A bicycle odometer (which measures distance traveled) iszb* qcgie9xn p +9qu ;9ug5f*x attached near the wheel hub and is desi9ez 5 *nqugxqfpu big; c9x9*+gned for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on thex46 pb hs5-/djxznm9e x;t o:j rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangentj-4;9 xtpds:bex 6/ zj hxmno5ial acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body5qe5a (a j-rmg be described by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert rmm0v5 0; 9yts lnc9tno onu-(a greater torque than a larger force? Explain.

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

Why is it more difficult to do a sit-up with your hands behind ya)jlh3,ph k tq,n-x2m :nbf5tour head than when your arms are stretched out in front o-jl,3nf b)h: k tmtq,xhp 25naf you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprocov/+apd)xdy 6q w2k0kr kx9) pkets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear spa )p+kq0xp v/kxw6yo9d )dkr2rocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower legs witt8:jr s,.tz d.ukua*t h flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynzusk. u:tjta8* . rtd,amics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrowciw*y9o gwk 0+ beam?

If the net force on a system is zero, is t)y-cfa8ci.t6dz0) qh e yit)u he net torque also zero? If the net torque o)q8uth.ec t-0c z i)iyf)yda6 n a system is zero, is the net force zero?

Two inclines have the same height but make difphq /(2vlqv2s/p z5q fferent angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bvsvp/qfp(2 l/5q q 2hzottom be greater? Explain.

Two solid spheres simultaneously vfokk340e g9 y0b*h9kg ml 8vustart rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of thvbf g*k9u e0gk3o0 m 4hv9ykl8e incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. They start from+gouy u (vy-r5 ie/ed(j 9dze. 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-z gvd /dy95e +oue (.(urjyei bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum i/ c61,uvxah; o c.zqqare conserved. Yet most moving or rotating objects eventually slow down and stop. cohz1cavx uqqi6, /; .Explain.

If there were a great migratio4ovex8ugre30xfe6v k8w lyj8:)d1 *t 0n ygnin of people toward the Earth’s equator, how would this affect the lengyyeo k0 1njv xu03*4lxgtwv8 r) n:d8f eg86ieth of the day?

Can the diver of Fig. 8–29 do a somersault without having any initial rotation68 *pe:9w2n48talgi: c+ix ,d kdt8xcqlbgtd when she lel*t:t+a88dw c6xtg 2,l:dp c 8xqdibk4e ni 9gaves the board?

The moment of inertia of a rotating solidfz 89pf. j(jkk disk about an axis through its center of mass is 9pkjfkf(z8 . j$\frac{1}{2}WR^2$ (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 onhcl- 64(tmih(0 a+pt(f wrt9vosuup9 a rotating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, w6opa 9+sm p wt c((ifht (40ut-lvu9rhill your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is hollow and-zp12p n yai 94clymr;m/wy ,g the other is solid. Describe an experiment to determine which is4ny ywp/a1yr;, i -2 cgzmml9p which.

The angular velocity of a wheel rotating on a horizontal axle points west. rs( 2 kz.dm5daIn 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 wheasd 5(k d2mrz.el. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge ;u hc8c-yw . ce)pochq*:syvk 54w1cvof a large freely rotating turntable. What happens if yov.) 1sk 4uc5-q*w yy8ccpewhc; :ocv hu walk toward the center?

A shortstop may leap intoa.enu ((p ij g-4os8ui bx+1vb the air to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legsspg((+.ju8xvibo b 4aue i-1n rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angrbxl8op:8qw. yf r.8d ular momentum, discuss why a helicopter must have more than one rotor (or8br .qld rw:y8po.8 xf propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in au3/av- j3 qokkq+s(u 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, usc (.m+h kpt1 8w2cue nou*a:u/gfw5zhing the information inside the Front Cover, the angular diamh utznp.(hu cegu wac/21 +omwfk*8 5:eters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The stvfk .z)a:536ux.gp b-qzdiu-d*tc bx :m8pbeam diverges a sfpum6- 5gbpq.)zdx ixz* -bt c8t:k3uv.:dat an angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. Whede(9b ksjg0,sn r.kk*n the motor is turned ofjd b*ek9kg.k( srn,s0 f during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the sy q;/ hgk;4uex.uz/t ball makes 1 kt ys;4u/q/; egxuzh.5.0 revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter.33ksb ,je1ifatq *iov .gzn1 travels 8.0 km. How many revolutions do thqbi 3ajog* i1f.nzk3vt , .1see wheels make?    $rev$

  (a) A grinding wheel 0.lm. .w qn8z8e f0r:o5t jwv3ce35 m in diameter rotates at 2500 rpm. Calculate its angular velocity in o0ej3qwrwz 8f tn .lm:85c.ev$rad/s$ $\omega$ =    $rad/sec$
(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-roundt7w 8,irnye:d makes one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linewe7:rni,ty 8dar speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a)vk4ik.c,a d 1zil1y1v in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speedgkzq 9*16 kzrhxwkg;-iuf.(b 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 if a particle 7.g0xjstud 6:7j gt32ht 1j tioku 3ta;+0 cm from the axis of rotation is to experience an acceleration of 10:t ;2t 1tisotxg3duj h+37gauj tj06k0,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly abfluq76fh1 8vdout its center from 130 rpm to 280 rpm in 4.0 s. Det1 lvdfh 876ufqermine
(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 radiupg6p5 6jayr 0rs $R_1$ 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, astrona(kboum/ 7-ipp dhxqqm w:)d .3h4-ycquts 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 every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determi.owp3k(qxdic7um-)- dy qm4qhhbp /:ne
(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.buh 9(yf4cei l))wt494whatw Through how(wthblw)t )c4hy 499ufieaw4 many revolutions did it turn in this time?    $rev$

  An automobile engine slows down from.q2rq 2hm(rov 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 t+e:5ut;uw4k wy uqw -4a 1mdqnn7t/ a;toz4bqhe stresses of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 compleez 4tt:4 +/7uukam qwwy5ud q t;nb;-noaw q41te revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 360 rpmd ue19k+twogzs5 .a, jo4vd3r in 6.5 s. How far will a point on the edge of the wheel have traveled in 5e+ 3oa.zouvt1,jsr4ddw k g9this time?    m

  A cooling fan is turned off when 9o4pfia wup24 it is running at 850rev/min It turns 1500 revolutions before it co4pwof 4p2u9ai mes 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 ua3f d0d )qy:ejz3p6 aa 5w+wsas the car reduces its speed uniformly from 95km/h to 45km/h The tirw adz6 qs3 0j:+ au5efpw3yd)aes 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 revolutio1) qm y1nhpjn3vx8.2pymgx0t ns as the car reduces its speed uniformly from 95km/h to 45y xgqn v3ympx81. 2hj1m0nt)pkm/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 alx t)lf/r+8xl- 2 e8nzb gad/zsl her weight on each pedal when climbing a hill. The pedals rotate in a circle of radi+elr) 8 - asxzbft/8dxl/nz2gus 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 u6gumzh,ko.qo11z p :dj:b4c of 55 N on the end of a door 74 cm wide. What is the magnitudu1k:zg6mz co .,bjq:op4uh 1de of the torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the sffbt;a6/h7 h wheel shown in Fig. 8–39. Assume thfh/t;hf7 b as6at a friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are atth ,fgsj (y+ y 0n3n,qzto5oue0ached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque , nyfs+0 y ,uo 5t(30eqgnojzhon this system.

  The bolts on the cylinj)9lwzn.mq ,4f90mr u j+h kk/ zbqeg0der head of an engine require tightening to a torque of 38 mg,4 zw)hbrqm9uf9kkj. nq0 l+j e0 /z$m \cdot N$ 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 inerti3cyfl (7z mvg)a of a 10.8-kg sphere of radius 0.648 m when the axis of rotation is through its clv(37 z)f gcmyenter.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicy )b z*h(c0me o:umhck8cle wheel 66.7 cm in diameter. The rim and tire have a combinede(hhkmo )cbc 8 mu0z*: mass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a ko sz7 lt8dq7o6dyl93 thin, light rod is rotated in a horizontal circle of radius 1.2 m. Caz qkl6st8 3ol 7dy9d7olculate
(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 consto pcj*l9,wz+ k pk3m5yant angular spee pom9+, c*kkj5lpzyw 3d (Fig. 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 ai0gy/8 ir4 b:xn8v9 lf hrrx+crray of point objects shown in Fig. 8–43 abfci8ir g8vlr/+x 9b4r:yhn0xout
(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 cofbr0-i1p-6w 24it)x f q xzcujnsists 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 (/(g:8qbcu05;bikjkf4nbpgook 5xsh +u. k tthe 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 i5 ug 5 tk0+8bokubj.sghpq ( k:fxonkcb; 4/(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 a ry l cf*li8:ba5g9gp :uadius of 8.50 cm and a mass of 0.580 kg.y:gifc l58abp9*:ugl Calculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerating i mc l8mwmk7y1:t from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven merry-go-round and is abh(2 rzl u+rfn-m) 2jxk3zealjr2p ,.ale to accelerate i a( fua+)2p2rzlkz3nh 2 ,x r.l-mjerjt 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 torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm ish m(kp97o(spc shut off and is eventually brought uniformly to rest by a friomp c(9s(pkh 7ctional 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 acgo0pt:)/ gon 65y jpvpcelerates 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,c( 3(hx3p f;:ys(xfwhg7z ciy which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from r7c pixzygff3 s(yh w (3c:h;x(est to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, as shown gzd2sbb*5 giz, *yrm 6in Fig. 8–46.bb sy6m*z i2g,r g*5dz
(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 cw6 04zkjgbodhe1 ::nvonsists of two masses, $m_1$ 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 hwo9u3 / w.:at1/adpcs ijkg +.mojzr;ammer from rest within four full turns (revolutions) and releasetu owazc/gs;.1 mji +oa.p/:j3w9rdk s 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 a hv4j,8rswo6.y h7c rinertia 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 developd9ra8c5(e -2vghvwu tmji io2:r2l;l s a torque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7 62uw k,l;ooupmdrb); .3 kg and radius 9.0 cm rolls without slipping down a lad lb ;mo,u)wo2k6pr;u ne at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic enero(; oj,*7,r ejt1debez t *jrrgy of the Earth with respect to the Sun as the sum of tt do(e*r;,r1 zretej,jb o7*jwo 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 of 7.50 m. How much nk8o8kau b:*xa 2a2rlq et work is required to accelerate it from rest to a rotatbrk82lkou* a2xq: a8aion 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 1.8ni:, se 2u2gjo:j *wc*en9r gq0 kg starts from rest and rolls without slippi2 ejcr:gi w*2*j:9nngou,q es ng 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 ba:kqq.0q8 le v*oha (malanced vertically on its tip. It starts to fall and its lowerhkl(0evo :q 8qq m*.aa end does 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 0.2(cax4s -ept7u10-kg ball rotating on the end of a thin string in a circle of radiu74 pa(- xtescus 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular v eb q.pe6iw.2momentum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotating ati..pq6eb 2vew 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 r7r6sf.eev so g tt093hotating at a rate of 1.3rev/s If he raises his arms to a horizontal posig096o sf7sre.v htet3tion, 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 ou cfn.bw2/)vone shown in Fig. 8–29) 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 nov.u)bc/f2w in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her w*1- sp xmpa0e16skha d6pfm(spin rotation rate from an initial rate of 1.0 rev every 2.0 s to a f 6 1mmakpas-*s1 6xpdwfh 0p(einal 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 vertical axis through its center afcl/ e:* z nwuf81ofu/t a freq1w*o 8un:l f/ff/euc zuency 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 the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momek7cksvgfxn0: br)(2 z ntum 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 angulasgyzw7*1 q+ uzlpr7j5da :jgu*w 94cpr momentum of the 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 as 6wh15xq9daei6k z p)n identical disk rotating at angular s6x)kzida 65 seqpw9h1peed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2v,do2 g1xebjxn( f h*6.4 $rev/s$ 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 kgl5gf,ty anyn * *j9m*m stands at the center of a rotating merry-go-round platform of radius 3.0 m and mom antlj*gy *m,m*y59 fnent 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 fa7hlgdguqe0. e.1y5)1+d xx5 zly vb0.8vee1 x5.uy)0d zl x1g5 ag 7+dbqy.lhf $rad/s$ 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 a, hki4,gpey* ap(7wl:jl uk -about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotati4lau7wih ,kg -yjep( p*kla :,on 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 wd3. rnrdu:j 4winds 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 * v* nc.f8 b(ur9nznzz$ 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 platformo82z fx6 svq/a, initially at rest, that can rotate freely without friction. The moment of inertia of the 6aoqsfx/8vz2person 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 diameth+b ,nq- 5zzpvch59ryer merry-go-round turntable that is mounted vznp y- z,c5+rh5qh 9bon frictionless 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 th,jee-rn+feul.c l5-:au y9uqdo6l yx ;9c3aie 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-;ur6au93cxq llu:c+lfdjne 5ya9e-i .e, oy onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the 5ihsnuq/ ;n4g b0/y3 wr azgb;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, h g;nzb//0y;urqsba wi43ng 5 treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest thefd,*ub36r7q+ zr bat 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 uniform3h2ja vi 3h2ae cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered he3i vha2a23jby the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 kg and dieun8h ;oy81bdameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use du8he8 1 nyob;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 of the rear whesxu79lo; s) x6.yigybel ($\omega_R$) 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 massco/46axbz 47 ak h9iis 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 k 4i4 7b/sxzic9oaa6hcarries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution auq5 slsstj;rsw-2 ;e (h(xbzyy.lw 9 7ptomobile 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 diameter 1.50 m and mass 240 kg, and should be abls s5;2s9; weq (ty sj7yh(lz.b prw-xle 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 illustrateso(ydp ps ze :28;1cfin 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 surfasx;e+6;ogda9 -iuc c. lsqg0 zce 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., 0c p/:)tfbjjpwe ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of t)t /bpp:0f jjche 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 rbgz*tm/,sa1spro;ca g;)ue/ adius R. It is standing vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has hu;/*st,gze 1r /cgabos ;amp)eight h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is makingrgrq(bm wlutu4a *)vm q0*9ff j 7.+wd a turn with a radius The forj04fq)*q vug a+u.9wm*lfmbdrr t7w (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 1.5 hfbtpy40id;h: bc+xtb5cb,. m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve thistc tdch5,xhp b:b;i 0byb 4+f. feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her d(g1 bv**qc:c,6fman ga ac . guioj)4arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skatqocaiag cv g6b . amc,(*:gd4jf*1)nuer with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylimhnu qr x4gt t m55q5b/kb.++rndrical plates, of m 4rtm5g b 5+ut/nb5xq+kh.rqmass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 loopibk9jd(o x.1: 9t akuwed 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 reacho:d9j1 ix(watu. 9kk b the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but -wwiyi5q)xa( 7 /:p lg8ffcpxdo not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car(xt ezrb0p:i8 reduces its speed uniformly from 90km/h to 60km/h The tepx 8rbtzi: (0ires 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