A bicycle odometer (whiche ivze*+o u+4g measures distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens i i+ovg4z+ ee*uf you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a poinl. u2pc;/jsc.a 4rxd4gn :xytt on the rim have radial ancgnyj a/4txu4:sl ; ..cd2r xpd/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a siirf6a 3 7:ha).y oalhwngle value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Expl -a99 k5 ley ash*iu5vbqsepol*.y8.iain.

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 don( 8a7,l4tcewe+:tc diidok :xvm+, x a sit-up with your hands behind your head than when your arms are stretched out in front of you? A diagram may help you to answer t tdil4ekd: xo8x +:t7c+,v,i mn(wc eahis.

A 21-speed bicycle has seven sprsjibzlw;n.o: 0zr z0,2 wh iv7ockets 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 sprocket? Why? In which gear 0ob,hzli0r z.wnv :j7sz 2;iw is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on beimz,yqjpkvi4d+ ru0,h*7aq p5 ng able to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distr+uz0 4 ,qy7i5 hm rvkdq*,pjpaibution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrofj+ks t 7bq azu(a8f j7bp/d6f9aym80w beam?

If the net force on a system is zero, is the net torque also zero? If the net toz,xf. q4mzgi 9i(i4f* (go+vorm7 dbj rque on a system is zero, is the net force z+jvi fzb ggirq74oi49fxmzd*o( m(. , ero?

Two inclines have the same height but make different angles with the horb dl36 2*r y5rwj zuy(0mg6prbq6p6uq9n cvm,izontal. The same steelc 3pym* uqu y d56blr,r66mvbj w6zrq9g20( pn ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from red4zj 0-9vrneh st) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom-04njhzdv 9r e?

A sphere and a cylinder have the 0 n uvuy4gzxx:kc3 d8m;ma:q ck+m(/ ksame radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? dvmkumk/xqcgz u(m::c043a8; nky+x Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most moving or rcv)8f lj3isoob pl9 q+m2(- vw z4a0bzah ak.8otoabbop4ql k)2f9 v wczv+(hmaj z0.i 88as l-3ating objects eventually slow down and stop. Explain.

If there were a great migration of people toward the Earth’sf y wbr(*,ae5r equator, how would this affect tr,ra5by( *ewf he length of the day?

Can the diver of Fig. 8–29 do a somersault without having any initial rotation wn(j5o 4;umkz khen she leaves thojuk nkz 45;m(e board?

The moment of inertia of a rotating solid disk about an axis through its center ,shj;0 )ncmx25d 2u ujth vb,knzn70sofc2s0un d5b;txn zsmj2)j hn7,0u, vkh mass is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2-kg mass in each ou n;hy9d)zb 1x5t 5/hzhj7sypotstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the sa7 hjds5x9;ybhp/yn15otzhz)me? Explain.

Two spheres look identical and har,u 5+w :1xg*yz rz ezp2pkd(uve the same mass. However, one is hollow and the other is soldgzx1*,pruk2e:w5 +p z z(yurid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotavul1+twq/t p 75gyt2 iq2qi 9nting on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or +t9viluq/ t5p7qwi12q2gynt decreasing?

Suppose you are standing tmab s0n2y -9)jki)fnjl gu3/t ;6 eison the edge of a large freely rotating turntable. What happens if you walk)g 0j;m tbis 3ks9ef/i6aj t2n-uyl)n toward the center?

A shortstop may leap into the airkym:e7c y,f,r 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 legs rotate in the f7:,yce myr,kopposite direction (Fig. 8–36). Explain.

On the basis of the law of conservi/s5;v jtu zj;ation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways thvsu/j ;tzij ;5e second propeller can operate to keep the helicopter stable.

  Express the following angles n:xt v4s.bbg z9q0)qo 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 becatv1:lb:op:*wcq*a p sm 4dd r4use of an amazing coincidence. Calculate, using the inforptq 4*dpo r1:s d:wvab *4:mlcmation inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directr2v +qb2;6q/ e 8o3pqydjip xzed at the Moon, 380,000 km from Earth. The beam diverges at an3eq b+yox z i 6q2pvrj8;q/2dp 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. When the motor is tur q t*hyd6denvlt3ye(*r48 n7cned off during operation, the blades slow to rest in 3.0 s. What i qh(ddnyr 3l tee4yvct8*76* ns 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 chin-s lm(u km47zb g* )k08hixjvld. If the ball makes 15.0 (u-mlj*7gim kh 0xb8k4 vsn)z revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameterjw +v5pc xd57.by (xsvln31te travels 8.0 km. How many revolutions do the wheels mw +dyj5 1p57cb lv.esnt(xx3v ake?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate its angu qt.pf(0pjc7hlap.0fcqp jh 7t(r velocity 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 mmf o7ctl 9/4r5hpvir1srs+ -eakes one complete revolution in 4.0 s (Fig. 8–38). (a) Whatft4srmcpl/ 9o 1+7v h r-s5ire is the linear 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) in its 4+mi , a8yx;fhrz5baworbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a poipg/nov 2 x n6kv(z9(gsnt
(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 if6 / i t,6n ,oqvp,jlo.cxtuys2 a particle 7.0 cm from the axis oftt,,ox6.yiou2 n6csv q/j ,lp rotation is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its ceazv,mlhns . vj lf0-9*nter from 130 rpm to 280 rpm in 4.0 s. Den-s,l .m*jla vf9hzv 0termine
(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 radiusy:d b0yfth9o d4,um5y;9 o zge $R_1$ 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, astrona3lz so+-1y/a8av yu;nrzo 8n puts 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 duringo8vy nz;1zaoyua8l s+ r/p 3-n a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15 iw xn);og h98lhxvzk7g a9nb h-3x6+z,000 rpm in 220 s. Through how many revolutions did it turn in tg-a k97 )hwvnxxh;g z69 nz+b83lxohihis time?    $rev$

  An automobile engine slows down 8ib,fip*8av yzpu p:u( f7gz4x( :zw sfrom 4500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeed jetstmr-9;w 2f sxe in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions f9er-;xtm w2 sbefore 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 accelerathl /n4gfxnp,l.:c2 qa1.i cfves uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edgenc f4qgal / hxcp.ni:1.2l fv, of the wheel have traveled in this time?    m

  A cooling fan is turnp8n mvfw y/f/:ed off when it is running at 850rev/min It turns 1500 revolutiof:pm /y/nfw8v ns before 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 the car reduces iik k.y i2j4rr3f(alq.ts speed uniformly from 95km/h to 45km/h The tires have a diame.i2 k 3ri(.fkq al4yrjter 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 -3sb rfroy 3y6yclej q+bw8+ 9revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires hs3q yrebyj8r3wf 96 o b+-+lcyave 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 a cnq;mo 34ic/;uozip7ll her weight on each pedal when climbing a hil3uzipq ;4omnc /; cio7l. 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 forcegfio hh8mv/q ,ij+i+*jd9q1 g of 55 N on the end of a door 74 cm wide. What is the magnitude of the torque if the idgjgf1j+m * , 98oh v+ih/qqiforce 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 axl rm (go ;/cb/cgzc9m.+vho .zse of the wheel shown in Fig. 8–39. Assume tgh ;coc/.o(/+m 9 zgmrcsb.z vhat 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 attached to the0/pk+ ict al;h ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position ck h;pl+0ita/and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening to a torque of w ub( gjlza4*jg *talx*,+2k -jr4 fix38jj(r4*w4,auxjlbt *k* ilgfx 2za+-g $m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of radius k/nd1h .ypc8l/i eem( 0.648 m when the axis of rotation is through its ce(nc8ey.m1deikh/l p/ nter.    $kg \cdot m^2$

  Calculate the moment of mueq 3+slo+s5;bbdh6e.n9ib u 7b 6keinertia of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.smiulkeb6 96b3d sb+hbq+e.u 7oe; 5n25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in a horizont.qzu ub7ze.7v alzzq7.7 u bev.u circle of radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping autt7-px d:usl: +3fzo bowl on a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between + -:zf7u3oxtptdu:slher 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 objects15:3y iu,dbv yg:ziof shown in Fig. 8–43 aboufvyo,1iu: big5z:3yd t
(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 t lzitu(dnl780q 8wo*qp tp .v,wo 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, unifo) f ju4z/1t0b-f0c ycldiivj+ rm cylindrical satellite spinning at the correct rate, engineef cbif zydj14/uvc) li 0+t-j0rs 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 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 radius of 8.50 cm a9 w,m- biqo/npa wg2c5nd a mass of 0.580 kg. Cal, ib 5mw 9gqp-w2oc/naculate
(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 it ;spu5;whm5)3 jzrpw h 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 tan gwdolg* yf:32a1a1px gentially on a small hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frica 1w32x:gf d*g1alyo ptional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm fg /en4/9nix. pkb2u s;5sdyois shut off and is eventually brought uniformly to rest 2s;4pbe. iy9/xnf du 5/n goksby 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 3.6nv 9fsm0mz64m-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, whico/5 u9nuu9sx mo 8v1tkh rouo5o/ u1n99x8 tmskvu tates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade 08v. o5ppyq; x qky-vrshoj6,can be considered a long thin rod, as shown in Fig. 8–46. ykj06 vp.qvy5,qsxr h 8op;o-
(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 twnlkwmr 8h: ;x+ig-66zlf uzp+o 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 wruk9 )7dfcg5 qithin four full turns (revolutions) and releases it at a speed ofc79urk 5q gd)f 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 inertvdyc /m8 cq +i;w/lh,iia 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 torqu q- (m4tsrjw 72lep1jre of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and rbj z pdk2v, 5uuw /tckp:1ft92adius 9.0 cm rolls without slipping down a lane p,fj pk9b:cd15uttkw22 /u z vat 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic *r )ck j+:l ck; b-(ykz5nyufzenergy of the Earth with respect to the Sun as the sum of twynlz- (yb5 k*ju) frkczc :;+ko 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 ck5h fk3zqahrqa5z)qmj ,ro(7 n35p6kg and a radius of 7.50 m. How much net work is required to accelerate it from reqk5fc5zkzr)ahj 7,(m h53aor36p qnq st 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 mass 1lywes9 . dt hz0yj1o4..80 kg starts from rest and rolls without slw4sy jl9e10yto .zh d.ipping down a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

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

  A 2.30-m-long pole is balanced vert udst:*3k cdp-ically on its tip. It starts to fall and its lower end does not slud*3kcdst p:- ip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the en9 oos:21wnoved of a thin string in a circle of radius 1.10 m at an a21s wn:eoov9ongular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 31d0x( ddyaox8-q) hz3 xgzyh2.8-kg uniform cylindrical grinding wheel of radius 18 1dox0d)3z8xh3 xyyhz ag (- dqcm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is4yv (hdqh z1.m rotating at a rate of 1.3rev/s If he raises his arms to a horizonta1v(.hh q dy4zml position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce her moment of id2,n.nn xl jsu--3yotnertia by a factor of about 3.5 when changing from the straight position to the tuck position.d2 y-3ls x.oj,nuntn- 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 in ( (mwe2ysou 4l-a,-i9is ps*4qrajupcrease her spin rotation rate from an initial rate of 1.0 rev every 2.0 s to a final rate of 34 4qsarp,u iw amei-9- psu2(o(yslj* $rev/s$ 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 rotatinh 5c 4ms 7ks78:t.hiyq a,thyhg around 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 is the frequency of the wheh8sm qt s4:5y7y,i kca.hh7ht el after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figurecp d/ a8f930ba vsnqk(l k10rl 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 momentumg k*siq i7y,+ 2w jxbim,)5cix 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 an a9q*jrvv vw13 pp/svo54,qifidentical disk rotating at an*s q9vipoa3, /5fjqv4wpvvr 1gular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4n iepwv-:, -.-sm lhft $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotating merry-go-round m):pl0e ri;ipplatform of radiiip:;0)rmle p us 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 wiu)44bkn b pys62hgo :0msnxa0th an angular velocity of 0.2 sk4pu40)nyonabmsh 0gb 6:x8 $rad/s$ 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 collapsesey)k7ezno-wma+1k fo+ y6u,i yqz 8,p 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 Sunm71ok8o+ yi,ze ky)q6,z -+p nw ufeay’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 winl8nrd+ ( vbh.jds 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 pdq-m,3 njgs:$ 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 platfd:x .g8 2krctform, initially at rest, that can rotate freely without friction. The momen kg.8:frcx2t dt 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 diam- qic dj9:-kyfeter merry-go-round turntabi9 y -kj-f:qcdle that is mounted on 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 the end of the rope a)gngmi.bk*+5 dubdd6z(v f9l ik/p gl1 0(bs lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mbf*kzgi)dbb/ 9i (lpn (u dkd+m01ag56s.lv gass move?

  The Moon orbits the Earth such that the same side always faces the Earth. Dete *l*u07gcrfb d l(xk+yg5q 3rxrmine the ratio of the Moon’s spin angular momentum (about its own adyfb5 *c07k(g r +xxg *3ullrqxis) to 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 raq-yn9ssga ; 1 vg2:+k axlk7kqte 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 ofn3zxzm g (: ivw4;w-s5mzm p4o a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval 5vs;wz-3o44mmw:g z mpi z(nxat constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disksn pp0 -rxt2vt/, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cytnxt v02/rp- plindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it 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 ro/hfd0ii 45w u;4s,ldcz rsw2ear 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 87xdb3g6a4focvri; t 4 +f-zyi.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;i fboia+3y c6r44xz7dvgt f- no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use energy storem x:;skgrh8j5e5 lb(h aq 7,:; wlvtyxd 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 ba h8lm rh5:bxxjg;ks,v5ey7 w;q t(:le able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates a79i3ms wt9p r(fr pyb6y6k) cqn $H_2O$ 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 h82 9*ys 9tfnsz fbyb6 t7rx6srolling on a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freel0pug9z;2w wsrf l205 xq:jd piy (i.e., we 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 the rod. Assume that the force of gravity acts at the center of mass of the rod, ap0u29rsf0w:;id z 5lwp gj2qx s shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radiukk,, xi9cya-/htk)sn i-ayg6 s R. It is standing vertically on the floor, and we want to exert a horizontai,h-cs9a)t,/ykn xk i ykg-6 al 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 makin;w6mybs.z 4ri:ys r;hg a turn with a radius The forces acting on the cyclist and cycle ar:4zs;ws ryb i m.rh6y;e 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 m and begins. ) f1nkjx7wz btepo7u-lgs (3 whirling the rock in a nearly horizontal circle above hisgn()1ujo-.ktz7xpe 7swfl3 b head, accelerating 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 b jfu4sip,6:i nody as a solid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angulaif pi,:n6s4 jur speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists o spt/q: kj/1szf two cylindrical plates, of/zqsp:s1kj /t 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 rolls alj m/5 ssqzmm+ or(n3m(t), fnhong the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the mmnf(+ zq/r )sj,htm nms3 (5ohighest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not ass /1be6rp9x v oh: k8z9unksd(yume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uniform 2rgskh -qdtx-kbz.6(ly from 90km/h to 60km/h The tires have a diabx(2-d.zk-r hktg q6smeter 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