee lejv1.v(ou(hvv0y 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 a8 kxinqna uld-2+;x fy, 27pho 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 ty: qc rb97--/johu bvzhrown solely by the action of the forearm, which rotates about the elbow joint underj-qvy:b 9h-r7czo/bu 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 consideredauev-ky x xxz7i, 9v*qs01;ta a long thin rod, as shown in Fig. 8–46.*yaxt x, ua1v0se;kqvx 7-zi9
(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 ma l+ami uhz/8 mcu+*ps joe57s.sses, $m_1$ 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 accelerat4(rw: ja.3 awcu:rc/xsc ok-ues the hammer from rest within four full turns (revolutions) andwa-: u/rko4c 3 wjr( .cusc:xa releases it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a momen-mg2nu((e obbwnb 8f0 t of inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque of6y8:.8jqxzw kqyelx1 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 altlr*p7*1u5d;h iud 9.0 cm rolls without slipping down a lane*h1*udtai57l p drl;u at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth witctl227k 2xhzch respect to the Sun as the sum of two terms, x2722kct lzhc
(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 ru pq/.e4bvnn r+ p/:k3yar)sp adius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolutio rrqu:yn/a4p v.pksbe n/p+3)n per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.vf*tbva (cn z .fv)*t-0 cm and mass 1.80 kg starts from rest and rolls without slipping down a 30. zv(f-v* )vbttnc. a*f0 $^{\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 xsxha+-eed-:f , av4tvertically on its tip. It starts to fall and its lower end does not slfx+de v ea,-txa-s:h4 ip. 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 of a 0.210-kg ball rotating on the en55*9ko e:i spikk.b, ;qeh2aq.ya bevd of a thin string in a circle of raik ;2. *ekq 5ya ve:ispaqhk9be.b5,odius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8 fo1+tr*n 0cdm-kg uniform cylindrical grinding wheel of radius 18 cm when rotating at 150don 0 +rm1ctf*0 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 rate o:7s spzeb-:dho5* hfp f 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, t7p5fo-:ph s s zed:b*hhe 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 reduce her moment of inertia by7a2tg3ygphj khx7 ,z/ a factor of about 3.5 wxy 7h/hgt 237jg,zakp hen 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 rotatio8komr q3 qsf9 73+yyr51zktci.g vz5xn rate from an initial rate of 1.0 rev every 2.0 s to a final rate of k95sz vck33rqqfx 1yzg.r8t7m 5oyi+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 arounde.qb6o ;+ ;idaybklh 1 a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat dyk b6da 1.eoh+q ;b;ilisk 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 mow e6 atsd.tk sj ;m6- j72jqjl)y+n,ijmentum of a 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 angular momentum of the Earthd dz0. ru9j(ct
(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 zs4vydso*9sqn 9m) 45plms m7an identical disk ro*7pmnm5 9oly)ssd9mq s zs44 vtating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at bk ()iajob+5i wz-lq b )uavnizg// ,8g(/ ssp2.4 $rev/s$ 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 stands at the center of a rotatinnco gf/7d2i j;gv-em8g merry-go-round platform c2fm/;ngv oed7-ji8 gof 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 angular velo6und0sejk(.g. ;8 jjxxmw 4xacity of 0.;ejxnxm 04agd.j6 kju(sx8 .w 8 $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a wh 2b(hq+w+th wiaf:kc/ite dwarf, losing about half its mass in the process, and winding up wfi /c abh:wq t+w(k2+hith 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 winds in edfe.1flevppmz(:l3c 56t ag kqm7*k:xcess 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 gs m4 goa.tug8 *y+we2vk x-.f,0wky i$ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, that can rotate ozs6t //beyoyo w.qafi26e 2( ken-p/ freely without friction. The moment of inertia of the person plusby if-aqzo 62kwyo/6e/eo .(etp2/ ns 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 theo 3 d7ylpb2qu, edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertib,y 7pqldou 23a of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of the rope lying on tjl,p- lrt5f1sw-rv ;i 1o az-xhe top edge of the spool. A person grabs the end of the rope 1zvwa frjx ts5lrl-p ,io;--1and 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 Earth such that the same side always faces the Earth. Determssayt 5e u3,z8ine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbi8tys z5us,a3e ting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rateevw -w z));sjz 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 cyvbpj5r0 ea)j*linder of radius 0.20 m acquires a rotational rate of fr0 jajpbr5v* )eom rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylind q)f8 t6yhel44 6afywerical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.005fyh6et8a l fe )qy464w0 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 m*o2n3k ii(xuthe 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, werg0s8m kcsjp ( 76enxx4e 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-quarters48s6mg( nxp sk c0xej7 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 fo9i4:xs t ouazd4 oz8zn9 51dhtr it to use energy stored in a heavy rotating flywheel. Suppose such zdathi85zo n:st4 z19o9xd4u 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 illustratesovw-6fzu a x94rx n39,miua)l an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal surface at speed v= astm5w +zpw ,v3n.ux.wrlqoc)t80 )x3.3 $m/s$ 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 pivohs;9fsjxw ./w t 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 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.]x ./fhws;w js9

A wheel of mass M has radius R .t o2qzttz03,)4ox5izk:j p vur2xew. 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 agaiz px:eqwt0o, xt)o4kr5 2 3ztjzv 2iu.nst which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling wid 24.7(ltitq:gxj jmgth speed v=4.2m/s on a flat road is making a turn with a radius The forces acting(lg2:iqjm7jtx dgt .4 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 rock into a sling of length 1.5k:zk;5 n4cxjw 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 required to achieve this feat, and where doewn 4;kcxjkz 5:s the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylind6i -fcqmh8 ea)n8(trw er 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, 6fqam iet -w8r)hcn8(and with arms held tightly against her body.   

  You are designing a clutch assembly which cov m c.ekc76n;uc21k *1lk qiwvnsists of two cylindrical plates, of ma7kkuecnm *wkqc2 iv.l6;11c vss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 a ybs18of h(h)c7h o;ysnd radius r rolls along the looped rough track of Fig. 8–58. What is the minimum valueyh7bhc o (h;1)oyf8ss of the vertical height h that the marble must drop if it 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 nonx8n8 sbu7j (wt assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car bu oxj4tjebmp9 )co 549o ih79gx9l:jreduces its speed uniformly from 90km/h to 60km/h The tires have a diameter hxcbb947 euoxpj 9ito9j5o4:g)m j9lof 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