A bicycle odometer (which measures distance traveledv7q 2searkg mh; *rr1njq9l74) is attached near the wheel hub and is designed for 27-insk *er7hqvn 92ljm1 rrqa ;47gch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a j *, ,9yvjde+ugc,,j 07yaq7fxpa vbhpoint on the rim have radial and/or tangential accelercdgx 0fqj7ava ,yp,b 7 ,+,hujj9yv* eation? 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 thel*ihx zdxb k: )4x s, *nfq1ehb6jsw;3 angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque x r)n* zbcf/:kuyul :87t pc)kj(ee.cthan 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 hands e/0b;zd.npjsu;c c uo yk:w+; bu;-zfbehind your head than when your arms are stretched out in front of you? c ckw. bnyp;/;suz dubz0-ufo ;e+j:;A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets jsk5;jezj (n. at the rear wheel and three at the pedal cranks. In(5zsknj;ej. j 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 on being able to run fast have sle pc txjzzs.5+vk97:lx nder lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On ts:.9xv7 +l5tkj zxpczhe basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long,v2y.xq+x1 7b ags e:mf narrow beam?

If the net force on a system is zero, is the net torque also zero? If the nelwgg n2aj 431tt torque on a system is zero, is thew1gg n2lt3 4ja net force zero?

Two inclines have the same height but make xqzl 9j(lwi5b9 k+1ur different angles with the horizontal. The same steel ball is rolled down each incline. On which inuzrk bx1(9jl iqwl95+cline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an incline. fjq 8wzsyf;rq 4m) *zql7,d;lOne sphere has twice the radius and twice tmj,y;lqlw fs q48 dz;)*fzrq7he mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radiu06a59glvt+lzew s( hm2szwt7r+ oqe 0rvs.*ys and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has te(s 9tqmhtayz6*vozw rl2sw.v 7gr+0ls+ e 50he greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are c0fna zvv.5 /nkonserved. Yet most moving or rotat/nnfka. z 50vving objects eventually slow down and stop. Explain.

If there were a great migration of people toward t),c0,z:e psgqi j t9*zyzn0tbhe Earth’s equator, how would this affei ,z t)sz 0zp,bgnq*tec:j9y0ct the length of the day?

Can the diver of Fig. 8–29 do a somersault without having any initial rotat- r7s/ryei q+.i(s k-w b*fbytion wheny-*is+ bkb- sr/ q.ytir wf(e7 she leaves the board?

The moment of inertia of a rotating solid disk about an axis through its center peu,lqt 00:u fof mass is 0f:qt0 pe,ulu$\frac{1}{2}WR^2$ (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 outstret3 w4 ozt y4nfgiq9kub/j kn9+*6(b mwpched hand. If you sudden+j6mn/bw4gi z93 tp4qyufk9( kw nb *oly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and hav wct3u:m myh-unf7)g3e the same mass. However, one is hollow and the other is solid. Describe an experiment to determine which y tn7mw)f-3um: ughc3is which.

The angular velocity of a wheel rotat(p;hk8d54 zdyy /fnw a.jatu6 ing on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the w( zdyfh/ay5 ;pn6.uja 8tk4dwheel? 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 decreasing?

Suppose you are standing on the edge of a large freely rotatingg(dl 99i ov+ic9id)fwkky. o, turntable. What happens iik(yvlficddwk,) og 9 i+o.99f you walk toward the center?

A shortstop may leap intovj/b ,3w9,qk,jyrkg, fv ykc: the air to catch a ball and throw it quickly. As he throws the ball, the upper par3,y,,,q wvrfjk / gcbk9vj:kyt 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 momenmnbu/2 7bl 38bkfe6wftum, discuss why a helicopter must 8n6/2 uffeb7 wkb3lmbhave more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles2ucw,boyhg .h+hhq 7* i0 tl/e 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. Calculate, usi1nrtnzgk2-4y d gpe41ax/5 pg ng the information inside the Front Cover, the angular diameters (in radians) og gy/ p4r 2-1edpzk1tannx4g5f the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The bea i;c qp3f.6nbam diverges at annp;a b.cfq i36 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 te9p2 zf *czj,6500 rpm. When the motor is turned off during operation, the blades slow to ret,*efjz9 c2zpst 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 flooy m37xjwo) 6mtv3szu6 r 3.5 m to another child. If the ball makes 15.0 revolutions, what is its mx33s7 o6)yuw zmv6t jdiameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. m iq/jqd+;:ui How many revolutions do the whd / i;q+jqiu:meels make?    $rev$

  (a) A grinding wheel 0.35 (annh.2slp.n,hu+ fi m in diameter rotates at 2500 rpm. Calculate its angular vel+ h,2ni(p l.fu.n nahsocity 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-ro /uhwtkq+ -,wyund makes one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a chiyu+w -wk/ th,qld 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 Sun3q8 uq; iu o7em,aq(ur    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed ofuv+vas . n: .wimied;(i/bca7f yec3; 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 pv0 rav f:a2q ;wfx((iqarticle 7.0 cm from the axis of rotation is to experience an accelerava(2f:;qqvf rxwai (0tion of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center fr /( z8wxn*lfqy2 jeu2ko.f4 abom 130 rpm to 280 rpm ezxa*bj /o4lu ky228(qf.wn f 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 radiuso pi+lz j(x4dzyj((b0 $R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboard the Apolx-* 6vw aay*nalo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter -ya* xaavnw6*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 accelerambt/ 5 tdx6q 0 hi;4lkg-a9gootes uniformly from rest to 15,000 rpm in 220 s. Through how many revolutions did 0moxgl4i d-5t9;gq 6kat/ohb it turn in this time?    $rev$

  An automobile engine slows dow;r,k2tffn-f urg( c(q n from 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 f; wwj ifr::9r3;3n*lm pzn v ra+t9nbilying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before;r w:*nn3ztmibn9v9wj:;a3 r if+plr 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 unifo xw9 6 4pzr,n0mgq) 5csyctkz08u0cvdrmly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have uscrq8 xv0mzpg0z c6wny,9d c 0)45k ttraveled in this time?    m

  A cooling fan is turned off when it is running at fr pjdn;i tvmpy80: +,-5pnvc 850rev/min It turns 1500 revolutions before im,tfvj8 0n5+rp -pp idn vcy;:t comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces itse +4qstonw2drlz/okr6+*;i7 hmyu r 3 speed uniformly from 95km/h to 452t6 *zr4d3+7 s rm/ ienkhuwyol;or q+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 reducn2ahbm3idaomn 1 0x8 1kdk4z*z*ecf 1es its speed uniformly from 95ko4 1znx*d1 h nf d2 bme*1kzm8aac3i0km/h to 45km/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 b k .7 t,o2lo5n1idaes(slh/s her weight on each pedal when climbing a hilsbde,ls. h/i2 (aslo 1kt75onl. The pedals rotate 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 oi::f7-tjbxk f2t d:sif 55 N on the end of a door 74 cm wide. What is the magnitude of the torquex:fks i2b-d : tifjt:7 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 whcd bl6; 8j(7hlwt jz/keel shown in Fig. 8–39. Assume that a friction torqljk8bt/dc6z 7h l;jw( ue 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 whichu.: v 1xsyn0rfqc*8ux pivots as shown v:fyn.x1uqc x 0r8*usin 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 engine r-wg (rh kqg motq,*7g6equire tightening to a torque of 38wmt*g(6hgk- r qoqg7 , $m \cdot N$ 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 1r+gwi+ zay7pf,:6 xna0.8-kg sphere of radius 0.648 m when the axis of+nrpgawf ix +y:6za7, rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of invvs u 9d;wf3k5ertia of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignored uksvf59d3 wv;(why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in a horizon(rxs9.j: iyv* xy-fmr tal cir.-:rryx vf9 *si (mxyjcle 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 constant ang v -beh367hbz,hlr,jm.zq a+sular speed (Fig. 8–42). The friction force between jab3h qehbz6 h,s .7rm,v -zl+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 objecoh(uf9 qji:b ,ts shown in Fig. 8–43 (:9ihj, uo qfbabout
(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 consgdnjelby0 z ncgj:*uy5(bd2i( z 8.;rists 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 correc4ict )yy20 wwdt rate, engineers fire four tad0)c w2y yi4wtngential 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 uniform cylinder with e1fbdqqbrz8 qm:b9jzx.n,( ; ht3e3h a lx */ga radius of 8.50 cm and a mass of 0.580 kghf;q1r3xbez 3*h.9 x z:t nqja8mb/eb gqd(l ,. Calculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat ps,7n /wcb3yxlip9i4, 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 ha /1y x:xk/n. khiv2pcfnd-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,n/yxi .2 /hxvkkp :cf1 and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shafe8 z -*6io+m-jv pmgut off and is eventually brought uniformly to rest by*jvez-m-+8pigm 6aof a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a 3.6h;s( upux8,(iz0 bj fp-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 t2jy9 a rzr,e-by the action of the forearm, which rotates about the elbow joint under the action of the tricepsejrtya-z9,r 2 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, a q1u9m3ri3euny5 +cyjs shown in Fig. 8–46ny3irj9c1u+my qeu3 5 .
(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 massebg obe.,,/a9lgw f7ot s, $m_1$ 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 fn ;en6h4a7pujj4 g yi0our full turns (revolutions) and releases it at a spee4j0n4jen6 u7p;ag yihd 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 inertia(ix *jgxvkr -gd,2kom- .k-jk 54utng 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 58ucb(q, uu i p hk8)mlwdo*2e lsmfy*89gy9dtorque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 76zxp7;9yu .8okc o ljos bn:h3.3 kg and radius 9.0 cm rolls without slipping down a lane atxy jkzb.so h;9p 6cu7:lon3o 8 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as thex; ihe-6bk*.ng glxe2 sum of two- hxk.gebi6xl2;nge * 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 massm.uauw5.rg: d of 1640 kg and a radius of 7.50 m. How much net work is required to accelerate it from:d 5.rm w.ugau 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 and +a5fj92d fe yvmass 1.80 kg starts from rest and rolls without slipping downf+y a5jf ve92d 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. It starts to fall and ikqn 839thjp.p dywdza o+720nts lower end does not slip. What will be the speed of the upper end of tjdyp+0qak7d .2hp93 zwtnno8 he 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 c ci 0,+b4d;vjwnhz+non the end of a thin string in a circle of radius dncncij40zh+ vb+ ,w;1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum o)ww ltdr2ru/h igg:: 7f a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotatl2 th gg:7/rwurwi)d:ing 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 him;49lcb ma cd/1g wzd2s side, on a platform that is rotating at a rate of 1.3rev/s If he rais;2wgamb dc4/9cl md1zes 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 redrxz4an2 8lq+r/ qhfk: uce 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 positi2 4+/f qlq:znrkxrah 8on, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rat*-vlu hg-x/y0ro0ya- oy w zo(e from an initial rate of 1.0 rev evgz y/au-loo0h( x ry *-vo-yw0ery 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 rotating eu mumdmg +/kgt.7m0 6giui am+g.a 4n1u*p,qaround a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What ist/g,6ug*++.1 uia0mu mieammguqdpng.k 4 m7 the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum /aj )/3zhslzsp 91 tryof 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 thedr6k 2rmawil )7m6 4vi 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 of moment of inertia I is dropped onto an ident/)u)g:xvz pac4o ;:plkam l8 qical disk rot) copuax:p;4:mlg zv/8 aql)kating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.44arc1uo q*6em.h8rt0t v4vt y $rev/s$ 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 zl mwt/.4v uu;merry-go-round platform of radius vtzuum ./4lw; 3.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an angular velocity of 0zt xi*k-m-h; oy 8 tsu/lw,oel/19lf e.t ol ewtz9/y 8msl/oexklu;,i-h-* f1 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 white dwarf, losing about half 1n6xcr;ogzc3p ,c, g)jc3nmbits mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the gcpj,ngc6 mz nco 13r ),x;bc3Sun’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,xwexg ujj,.. winds in excess of 120 $km/h$ at the outer edge. Make a crude estimate of
(a) the energy,    $ \times10^{16 }$ J
(b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 $kg \cdot m^2$) of radius 100 km and height 4.0 km.    $ \times10^{20 }$ $kg \cdot m^2$

  An asteroid of mass ; ao4bn l0v/v9qvudevv2yl; 1h;.y xb$ 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, initianz+4yo u/6anb lly at rest, that can rotate freely without frictin 46u+yza/bonon. 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-m diameter merry cdql2:a-bpkpcqn5mg-j ; (c;-go-round turntable that is mounted on frictionless :-qjqb5dg-pcl;;amk 2 n(pccbearings 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 rop sx3ikrp-y u4/bhk414en w99xi7n px gi6:wqce 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. -:cg1inx 4xxkeh 79pnkb s/6wi4u 4 rqyp93i w8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same sikm pv,5d ; t 2qlmo5+en;;mgmsde always faces the Earth. Determine tms nm+q ;g;2;kmdpe 5,m t5lovhe 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.)    $\times10^{ -6}$

  A cyclist accelerates from rest9ejxl8e8s )kn(g* 6ub)h odhs 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 shabuf w e tiu4b)fg55+hy )4*ciape of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculiua4gt)u)*eh5+ b5fw y fcbi4ate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cyr(ybg )1bs9 c;xr64;p idt rhrlindrical 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 0.0 (h9tgbp;s6xy4r c rr)b;rd1i10 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed ofgq,x gwj; 1(ky 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 siep2c o*m,9vtmze 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 carries off no angular momentcmp emvo9,* 2tum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-polv5prr-m0hts.fo; 4m p lution automobile is for it to use energy stored in ; rr-50mtosv4hfp .mp a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustra otju1subp*-y k u(n6/)m/r istes an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal surface at speed v b8,;+provxlgqr86 g lyi4w/m=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 an zq*lo +(0nj: f(w5 ezmd ,uqti5nkkc,d length L can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontallzkwqn5l: q0kc df+nei5t,j(,*zumo ( y and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. Itmxq zg7 83iv.wo( 3igv is standing vertically on the floor, and we want to exert a horizono7ivimw3q gvx(. z3 g8tal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a turn with aj2-uih;i t 7tql5g mb;. dygt: radius The forces acting on ttm.;gyd27-l5 j bq;thg:i u ithe 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 o9y r4 ;d,xnhpki80 fb7ssu8kr f length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelera dx;h0b84s7 p 89,knkrfryiusting it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder au-;o;jkthkm p0j942gj s0wvjnd her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular 4- wm9u kotj;v;0j sjghp0kj2 speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindrical 0jphfmvryw3x-uf77 t6,s -bbplates, of mass07-,hf 6supf tym r3wb-7bjxv $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 rollsxam q 3tl1 u9/u+zbb;usw(bays 7d+0 jggf:c( along the looped rough track of Fig. 8–58. Whatz g/fl 0;m+wqbyu(7 b:+s3uug jdx 1bc9 s(aat 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? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assume3llm( wj+pdph x+mug:t ks011 $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its spuj )mq :m*ndc(eed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration o :)un*m(m dqcjf 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