A bicycle odometer (which meaatow 4i ep-j+.d (w;+uy3dxausures distance traveled) is attached near the wheel- j(a oyu3ew+i;dwt + x.pd4au hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on the rihed80l (x2xse;jyxd c 8v1c4 blc2cs9l/oht.m have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point ha0 cx;lyhc b98 /jt 1lo(cv2ce8.sl4exxhs2d dve 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 val4fil z:aup5e lct4,t)ue of the angular velocity $\omega$ Explain.

Can a small force ever exert a great;f50 rjw4nwi l1uldl (er torque than a larger force? Explain.

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

Why is it more difficult to do a sit-up with your ox0w2wo m7rdn 3kz:on 1)qfx9 hands behind your head than when your arms are stretched out in front of you? A diagram may help you to answer thdwonx731zox)kwmrfqo 2 :n09 is.

A 21-speed bicycle has seven sprockets at th1nq6lh/2gog)lj kah 2r- l, hxe rear wheel and three at the pedal cr1/ khon ljga)-l262hhg,qxr lanks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend oq h1v24xr- mqpy/f(z9crnk 5on being able to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On h4 (vq5yqm9k c/ro px2nrfz-1 the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (9xzjj g+b.: oc 5/bwhvFig. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the net torque als1ixwr uvpb7zzl h3 9c) v, orq)67m)poo zero? If the net torque on a system is zero, is the n )qcu1lizpxo9o 6 wpvmhbrv)3 r7 ,)7zet force zero?

Two inclines have the same height k0*6/ylpq5 ( mjzqsmm r*mabzvm3/ *fbut make different angles with 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? Explai*y6qmlm(v mzz r*3m 5fjbs ka0//qpm* n.

Two solid spheres simultaneously start rolling (from rest) down an incline. pz1u (s+bdsh5in*j0 rykl9z /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 has z+n/lsbj1 i* pk9s(zurh05 ydthe greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. They starz h7 /tv.5sbt*jkt d3wt from rest at the top of an incline. Which reaches the bottom first? Which has the greath5sb v7z /kt* t3j.wdter speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most mov hbxd mwwqb;2 :8nhi24ing or rotating objects eventually slow down and stop. E4dx 22 m;b8hiw:b qhwnxplain.

If there were a great migr2a h33xi vy ajuj4r;fq2u9:lwation of people toward the Earth’s equator, how would this affect t;fax4v2h 3u2j3 9 rua:wil qjyhe length of the day?

Can the diver of Fig. 8–29 do a somersault without having any x,sr-o i3k/a t0yki1 pinitial rotation when she leaves the xa0-i3,oy/sk1 rit kpboard?

The moment of inertia p- 2vz55ve mitof a rotating solid disk about an axis through its center of pt5v52 me-ziv 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 bfl w bkf)z /i):6u-rya 2-kg mass in each outstretched hand. If you suddenly drop the massesbi wfb kzf6- )yr:u/l), will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. How wa 6yrt1d0rf8ever, one is hollow and the othea01dr 6 ft8rywr is solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle podzt8 4hvque xw 440plr/*-yxj ints west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing 0r*/p4 jhw ud44eqlvx8-yztxor decreasing?

Suppose you are standing on the edge of a large freely rotating turntable. W9(ia tt/mkl ybxa)c 2*+fk 8cqhat happens if you walk toward the tlyka* 2k+ xb9c/f)8ma(tiq ccenter?

A shortstop may leap i) ( n5jdlitd:bnto the air to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice th(nt didjlb:)5 at his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of 4wx3r 3 pp:prdj;/aez conservation of angular momentum, discuss why a helicopter must have m33rppj/; xpawz d:4reore than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the followinmifrg 3 z+ellt*9r(7th(oo+ whv 37bsg 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 because of an amazing cpxc y7j9v9wcz yggv /v,:j/5 toincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on E /9v xz9 jvgj,cvyt:/cwgy 7p5arth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed , yuj.vaf.u7ih .hzi2u;c (ufat the Moon, 380,000 km from Earth. The beam diverges aih(iz7.j.,fua uuvhu y.f;c 2 t 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 jj7+nt m8** siws3key7 rlb0srpm. When the motor is turned off during operation, th *m *nk 0w8 7jeilyts+sb3rsj7e 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 ball make9m1e. rem6w+m2+kljfz(d n cy jcfdtv a1*6,is 1z+t 6c +j1e f wjamr6idmvyfd9(lc2 .,1 *emnk5.0 revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revoluti +mleuwa-u 9j-ons do the wheee-+9w u j-lamuls make?    $rev$

  (a) A grinding wheel 0.35uwm0 -bh :7vey )7lyo7eyq ,tg1p ee;dr- rh9b m in diameter rotates at 2500 rpm. Calculate its angular velocity77uvmte,ohhbyp-wq-y :0; eerb) d79yreg 1 l 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 4hazt/-ch8 i-l.0 s (Fig. 8–38). (a) What hl/t-h ic 8z-ais 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 Sud.o+ ja54+uvc:9-r 1ts eyxxakl fk9un    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of anm0s9 cl1r9i ab l/0jk 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 xi2 z5cdri0w3from the axis of rotation 0dcii2xr53zw is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about itsm2k66 jrrd2nt center from 130 rpm to 280 rpm injr mn6ktd r262 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 radiuj /6:t)tqcg-z+ds uq ps $R_1$ 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 a-in bufnay3 c)qy92 s6board the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At taufn3ncqb 2iy6 sy- 9)he start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpm in ef;o/w:5 . hfhaev) zq220 s. Through how m:ahfqwv)h z5foee ;/.any revolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calcul4v6xd8t ow mi7xqgs;9 ate
(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 forqyen6 m.+o z rk:;ms: rnh6e3c the stresses of flying highspeed jets in a whirling “humr yz6+q6e. cs3kn:m emn:hor ;an centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpmjtrxu t320.e.h xbf l*g6jmp7s ,5dxv to 360 rpm in 6.5 s. H.vdfx0 tm,.xg 2l*6 xu5 b3t7hspje jrow far will a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turne) 9r808 .glmfe1 hf:gtpbuhd xd off when it is running at 850rev/min It turns 1500 revolutions befor :e98pb0fhhxtg l1. grfd )8ume 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 re. ffgbuhyr/ : /5ue;hq--sxmiduces its speed uniformly from 95km/h to 45km/h The tires have a di: m sf5fhq.ue/x;r-gyh u -b/iameter 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 6(,lormywl8 :jy61xj t5 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter(6,8t1 ryjoyxm :lwl j of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each pedal when climbing a 6nbz p doj6( *)ho(4+naoylrphill. The pedals rotate in a circle of radiu jpbnopd )y rzo+66hln(4 o(*as 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 *7-uj( ts32 qehkiqe)6wtwz pdoor 74 cm wide. What is the mag z7wett*2k) qh-3pw6ie j(s qunitude of the torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axlii1r69 j2t qkze of the wheel shown in Fig. 8–39. Assume that a friction to9qt6 ri ikzj21rque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of 3a rgd*bb/9,s23wrfkpi wf33j .c rov mass m, are 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 re2/j. b ibdvprfa3r3 3 srowc ,*g3fwk9leased. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening to99rdl8 1bssn pe2erp+b agu0: a torque of 38 eass: r9dpn0+gpb2 81lub9e r$m \cdot N$ 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;u74lncsfe e 1s wr.(i of radius 0.648 m when the axis of rotation is through itsc41 w;s.e(l e7insfru center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm inmstjb38) j rre ak9w,)he*x l2 diameter. The rim and tire have a combined mass 8re amt*x )j)j,er3w9sl2 k hbof 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, u9k( )q qzy,nr u(kf-jlight rod is rotated in a horizontal circle of radius 1.2 9kuu,jr( kqq -f( ynz)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 ro7lrx6)s 4ie ghtating at constant angular speed (Figrlisg )64e7 hx. 8–42). 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 of point objects shown in Fig. w/ d,;m on)rqi8–43 aboutq wonir,md); /
(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 c jv( yk q/4b8i6- kve;vte;bmxonsists 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 satellitehrwcfva5 v k (k3np6zm9 7;jb3 w.8jtn spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is t nnwkc tw(3.65fvj 9k3h 7a8vzjpb;mro 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 mtgiser xo8f367*4v+abmo - j 8.50 cm and a mass of 0.580 kg. Calculi ftg ovj3mb76as mr-e +8ox*4ate
(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/rm2ji nbnffh s* 9,f cp+h:4x a bat, accelerating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven m .b4uflk8z+mavuwue 3b6 1ex.erry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit 4zum+k63l. awv1xbefb8 euu. opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is eventually broupu8ybu rnpk7w+(: tr 4ji/s.d 8(rc mught uniformly to rest by a fricrj (8c/(ukp8su pw +rdut.n4b:7 iymrtional 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 accelerat t)2u/u0pd qn4iaqyf2)y x.uues a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the action of i6jcr9 lb1: /nwe4.rpzfty 5rthe forearm, which rotates about the elbow joint un nr6:5f1er4iw9.c byz/l rtpjder 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 Fi/l5o 1d tclp,ng. 8–46.l d5c/p nt,ol1
(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.;j utd+vzyt ;eq)) h. lzk,bnsses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest within four1y4;n:q/xjdn 3y. feufz *geg full turns (revolutions) afy*e ; zj 1nfd /4e3.q:uxgnygnd 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 moment of inertq0e5l m0p5is eia 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 z z16vxc za(1c4 2fr5lkzg,o6e;gazma torque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls wi0 k :n8pgks ppjtnj;3zqo28a+3vpw 8 bthout slipping down a lane at 3 w2vz kppnp os:0 paqbn3j8+3;tg88kj.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respectgvnwxkhn 08+: to the Sun as the sum of two terms,kv :0n+g 8xwhn
(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 7pxf2gbs( lhya44fy: and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotatip2f 7afyxlg y:b4sh(4 on rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mwinhu nq 0n2r zxxnq8;829 .vaass 1.80 kg starts from rest and rolls without slipping down a 30.9anxw2 0q.unh;n82rx8ni zv q 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 its tip. It starts to fall and its )cr4zd vjafe h3,v2(rlower end v) jz vd(hare32,c4rfdoes 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 of a 0.210-kg ball rotating on the end of a thin stfo lr8rjin-d j2m )g7gc8(,lfring in a circle of radius 1.10 mm,o8n 2-frldl f j(i7r jcgg)8 at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grinding wheel hoxwf dt2hp.04yp7s b r(vo2+ of radius 18 cv yx4of+d w.trb7o2 s(20hph pm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotati v3: qyd(ub-.)xntug zng at a rate of 1.3rev/s If he raises his arms to a horizontynzx v .gt buqu)d:3(-al 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 reduce her momf - g *fnbo+; z3dqq4k.di9qsjent 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 in 1.5 s when in the tuck positio+n gj f9qfd q*zq 3kd-.i4so;bn, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her qdm,*q o/*kv4z/p+u oh-m jwlspin rotation rate from an initial rate of 1.0 rev every * h+j*,wlvq-/ oduozpkmq4m/ 2.0 s 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 rv7kk o*96rrpce, 8vpda8gt8ndo3- yuotating around a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and dvdpcdetop v3 ,96 nor*87-y88 akkgruiameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinning at 3.5 r 5.;wmykn4fa $rev/s$ 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 Ek q/irth( ds::arth
(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 an ideni lnk8(p udk/4tical disk rota(k8 k4i lp/dnuting at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns aft8rb i4cn-c7 t 2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotatingrme ac 3e3 9 3e4xvzo)y.sz5cg merry-go-round platform of radius 3 eryz.z)9ovse4 ga33cec m5x3 .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 velocity of tqn8r7t- ) .vnn8wd kkrcf00l 0.807q.n8-rlrknn tvwt kfd 0 )c8 $rad/s$ 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 collapset.28gxv )zd j*xls4cw s into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular dvz* 4l)ct .jw gsx8x2momentum, 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 iwu+:2wo rrodgjibp+1mm(q3uz61;mc n excess of 120 $km/h$ at the outer edge. Make a crude estimate of
(a) the energy,    $ \times10^{16 }$ J
(b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 $kg \cdot m^2$) of radius 100 km and height 4.0 km.    $ \times10^{20 }$ $kg \cdot m^2$

  An asteroid of mass bn9gg0 m5b h(t$ 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 u 4v;0-ggecq kpy y7+irest, that can rotate freely without e70;yp+ cv4gkg uiyq -friction. The 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 a 6.5-mv39s 2zr)cp boawl9d- diameter merry-go-round turntable that is mounted on frictionless bear9d l9wps3 ova-)2czrbings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of the rope lying gtd pa+o igbq3a *1w1+tru78oon 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 behi71u3td18b*a+aotrp+qi wo g gnd 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 Egwk gmg9j ,zb m;,rerb2k3r;- arth 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 a particle orbiting the Earth.gzmmrb3wgrbg- kkr9; e ,;j, 2)    $\times10^{ -6}$

  A cyclist accelerates from r*/ag *i9p c :2wb.pn +esqib1dy3xinvest 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 ca6ft13hx -jl of a uniform cylinder of radius 0.20 m acquires a rotational rate of from f6t3j1-xlcha 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 cylindric *f 3:0wqy cq6mim tpy:.4awbtal 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 reachewfyya mwt6icqq:0 :t.b* p4 m3s the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular spee3z vnvyunq62yn)p5 9(q3bf fyd 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 wskbo q82g9trc h 6f:;; n6,tcc,pja atith mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse9ck;,so fcq68 b6: r na2ph,tattcjg; 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 a94dtmnct (:sdp-( *dumt mecd.ax.6 automobile is for it to use energy stored in a heavy rotating flywh -m.n (edtmtx4 :c*.uacmd t dd6pa9(seel. 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 illustrates a /j1rzr84 w qa(v5tf/zmykr(un $H_2O$ 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=3.3 m14 *xkbs* 1s/ acc,p qe/2/ldxbcrpaw;kw t:$m/s$ 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 pivocy:v; xmg2xmvw0c)y 4t 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 )m0 gvc2w4vyx xcm ;:ythe rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing5 )dj)8namz jf 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 stend5jaj fz )m)8p has height h, where h

  A bicyclist traveling w fxo )cs*4 hdkbo;:r2mith speed v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist and cycms2k)r:cxdb;*foho 4 le 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-ik,cz/uuj2 ) 0 e;vmalkg rock into a sling of length 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 required tzkjv);u,/2eum0la c i o achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body acc 6./ 2rzsy)vk5 k;m wnj*sbds a solid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinnin2wr5 mdk.bzc/n)vky jss* c 6;g skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a cl7 0dv36hgzxaq utch assembly which consists of two cylindrical plates, of masdgvah63z0 x7 qs $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 :)ub)r-ok/ton42xur nn4r lr 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 of the loop without leaving the track? Asrb/k x4roo-) ur42lnn un: rt)sume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, butdbxwb ori7 1*c(1-aowgt*j.v do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its s mvo;:z*wvpu(,dt v 9ok f-hr;dw;5 sspeed uniformly from 90km/h to- k zm9:fd;(prwtv uw;dso;s vvh*5,o 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