A bicycle odometer (which measures distance t-h**yt/ lrst97 j gy-3gty6 (l9cnu)ai ywb ytraveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with y*u/ytn3ght9r -)-c(ytyygs 9 *t bj76walli24-inch wheels?

Suppose a disk rotate *y9yzykw7ds1,. cmf c udtc8.s at constant angular velocity. Does a point on the rim have radial az.*us8yc9dc.1f7 y mk y t,wdcnd/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 bs. nyl z z,d9d1 re1rm*hcm,1prh;(bz y a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? j(jrg(7v 5q i dkm;z0/pzuu;eExplain.

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 your head thaq. f9su 1rd8tnn when your arms are stretched out in front of y.d1 f9rtq8usn ou? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three .ei9d0 .pn s2bnu0goh at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In whichh e.npod9si ug0b2. n0 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 ;z2ea6/r3dyw2e4ryh ;of si*/zqkg-3kr uv qslender lower legs with flesh and muscle concentrat2yke z*3r/vd4;r; hu3z/26o rkwsa-ef g yiqqed high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fia0y m+25dx- f*8hghtcw hyyo/g. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the net t76g/cpn1.k lrsyen ila;) 3v porque also zero? If the net torque on a system is 73;rlg/6np va1 kep.)ycslni zero, is the net force zero?

Two inclines have the same height but make differ b-wwtj :nh.x u5x.:olent angles with the horizontal. The same steel ball is rolled down eat.ln uxbw-h.wo :j5x:ch incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from re6r5u3/am zjw19y 7wpinlmtwk 5 /fi 6mst) down an incline. One spmy6 mw zwp5/jk1a7uf65til3 mwir/ 9 nhere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and d1k ,.:crgug qthe 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 totaqc. , g:ku1dgrl kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular 5i y0 o;gjua zpmaf8d-8 bc6:cmomentum are conserved. Yet most moving or rotating objects eventually slow down and sz m5 8cy;ai:gj-fcp6 o0 ba8udtop. Explain.

If there were a great migration of people toward -ov64y kewyei8t( 13 fzw pkp8the Earth’s equator, how would this ay618 -iee84y3 otwkwk z(fppvffect the length of the day?

Can the diver of Fig. 8–29 do a somrsw v54t dq,j*ersault without having any initial rotation when she leav,v 4j tr5*qdswes the board?

The moment of inertia of a rotating solid disk about an axis through its cente l*0 e3razi58jw)i- bc hme5ekr rbih 358zlc*e)kmwiaje -e50of 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 jaa2leib 55 j.0euf4a2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the s2auf5 j5e4 a.0aljbeiame? Explain.

Two spheres look identi)1 i, ripxfpcmk7 t8w22t mfc7cal and have the same mass. However, one is hollow and the other is solid. Describe an experiment to deterp2i)f cpm27wim1k7 frtxt,c 8mine which is which.

The angular velocity o76(d g2 .z lqz/q0c cc6wcdlhhn41 jnqf a wheel rotating on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wqzc7lcj2h hd q .1qznld/cg 64n0(6 cwheel? 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 rotatmb;; j,pm3fksing turntable. What hap f;3,;bs jpmkmpens if you walk toward the center?

A shortstop may leap into the air to catch a ball and throw it quicklyle 9lx3fu 4 d5gd:4hol. As he throws the ball,3df:o l4 dl5 xulg9e4h the upper 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 l)ptjr:hh6rbb (6 mp2jaw of conservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propelle2 hr(r6:jh)b 6p btmjpr can operate to keep the helicopter stable.

  Express the following angles in radians: (a) 30x n)b o4/o2los $^{\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 t yi7ehold1u 0-he information inside the Front Cover, the angulhu y7-eo1dl0 iar 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. The bea kfp71cgxx*6d 77m(rrpdkx y1m diverges at an angle 1dypg(*6 p dkr1kx7x77mxr cf $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. When the mol m s+fcx8.lce) 0so:;uj:wxv tor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleratiox+:; 8fjlu os:w l)vxsc0me.c n as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another chiry+)*5fywf mq,2y-sq z8m hw )kuiqyr8 f14kkld. If the ball makes 15.0 revolutions, whaqfrwr)uqffm),s* 8mihw kyk-zy+48k52 yyq 1 t is its diameter?    m

  A bicycle with tires 68 cm,(zqd7j ,:b*zynycs d in diameter travels 8.0 km. How many revolutions do the whee *sq,b(:dnd 7yyjcz ,zls make?    $rev$

  (a) A grinding wheel 0.35 m in diamr8u4e.e ,wquo eter rotates at 2500 rpm. Calculate its angular vel e.ruo8,wqeu4ocity 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 7w-m3yd.p sgqwj(6vn one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 6gdv(wjy wnq.- p 37ms1.2 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 veiqxj-vw4q/ b)y 2vci 4locity of the Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a pi3frnyx(, 7 afoint
(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 partiy.;y wa*tn7e2u ,i qrycle 7.0 cm from the axis of rotation is to experience an acc,r*t aq2;y.wu7yeiny eleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about j 4afe)wu7;mgits center from 130 rpm to 280 rpmem w);a7fuj4 g 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 radiuy jevj7lzuq: o9o:z9-r)fore; t59 n*f udz7vs $R_1$ 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 Apollo spacecra:6k697r00eo s fc* ,rdxze3vyihtf dn ft 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 06rd dy he:7encf 06kxov3sf z,*9itrspacecraft 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 frdtxgh(h73d-ao hxi( .*i t;cbom rest to 15,000 rpm in 220 s. Through how many revolutions did it turn in thao chix*t7 dt.3- b;x(dg i(hhis time?    $rev$

  An automobile engine slows r/izog*5x )a ox1i l7odown 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 stressew 9 ,tgu36(h+kgrpnt ps of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its fi,utrkgpg+t npwh( 6 39nal speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 360 rpm *jo al 5wmc3t-n: y8flin 6.5 s. How far will a point on the edge of the wheel have traveled in this w*lmnayo3-8 lj:5 f tctime?    m

  A cooling fan is turned off when 8c/5kbl4gjyo h xddxe51u-+tit is running at 850rev/min It turns 1500 revolutions be5d/58 gxkxc1you+bhel-d4tj fore it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as t9u z2 p5)lpiwgdzc4e3he car reduces its speed uniformly frou4c9 2w)3ez pp5zg dilm 95km/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

  The tires of a car make 65 revolutions as the car rev+s0vcl jx. w9duces its speed uniformly from 95km/h to 45km/h Thelvsj+09 w xcv. 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 bikejo /p uh)5jwu i(-6wxu puts all her weight on each pedal when climbing a hill. The pedals rotate in a circlh (pw 56j uiuu-/xo)jwe 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 fotz2kd)uk4 fd /a j1+tbrce of 55 N on the end of a door 74 cm wide. What is the magnitude of the torque if the forck / z2tt+dfk4b)a1uj de 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..x( 1fsvh ebr3qrs;cva e:: i: Assume that a fricti ;av:rhf bsvece1 3.rq(is :x:on torque 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 endm7-ne tz.u:ku1xg0 b1d j,jlfs of a massless rod which pivots as shown ie.7,mbkt 1l z-x jjug0d1f: unn 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 hebz *5d dx.b-zoad of an engine require tightening to a torque of 38 bbz*5ddz-x .o$m \cdot N$ 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 inert ycr gh64x62d/d*nzf oia of a 10.8-kg sphere of radius 0.648 m when the axis of rotation is through it /6c* dnzyd 6o4rhfxg2s center.    $kg \cdot m^2$

  Calculate the moment of inertia y s33*jzjfnz2 z1nz( cv5f43;pn rm uoof 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 (zr j p3c*zv3u3os1 ;nf(fnnzjz2my45why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in a hory8 c+.7x9cprd/i7s9znoae t pizontal circle of radius 1.2 m. Calculat 7/c9+z.npso p ye8x rci9t7ade
(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 when7ju urn7zzft,*iy(0el rotating at constant angular speed (Fig. 8–42). The friction force between her hands and the clay z(7n7,uy t* z0nu ifjris 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 arrau )lgr0w8jj( sy of point objects shown in Fig. 8–43 about(swjgl0 ru)j 8
(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 consists of two oxygen atoms muwq /sb*ifobb)i(q::bm/gs9 d+ex9 whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the correct w-etup0s ;b2 irate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if i sbuw;0-t2epthe 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 radius of 8.50 cmc;o kw 3qyp/r8bj56vg and a mass of 0.58wck5 ogj/3;yrbvq p680 kg. 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, acceleratig ,8 (t:r7py-gc6,zqus rxkrgng 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 /dz *uvm7 gcjkig;t)e199osfhand-driven merry-go-round and is able to accelerate it from rest to a frequenc 1 *m9jc)vzfot;ke/gd7s 9guiy 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 1w osm0qy1 u:7 san*(psmm/9ccrk(xois shut off and is eventually brought uniformly to rest by a frict ramk:0qpmx1 u*owc7ns/s 9m((o 1y csional 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.p/l.*u r ,pze2riacd 96-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 solelr9o4siu tbr q+* m v*:o/zn2uky by the action of the forearm, which rotates about the *nrotq 4v9mr /:i2k +ou* sbzuelbow 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 canh-q)b4m8 npb2 uk32rs qe /3 3uzbzxbx be considered a long thin rod, as shown in Fig. 8–46r/3 )2ns2 3buxme 8uz b3-kqzhpb bq4x.
(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 c1zx 4m1c x*toxonsists 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 hammer from rest within four f p+zkc6t:qzz )wf8r x;ull turns (revolukw) x+f :pz zc86qr;tztions) and releases it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a moment of inertia o n lrs1/ou1bg*i10o iof $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque onm 7ysz kb:-ri/qz,k c 6;x;dhf 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radiu93+) trlji rs0n kbmr/s 9.0 cm rolls without slipping down a lane at 3.33n itlr ms/9 0j+brrk) $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the sum of t d8mc(:.v6bh2w5rum ep,py f qwo te pf6.58me:cr, (dwvu q2 ybmphrms,
(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 mogco8. sjy 5f)uch net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is y5og. )scfjo8a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest s ,jd8mz7mwex.zfk ok3w:;a zfp p0* /and rolls without slipping dek /z;oja3:8x* fz0zpwksw,d7fp m.mown 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 falo kq;,uhzs 63-g1l h(hogb5hsl and its lower en6k(ol hszo5hq g1gbhh;u -3 s,d 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 mo3u i 4bmfxo j7d29)hr6zvfi+jmentum of a 0.210-kg ball rotating on the end of a thin string in a circle of radius rxviffhzm 3)2u 4 7d96bo+ji j1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform 9naz/xh- .6 rnuj*tzv 2gxdg4cylindrical grinding wheel of radius 18 cm whej2 t6/.uzg-vn4 h9rz*na gxdxn 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 zifr-azj, u ei c*8-qx+,o 5kzhis side, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–4rzu8i,fe-xzqk i zo +a-c ,5j*8, 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 moment of inertia bro6:og x;6b ruy,q+ad y a factor of about uqdy6;,ax gr+rb o :o63.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial rate ofa6)j zhvh-vpz;fcm23g9e m4 f 1.0 rev everyhm)- h4g2 ja6ze39zpv;vc mff 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 rotatingmaq:h(zdj69d ftjw (mmw 9-k- around a vertical axis through its center at a frequency of 1.5rev/s The whee9(t qw: j-6jhdammdwk 9-m z(fl 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 mom,o70dnzzp m2p entum of a figure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum of the Eart .2hrrpa y)ey93naz3 kh
(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 onic(yp: fy8sx7 jzm :/q0: fddh:7ex zcto an identical disk rotati c:yzfdh0j :qzs:eid8c :fpx7/7x y(mng at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2sv)n:19ige n* h:s mjf.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center goj8 (w 0o9+ anomv9s+an1ptq of a rotating merry-go-round platform of radius 3.0 m and moment of inertia 920 sanj9vo01o g9( p8+an qwm t+o$kg \cdot m^2$ 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-rouu5 gm uvp0e1. 2,tmrflnd is rotating freely with an angular velocity of 0.l1.2trmp 50evf,gmuu8 $rad/s$ 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 co. )n-wx1t w zt3( n8hs9h kiw,hmh(ndollapses into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotati -h 3h)ownis81w h, htdwnz(n9m.kxt( 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 windsn/:8fp87a ngwz3mr;/p hqzs 0g(aam yiqmc9 ( 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 9lq)m/ 3a r(wz lw*rsr$ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, that can rotate flbj4ad3v3rxz*vop itmqq / 54p2 8;gcreely without friction. The moment of inertia of v4t5pd4cg2q83/m;zpiba x3r v o j*l qthe 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 seu+i:vykv 9,5i2h dmtz nr:6merry-go-round turntable that is mounted on frictmse5,296dvrkn i hy z vt+u:i:ionless bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of6+przx7 9lzes the rope lying on the top edge of the spool. A person grabs t6x7 ezp9 s+zrlhe end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same bs/-g36 b(qanxcl n r*side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) cb*/rxa(q -3ng nl6sbto its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate njwc5rz ou:c/e3pzq/4/wb(5 djb z)yof 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 thef:erefa22zw f)x( zh0 shape 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 accex 2) ar(zfzh 20:feefwleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical diskq0hok2j8y w5yh/sspf i* w 65gs, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical huh5o *qpysiyw 0/hks58j 2 f6wgb 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 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 retc2 vx/dgw ,0bar wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times as gr,eqp*3(e6 d1 1 yjessfhr ut1reat, were rotating at a speed of 1.0 revrh 1u, t(q j1e16 dpe*3syesfrolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for i35h eiz.k4qkt sd )ns6t 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 able to travel 350 km without needikn 4st. 3)dsh6 5zeqikng 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 ag 6e(kqxlovcv59 h;yi6bm(wd ma6 +r)n $H_2O$ 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-k ad7+lwbxwh6xy* jrw pq5-8ace 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.4tqg- (ra t k5j.fgbn:e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held hor 4kgg.rfj :tqtn-(ab5izontally 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. It is standing vertic7v* 6:ovfs ekyally on the floor, and we want to exert a horizontal force F at its axle so fsy :v7k*eov 6that 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 onv6l36o3 xqeyddv)j: s a flat road is making a turn with a radius The forces acting on the cyclist 6:eq x33 v6d)yodvsj land 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 /dh8jv btduh*gv)9, 1o; gh9kmz s2 wnrock into a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque r w mhnk99ug sbvohdtj zhg,21/)*8v;dequired to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a f5 +-fqcyku ,jsolid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and wit f-c k5ju,+yqfh arms held tightly against her body.   

  You are designing a clutch assem /t 0,ml0ethjm w(cry7bly which consists of two cylindrical plates, of c m0em(lh0t7/rjtwy ,mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 roljd6mq 4oh9mg i2iw* t np;/5atls along the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that gq5int;/jhd92 tmim w4p*ao6 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 assume vfu 7jq(+c ku8;6xm rhfn6nf2$r\ll R$ h =    (R-r)

  The tires of a car make 85 r+.guynt l01iq+ o x)d1wuo l3n)e9umuevolutions as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the am9odgnyu+11u)oe)u n+t3wx l0 uqi.l ngular 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