A bicycle odometer (which measures distance traveled) is attached near the whecf3klgp6aj; 4rlc06 del hub and is designed for 27-inch wheelk03ral6jc;g6d lf c4 ps. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular v/p4m ampd+0-h1sb zk( a dhon5elocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangentm-zap/phdh am+kso4 1 b05 d(nial 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 angulijbnu ( cv*6r wi:*vo+ar velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a largd/v4u, m,mz :e5l*osv1cpx nr,bbc+y er 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 n(rn2 mxei ;l7hands behind your head than when your arms are stretched out in front of you? A diagram may help you to answerr7ex2nlni (; m this.

A 21-speed bicycle has seven sprockets at theyre q6i/qrr5)6r mab8 rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprockeib 8yrmaer qq rr6/56)t 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 sles/onkvp- kf8 ,o2a)xqnf4s( q nder lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of m4n8soq pvnk os2x ffk,q)(/a -ass is advantageous.

Why do tightrope walkers z4tgvy1 r( 4rmlw8i1 +vog gs-(Fig. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is,qj3)jm hkzyp wq8:;( dn3iov the net torque also zero? If the net torque on a systnp3y idj 8)3v(:omq, qw;k jhzem is zero, is the net force zero?

Two inclines have the same height-usokk :ghe kwn8g77 qj 45jb. but make different angles with the horizontal. The same steel ball is rolled down each incline. On which .j7u: bk7 8s- g5goew4qknhkj incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres sim/d reljj(r;:cxdv ta1r ri,46 ultaneously start rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incrl(rv 6 x rir, adt4/:1dc;jejline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylindg3ipsr(0 eu8 rik28+o yzvtg)er have the same radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the gre3te i8p(o2y80+rugrg v)sz kiater 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 moment hx;03ib.bo71d: -n yaku.tulxnw9js um are conserved. Yet most moving or rotating objects eventualnibw-1b ud37;:0 sxx.tn lh9. juaykoly slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equator, how wos-lrtd.gv 3 igcg*nku+*w+2s uld this affect th gsw*g3s *ngiclrkut +v 2d-.+e length of the day?

Can the diver of Figud-sim .j; j-u. 8–29 do a somersault without having any initial rotation when she leave .duj- msu-ij;s the board?

The moment of inertia of a rotating9 pwgxdc w8 6fy*.dakrv)w-ac6fu q/l5/1qj t solid disk about an axis through its center of massdrktp9xygww6/ 6uc *qf5 d8.j/a c-lq )av1fw 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 sittinga3.g ch.ls9cbm7vmwami ,e ; 5 on a rotating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or 3 9ewvhi;ab m5c.cl. s ,mam7gstay the same? Explain.

Two spheres look identical and have the same mass. However, one i s:un-ue+ ec ng3k 7l+aeed2 +d/lvo0cs hollow and the other is solid. Describe au +ced/ k-o+g :u2a+el3vs n0le7endcn experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points west. In whz.m s;j0bked3 at direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear a z ;sj.k30bmedcceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating turntable. Wha)t5y t6m4q2 0 f7bm7-w wtnx ppupjw(yt happens if you walk toward the c47y 7 w)wqj xn5(tywm2upb0pf-6p mttenter?

A shortstop may leap :0rqn s 8*y11yeamrhh into the air to catch a ball and throw it quickly. As he throws the ball, the upper p1r ha:0s8 rq yn1ey*hmart of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discuss why a l11d q :;1mdyxug-7 dpd6 pzllhelicopter must have pu 7zm 11l6px 1dydd- gql:dl;more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the followingr6yxot,bxk* /u+ie c9 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 coincidence. Calce- 0yaox,h2 8zlf y:bxscv,; lulate, using the information inside the Front Cover, the angular diameters (in rasl cf-x;,elh:ob0,2 y y xvaz8dians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 -hd4kmf w tz)0km from Earth. The beam diverges at an hd 4)ztwk- fm0angle $\theta$ (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 thxedan7l3*v*e o nw9 v;e motor is turned off during operationxeavno 3dn*9; *wv7el, 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* /a ,sep*sm;,vbn1mzj:g yed child. If the ball makes 15.0 rev*na:m1 vz ;/,je *e bs,dgspmyolutions, what is its diameter?    m

  A bicycle with tires 68 cm ing qgg ;2-l y/l:amnr*z0jx s8o diameter travels 8.0 km. How many revolutions do the wheels mrogz yl80jx s g* gl/amn:2-q;ake?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2ub;.9ibno q m8500 rpm. Calculate its angular veluibmqb8 n9;o. ocity 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 co39pl6sxp wqq, mplete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed oq9p 3wl6, xqpsf 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 angula*lzrx zl;u.gx) 27q.kopjv t+r velocity of the Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed zf6ihghfj :nj-l7) 5u l+k+ x*jbjym, of a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate iwiiykugfhf9d 1 -vsv g.g7)+0 f a particle 7.0 cm from the axis of rg h1idfgu-9+vyks gv70 )iw. fotation is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly aboujjs3h4/o5m v zt its center from 130 rpm to 280o v/mz34s5jhj rpm in 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 radiuru/lx-unkfi*bd ( s(( s $R_1$ 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 aboardq)ww,2fgxpd1 8ai; oxj o-g 3j8dqib+ 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- i2xw ji 83jbg q1f8apdq)wo;x,odg+ 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 a-t4j9vn5bry from rest to 15,000 rpm in 220 s. Through how many revolutions did it turn in trynb9va jt 54-his time?    $rev$

  An automobile engine slows down fas7vntap6 ;2x moh9c/ rom 4500 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 hb h5hf 7aj* k-b:ynt ,:v8*iae7iogemighspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reachingbe7 * 8tgma7nfeijbh,hi : v*k5ao -y: 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 rpm to 360 rpm fbs8l n/e(i3ein 6.5 s. How far will a point on the edge o/ fsie (8lnbe3f the wheel have traveled in this time?    m

  A cooling fan is turned off whphz ofs1: .* njn5eeh1en it is running at 850rev/min It turns 1500 revolutions before it comen5eo.p fsez1*j1hn :hs 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 reducyks(agm; v1w, es its speed uniformly from 95km/h to 45km/h The tirk;a y1,(w gsvmes 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 a7k2dmyhcr./ d s the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter oy mkc2dr.d /h7f 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.,f+ sd.wt.ij hvmas nf(kafnm6- 7x, weight on each pedal when climbing a hill. The pedals rota-mxk. 6w vhfn, f.i(j+mt7ndssa. ,fate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cm wide. What is the0 n u i 9u65pf6dmciwv-w,kh+r magnitude of the to5- inp+9d6vfuk6 uwr0 i,c mhwrque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fig. h )5s4*x t)pfmw pg)ie8–39. Assume that a friction torque ow)h 45pxs)g m*)t fipef 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 which p4d.gkqw l m4p4ivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the p4qw44.dgk m lnet torque on this system.

  The bolts on the cylinder head of an engine require tightening to a torque fzxti.- tc c5u3x+c3bof 38 ib53cx3 t +.ctx-uz cf$m \cdot N$ 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 in*e8a qmz y5p0kertia of a 10.8-kg sphere of radius 0.648 m when the axis opm5 *0ez8 ykqaf rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in dia 5mq qp7a2albj 6f/6zpmeter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignored (why?). qf 6qzb2a/j6p5am7 pl    $kg \cdot m^2$

  A small 650-gram ball on the end ofv8s3 .6n zwnri n)n5dsp f.mz7 a thin, light rod is rotated in a horizontal ci. n3)m zvinr.fsndnp786z s5w rcle 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 c4lxl9vo lti/*v1vk ww8m1 , ogonstant angular speed (Fig. 8–42). The friction force between her hands and thl*4lgit l, v9mw 1/1v8vkowx oe 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 f8+f) 9 b9k4 qil3s rsz1vagipin Fig. 8–43 abou ifs4aipr+s1g z8kb 9q)fl39vt
(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 conla:be2+q in 5nsists 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 f2k9 k1n. 5h /8xnvva1 ux1 fm)nliozgrate, engineers fire four tanvkkh 8n1xm5 nx2l1.v19uo af/ig)nz fgential 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 to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uni6jiaae7fi m4/rne h;;form cylinder with a radius of 8.50 cm and a mass of 0.580 kg. Calairm;h feea 6nj/7;i 4culate
(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, acceleratingwcr fa0/ot2/s 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 smale1;rmxi*h bs/l hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a sbr*1;mi eh/xmass 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 /; y*iaqh5fef8-tvtuat 10,300 rpm is shut off and is eventually brought uniformly to rest by a frictional torque of 1.2 faq yt / tvuf;*i8-e5h$m \cdot N$ 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-k 6a+y/1ggyl hf5lmw+h g 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 thek5ne r+9+ 4.bzb-.n ur0 k(x9baqcvmaf s d:fq action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly fb9rb e.a r k uxnsa+.:n4f(9bvcz5q-dfkmq0+rom 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 thiis a8.wi1mcxya( 0;cvv8 a0ky n rod, as shown in Fig. 8–4(1 v0kxs8awaic; yivcy8a0 m.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 majguf2d 4 v7a;8degx(w.k ij+g nrz0ia0ze v,,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 accelerates the hammer from rest within four full tu-t lk1lbw.3w tbs87xz rns (revolutions) and r81t-tkwbwl 7.bxl z3s 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 inerti*iu.:l *v;hnz d-yxax:xdb l :)g- vwia 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 torb7-*:vep1 drgboh1u u que 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 +vrvos+ c. cb-9.0 cm rolls without slipping down a lane at 3.3r+o.bv c- cv+s $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic /7 b ypxiwz.;vjas 7 5f2d)hcic:aq o*energy of the Earth with respect to the Sun as the sum of two terqd5y:jx ofpav) ci7iawszc.b h7* / 2;ms,
(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 malt7 /e/yye-)4r mfhsg7 cr5 duch net work is required to accelerate it from rest to a rotatioyr h5/ysmetr /) l7de-47gcfan rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls with f/xy-phvv uv j057it0out sli-xiy05 f7uh pvv0 vj/tpping down a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

  Two masses, $m_1$ = 18 kg and $m_2$ = 26.5 kg are connected by a rope that hangs over a pulley (as in Fig. 8–47). The pulley is a uniform cylinder of radius 0.260 m and mass 7.50 kg. Initially, is on the ground and $m_2$ rests 3.00 m above the ground. If the system is now released, use conservation of energy to determine the speed of $m_2$ just before it strikes the ground. Assume the pulley is frictionless.    $m/s$

  A 2.30-m-long pole is balanced vertically on its tip.24;kv *;joumvqanh 6h It starts to fall and its lower end does not slip. What will be the speed of the upper;;v jhk ou2nm*4hqa6v end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 4(:xdua6vsx 1kzquc1 0.210-kg ball rotating on the end of a thin strid:xz 1 6 v4xkcuuq1(sang in a circle of radius 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-kg uniform cylindwrrof(0u*do e9 go( h12uy ao-rical grinding wheel of radius 18 cm when rotatinu (o(1 du 0gyoa 9hr2*woof-erg at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotating yri- n43sad7sxl 9b 6,uzun7e at a rate of 1.3rev/s If h d-b 7n lr,y9su4e3u7nxi 6saze raises his arms to a horizontal 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. 8–29qf9a7ni8) zjn(s-gak ) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what ik( )j-gi z9a7qn8fnass her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her sfgd) :q5 g94 vllhkir.pin rotation rate from an initial rate of 1.0 rev every 2.0 s to ah.lvg45g f r:d9lk)iq 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 v htq85, t0w/ tvnkisp/ertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg v, wsq8hnt i5t//0k ptand diameter 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 angulqo+geb0 zw 80uar momentum 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 El -n4h9 fjnszo9h 2mm+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 momenp1tay(jn8k 17b sz2ck t of inertia I is dropped onto an identical di(1 7b c znk2sya1p8tjksk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns z dy5xx3ev2g0at 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 rotating m1m*wpces7 o ko w9q)x+erry-go-round platform of radius 3.0 m aneop xows1k+c9)wmq *7d 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 rodudr z.: r)lt/fxl-5 ;of: iwutating freely with an angular velocity of 0xl rwftod /::)l5d-. uuifrz ;.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 whibm+n n07 du.gfte dwarf, losing about half its mass in the process, and winding up with a radi 0nn.du 7+mfgbus 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 excessj+s- blntzm 6meaa71( 9,jjl n 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 ,bvjt0 d6; hs/ 3)yr u7unlmsiw1wck: $ 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, initiaqf /ofk)sskw,9 3o- n h:0piawlly at rest, that can rotate freely without friction. The moment of inertia of p qfwk si3fnsa-ko0hw)/,9 :othe 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-m diameter merry-go-round tuzos k. 3 qtvsausm.70d uz203urntable that is mounted on frictvksomuz3s0u d.u0 t q2a.z3s7ionless 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 of (+:ftxx5u7 n3u*opso3csptq 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 th3 cpssou:qnxt(x* t+p5fo u37e spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces the Eartm3*c+v up:0iwy uxxi6xixy23 h. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat yc3 xixpix0 wuumy :3+ 6v*2ixthe Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest atjs d7d,:d/c,zw cv+if a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniformkls: r9;vjn u6 cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s intevrs: ;nlj6ku 9rval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two s ke8*bd 2)st7q;w vwseolid cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate 2e8s ev ;kwdb7tw*qs)the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of the .* akxa c7f6 p+g, j8gc+ldwpirear 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 ig di6m,8xhg )great, were rotating at a speed of 1.0 revolutioim6x g, h8d)gin every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use energy store01 so1wts,xzdhd pet3)o)xq :d in a heavy rotating flywheel. Suppose such a car has a total mass of 1)qhsw1s0p de3:t)ozt x,1xo d400 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 illustra6vemn 2 16:rht2jkelso0vs d1tes 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 horizonjni :ev3t/3d na flyyx+i3qa1( o95mhtal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., we ignore frictio*h0tm c3ytj )qn) 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 Figyj qh*c t3mt)0. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on the zzmo9 lsbt9rcs3if4/8 xyi9zkxz o2/38qz 7t floor, and we want to exert a horizontal force F at its axle so that it will climb a step against wqf7zxrc3som /yos/lzt9i4z ti zz2xk8893 b9hich it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed iz(ytv jj a48)u/q1 zm m 7cbe:+mr7kiv=4.2m/s on a flat road is making a turn with a radius The for7c1b) ymkjz m4tm ir8je7quaz/v(i: + ces 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 rock into a sling of length 1.53t1z)-it uc qt m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of3tit -zc1uq)t 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 arm1b,gbzy cwz 9 5yk7pqmt ,ch00s as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater wit7,9gt , zcp1 mc0yqyw b5hzbk0h outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of tp,/yh r9qz ;y-l lzmk:21hahty xe8p8wo cylindrical plates, of mpzer;1k9-8q:x p/h hy 2mhtz8y,lly a ass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 -t6s:o r-trpqp) zl*hrlh+h;76 f4okmx d qa8radius 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 highet zk:ll+frqx-h6p*hp;6 7 -)oar4h rdsq8t o mst point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but dv98sxutxlm904x.y1dx-gr r jo not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85r6 p:vxvfib/o i2* nqn3 ka0(a revolutions as the car reduces its speed uniformly from 90km/h to 60km/h The tires hain6:rk/q a2 vi f0o*n3 vbxpa(ve 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