A bicycle odometer (which measures distance traveled) is a lgsp3 rh7v+6re3o)x attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch whee3eg7rlhra6x 3vo s+)p ls?

Suppose a disk rotates at constant angular velocityt 2ce)ofmc0ho8 i2bsosb3mr77 n2rf9 . Does a point on the rim have rifmomso79e c s2083 7tf2o)r r b2cnbhadial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of iak3z* zvu i t+(4j7n iil7a1zthe angular velocity $\omega$ Explain.

Can a small force ever exert au3 ebqnzwdp8z ze9 668vj94ljo y7.j7fy t8ot 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 to do a sit-ufqd k dns27fu 4m--u2dp with your hands behind your head than when your arms are stretched out in front of you? A diagram may help you to f2qu7--u md4sf n2dkdanswer this.

A 21-speed bicycle has seven sprockets at the rear wheel+ dg3euvkz+*v and three at the pedal cranks. 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 fron+ k+*3uvdvzge t sprocket or a large front sprocket? Why?

Mammals that depend on being able to ru(vp0svh5/ lyrl fq(q2 n fast have slender 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 mass (h yvr5(v/q0s plqf 2lis advantageous.

Why do tightrope walkers (Fig. 8–35) carry a lonzm;mwzkfa ; (zr)q;5j/ l4ava g, narrow beam?

If the net force on a system is zero, is th+cyy px84ex+s e net torque also zero? If the net torqyxexy48c s +p+ue on a system is zero, is the net force zero?

Two inclines have the sam9layi97qfzh27t:cg/l c og+ c e height but make different angles with the horizontal. The same steel ball is rolled down 7ly9q:t g+ccical7g2 hf z /9oeach incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneouq m8., qxoz,g kybzomw57e0f.sly 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 incline first? Which has th,x5zkoq ygbf0o.7 , .ezw8qmme greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the sa.8m b;w5xf i6tsa2 bzlme radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotationalasf5lwbim x2t;8.b6z KE?

We claim that momentum and angular momentum are cons .sgl /jztclbh +/qn)1erved. Yet most moving or rotating objects eventually sqz sb/hgt1.)j+/ lclnlow down and stop. Explain.

If there were a great migration of people toward the Eajtenyj;nsy 9d. t::if:.a1 ym rth’s equator, how would thn s.n jajy9:;im.:t fyd :y1etis affect the length of the day?

Can the diver of Fig. 8–29 do a somersault without having any initz 0 6-7bvqm/iv9(yatk8 1bv nq zbci,fial rotation when shyb 8 9az6v(vq1,bnc/b0t ziq iv7mfk-e leaves the board?

The moment of inertia of a rotating soc/j.h v;hz e 5gr7aga)lid disk about an axis through its center ofhr);vh e/5ag.j 7gcaz 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 each outsrss1z(y; w+a q lt3bs;tretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay t tzb(y+s s13qrsal;w;he same? Explain.

Two spheres look identical and have the same mass. However, one is hollow anau3ru(r/r4o xd the other is solid. Describe xr3 /4 ur(aruoan experiment to determine which is which.

The angular velocity of a wheel rotating on a horizondd2jy 2uzw,v b n5v3.u;ixs jq9d 4f1mtal 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 point at the top of the wheel. Is the angular speed increasing or decreawj y3 u2vfd.j;95 zmxi bqv d1u4ns,d2sing?

Suppose you are standing on tmf88zil1tr, gkca-64ylp r4the edge of a large freely rotating turntable. What happens if you walk towardr4arlygc 16tt884 ,f p kzl-mi the center?

A shortstop may leap into the air to catch a ball anl ; 7fuqw z:vj4ke4zs3d throw it quickly. As he throws the ball, the uuvefwl z:j3q44k; z 7spper part 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, d2jup m; 7( yy;y7cajfwiscuss why a helicopter must have more than one auf(y 27jp j7y;m;wyc rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (a) 30; rap*5z(6wn( dapn gf $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincidence. Calculate, using a41sr)oqlp8t+ r/e ypthe information inside t+/ 4rtel 8)rqpyo1aps he Front Cover, the 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. Thv ufomo5-g3ex7m ec(+ e beam diverges at an an5+-m o g3o(vu7em xecfgle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rmt ;:b0yici4gotate at a rate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blai ;t g4yib:0cmdes slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to an z bar6a8n79ndother child. If the ball makes 1n 9dz78a a6nrb5.0 revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutions do bfwj9 a08nb-ejc +e(:r qc*oothe wh*o(0 :b e8eroabn9- wjcj+fqc eels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate its ank,-utrxx37lym0 35 uudh/ lcs gular velocitx3 7c5l,sy3hut-0/kluru xdm y in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.0 s (Fig. 8–38). (a37 *;heq7gjq)tomigd ) What is the linear speeg;d o7t)i qh7e3*jmgqd 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 arougy8qln7, yj4 0z-nu)hcih/ xdnd the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed o j+yhefso 9j1*6z s*ihf a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.0 cm from the axv u,jqd 3: t5gne+y+vsis of rotation is to e 5vv : ,stygqjdnu+3+experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 13z(f)evl8til kkw555o0 rpm to 280 rpm in 4.0(kfe)l wzt5v8o5il5 k 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 5b0ss 9 w5ztvax5bd;r$R_1$ 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 Moo0w ip erfk. osd9/qh.3n, astronauts aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution everyo.d0 9qwif. k /ehs3pr 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 220 s. :r8+k)wh; rt6u;eawcc dh gr :Through how ma r6 h;kewdhr:+rwcc:g)8 uta;ny revolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm inauykz2xn k1 p2rf/ywy 6b5e4+ 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeed jets in a whirp-cmqe ud5 a-r +l9e+sling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its fin9 s-pl+-e+5 daqcreu mal 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 df8itr np0ad/ j,9 zf -;oy+cbaccelerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the ed+zd ,prffb9yctjn - /;iod a80ge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 1500 revolf( h4(*stgl4qzys; z fhvzj :1utions before it comes to a stop.h:*yts4 zv(gjs l h4fq(1;f zz
(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 reducxj*jl8j:69*bh(s(u i n;uzwil * tfoaes its speed uniformly from 95km/h to 45ts*j:uuoifz;nilwb(69(h a8 j*xl*j km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed v(ap9g:jjemu 3p :/jlqxk/7 5ohmf ).d y9hsd uniformly from 95km/h to 45khelv5m:kjg uaf p/p:oqx3s97hm 9/. j()jyd dm/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each pedala/b m h2r,ut 5:f6e(1mzrsmre1*pgdo when climbing a hill. The pedals rotate in a circle of radzr/ otfagmrh11se:*bm 5 p (er 6um2d,ius 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 the mag9dq9zw-z gr+v4bnmv upw: )9y nitude o-nwpd) q+:rgy99uv v mz9wb z4f 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 axle of the wheel shown in Fig. 8–39. Assu/-ehj;on x-q7/o jhy mme that a friction tj;ye7 hox//nh oj--q morque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a massless rod whikke4 mrdk,+ 6ealsgn- unby;;2; q g)ich pivots ;q ;2bmlai)dgnk +-nk;6gy4r,skeeu as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an e gj4 p;8wgb*wyngine require tightening to a torque of 38 jg*8bg4 ;wp yw$m \cdot N$ 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 c-ofy8 0m ucfsv4*wy yf,ke67-kg sphere of radius 0.648 m when the axis oy fk ccye08u 4vm,of7yf*sw6 -f rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia(1)o:q s5lnqic if ;rr of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kgcrso in 5q)r(1fqi; l:. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a (ly.ayly;7l29)jlqgr ij 2zlb rz -;nthin, light rod is rotated in a horizontall- rz7 abl.;nril zyl)2jj9 ;2gyl(qy circle 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 constaned ajw(sgfyy4gb,* 4:t angular speed (Fig. 4* gd gfbjwy(s4e:ya ,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. 8lxo c b4h-a6,8ohzo5 k–43 aboh,4bxz- kc aoool5h 68ut
(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 consirl.,hq0xj+0w ehwm;t6x :b khsts 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 satellbn1f pm6hfb* ;e /nn(rite 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 epmffb; bn6e1nh n(/ r*ach 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 gjbgo;90h51*hft o fa uniform cylinder with a radius of 8.50 cm and a mass of 0.580 kg. jbhtfa9ggoo501h*f; 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 swingqy/yry5 :,ymu cht0 m(s 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 smap2qj.)fvx;po sy, xf- - .iocrll 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 mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torv xo-iq.rfof2 yc .p)-pxj, s;que. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotatin rrsot2,.9 ;gk/cwo p kai3c:elxeh84g at 10,300 rpm is shut off and is eventually brought unifo:lske293x,8/ oa it orp4 w;cerkchg.rmly to rest by a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 i+)2wuj krst 4accelerates 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 the forearm, whic hfj l1;j qyubpa;kc10z09aaoh w.17gh rotateb9lf1u ;ah.wz1 j gk0 ojhq0pcyaa;17s about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be co j.3c jbyv-/)kz9bafk nsidered a long thin rod, as shown in Fig. 8–46abz.v39ck -)jjb/ fyk .
(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 masirgeyce*+ z3/ d l:6esses, $m_1$ 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 withs( ukz7zm8g 7hdnl7, bd06halin four full turns (revolutions) and rezzskag8 l7d6 0bl7un7mh ,dh(leases 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 mome pj:7g1 mken.eff j x1e-q30utnt 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 ob:kybo ox *w:jpxxxdk2 ;q8n o5xp+(4a9+ewff 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of massbv3l,u)p+(n de 6elz8v0*brj1.yb u euyr1go 7.3 kg and radius 9.0 cm rolls without slipping down a lane at 3.3jrb6,p 13)vu yoneb.ul +ye (ve*g 01r8uzdb l $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic enee3*g+,jnrhs/ pq dp1xrgy of the Earth with respect to the Sun as the sum of twoxgpp+e n/1 ,rj3ds* hq terms,
(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 *rotc0tjjm3h p( jp, -of 7.50 m. How much net work is required to accelerate it frojt pj*3( pjct r-0mh,om rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm anw;7xc-vwy81l0 lymn md mass 1.80 kg starts from rest and rolls without slipping down y-ynmm0l ;w 7cvx1l8 wa 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. It star ulyg6*hp b*dal5, ypjkmm 098ts to fall and its lower end doe6y**yjhb dp u9mlmp 5l8a 0gk,s 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 ok 1cg soa-v0y2t-t,nuq,jmg2eab r88 f a thin string in a circle of radius 1.10 m at82-a rog 81etjkn 0 gbu,smq- y,vac2t an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindr1xx;0g5ajswsrv;v hl -mq0x5ical grinding wheel of radius 18 cm when1 wv5; vsgqh5rxl;j sax0-0mx 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 rotating at-/j j b5al3j3a9x hczm a rate of 1.3rev/s If he 95ab3- j cj3z hajlxm/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. 8–29) can reduce her mom) * ak3q3lrdqq;a xzj6ent 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 * jda6l33;qkqzr)aqxn, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate :cto* y ch(jf:.l uim0k 1yz2* mufcs0from an initial rate of 1.0 rev every 2.0 s to a0zcu sfi :*ju cc y m.0:lhot*2m(yfk1 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 verticalh.r3z 8 af,yvr cbwv/ y6m7n:u 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 disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of twy 6y3f7nu8mr/vhcv az ,:r b.he wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinniy k9lq6ee6za .ng at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum of x)jbzd0tu rmi47a /7kr jp+2 lthe Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk y)3 wcyem ke+2nhn( 7hli1a4fof moment of inertia I is dropped onto an identicaa3 flm7cehnk1y 2)+ewnh (yi 4l disk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.b/t4vq9u*phv +h:t lt4 $rev/s$ 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 merry-go-rouj h,2k cu,h zga)zl,t/(bw4ecnd platform of radius 3.0eb(k wz2tc, hu/h) ,zag ,jcl4 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 i)jprd-qb 8z-ls rotating freely with an angular velocity of 0.8 r )8-zjqp-l db$rad/s$ 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 eventualh9 m2p y bwyk 6x28y(qm*y1kbmly collapses 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 a ykm8612 *9mbyq2yp(yhwxm bkway 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 w -zxp*lbwy:zwow8;h :zb*7 + rypv6tiinds in excess of 120 $km/h$ at the outer edge. Make a crude estimate of
(a) the energy,    $ \times10^{16 }$ J
(b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 $kg \cdot m^2$) of radius 100 km and height 4.0 km.    $ \times10^{20 }$ $kg \cdot m^2$

  An asteroid of mass s2 +z2: hdjc5uv uxc/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, initiq6ydfzl cd bk*8w )*o)ally at rest, that can rotate freely without friction. The moment of inertia of th*bkyw) cdf) z86odlq* e 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,jgokgij-6 7.w2 al69uguj of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has,ggoul-u2f7jo6.a 6 jwi kg9j 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 c qbpegdp o,4.e4n*;st9f;xk on the ground with the end of 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. W*.tp;4eg4ofq9 n kdc b xesp;,hat length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth sibinc 6),ub) fuch 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.c),fui inb6)b )    $\times10^{ -6}$

  A cyclist accelerates from rest at a ra))hg:deo.-) w)za6 lx puff4r8+quagrmhrp 0 te of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder ooln9sk,aw a -9f radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. konw 9a9l,sa -Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 k-re2xfo )yi+ wg 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 reachesxe2+w ri )f-oy 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 o3xxxhbu- uj - 3k-ty j6p; va99t8fxfif 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 526topqc4m0ghahsz5 of our Sun, but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass c 4c5zqto2am56phsh 0g arries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution azv0lm j7ma; u7utomobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of d0 a;um mz77vljiameter 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 m4p*syp3hum33v sei/p10y oen $H_2O$ 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 surf ue2*rlsf k82wace 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 iz38ny)eyi lcz3 e p )mw4nm,5rgnore 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. 8–21gyy,5 e3lcrzwnzip) )8 em 4m3n.]

A wheel of mass M has radius R. It is standing vee9w:g0,uz5) ohw))qtbcel qkt,z j,x )f:dwkrtically 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 (k,)) kwzht9xb:, )o ctq ez fu:dql)gww j,0e5Fig. 8–55). The step has height h, where h

  A bicyclist travelingg ssf2vp /nqf./hf) o2 with speed v=4.2m/s on a flat road is making a turn with a radius The force/hpqos.ffs ) n vg2f2/s 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 29y(;t9z kp y.hql3j n8ib 59xlw inlysling 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 i3ipq9lyjnihtxlyb w 9. 95 (zlnk8y;2s 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 armsr7gg 15x*jzbj+ush 0 f as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning f jbxgh*1zjs+5r07 g uskater with outstretched arms, and with arms held tightly against her body.   

  You are designing a sx5suk z k;(. j)vmrv;clutch assembly which consists of two cylindrical plates, of mass z;s 5 mxvk( j)sruk.;v$M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the looped rough n ojq/ y7,av5-z*q4 d9vdgm nftrack of Fig. 8–58. What is the minimum value of the vertical height h that the ma7 n4z5*vfdmd n oqy -/gj9,vaqrble 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, but5 l5ufdx*sb7 qz+x3o zt+;gby do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car rempd*tlltu9ec601 5efw ;gj) hduces its speed uniformly from 90k0ee ;d l uhg5twmt p*f6j19cl)m/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s