A bicycle odometer (which measures distance travelu*l x(( wou*,:y6q,l ognvm4pljoy1 bed) is attached near the wheel hub and is designed for 27-inch wheels.l(oyx *j qu,y4plw 1louo6* ( v,bm:ng What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at c 7szx6 0 cu/1cvf7mjmc. *qqcjonstant angular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have q.u/0c7c1mjq js*xc7z mvcf6radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value o2o k0-amg 2ezs9w+ws mf the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a largv.jht3yn 0px whu-m56er 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 behin mxx 4 zll3b syz9k.l0nv)6w6nd your head than when your arms are stretched out in front of you? A diagram ml) 9b.0m6 ns3yz4kxlx6v znwlay help you to answer this.

A 21-speed bicycle has seven sprockets ek/qiaqpuc/l 2;d5:at9r:.zh 5 zwyqat 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 sprockhd/ kplw 5yz/aau:qie qct :;29.5 qrzet? 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 lobeq15zea ipjg7r+,d 6o lj3h.wer legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On eg,jpdzah1+ 3qeji76 .5 rob lthe basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a loj7psx ,kt.vt 2ng, narrow beam?

If the net force on fnd m kf;9m74xa system is zero, is the net torque also zero? If the net torque4;m9n dx7kfmf on a system is zero, is the net force zero?

Two inclines have the same height but make different angles with the horizontg2+1lyacj iqsw-os +- al. The same steel ball is rolled down each incline+cw+ o ygq21 jl-sa-is. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously starz.qomr*woy 4+g;q: pg t rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which;q4zowo+ *gmgp :q.r y 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 i 1m fyetf9:nu,.cah *from rest at tht,efy: fhcu9aim 1n.* e top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angullf c-j +20 s:ahgc4jlnar momentum are conserved. Yet most moving or rotating objects eventually slow down and stop. Expl: -asc2g 40+hlnj cfljain.

If there were a great migration of people toward the Earp901fjw esur k,;zvh8 th’s equator, how would this affect the length of th jzvfhr;1wsk,p e 8u90e day?

Can the diver of Fig. 8–29 dohr02s: ksta)v j4x9gf a somersault without having any initial rotation when she leaves the boarksa4 vj:g2hx9 0)sf rtd?

The moment of inertia of a rotating sol0qa *u e7)fwp/dl;j3x ept(rxid disk about an axis through its center of maslx/;x3upt7 (wa e 0rd ef*jp)qs 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 9 h0c/5w mx ogiow*:1wfh* isg2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decreaomw/o5ggs 09i1hwi *: *w chfxse, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is y3g3b 9mjz s4 umjt;(fhollow and the other is solid. Describe an experiment to determgfjj4mz3(; u 3ys9mbt ine which is which.

The angular velocity of a whe ,.-jodj0 lrvrd9gnn)el rotating 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 point.n)d j r,dgrj-nlov09s 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 rotatf a)emnojt kdv4/ 9v;9ing turntable. What happens if you walk toward the cen;f4 ton/9 a9jdevm)v kter?

A shortstop may leap into the ap27h,zb 6r 5h j5mj-qnn30v p7 isi0 1asxkfmdir 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 w m-0zpj5v765 psm x2hi7fdsb1jqa krnh ni3 0,ill notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angul6i+ *:uakis+hw4kf qkn-i0x1 10ec o 0nrsvwsar momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the wu- ankfn61iwk1sqr kc+e0ho:0i i0+4*vss x helicopter stable.

  Express the following anglk2)5oshyur kbi)9 3lm)b ql)u es in radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincidence. Calj/g 2ibznu*.z culate, using the information inside the Front Cover, the angular dian*2zgibu. zj /meters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moo8 o o telgx2u0s *iidg*j*v0.vn, 380,000 km from Earth. The beam diverges at an ang80 v*g .ilvgdj*s2oi *e0toxule $\theta$ (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 motor6hz- u8j;ti h:jqh lio/eg n// is turned off duiegj8/uj/tln h hi- oz/;qh:6ring 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 6yx4dxdm ) +u3hgklnj1h 9iy1 child. If the ball makes 3g9nmh)ljh di+uy xy1x4d6 k115.0 revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 87 c 6rryv u.lc(+pr+/5lhglkw .0 km. How many revolutions do the whee klgwl(7yccu +5 r6+phrr/ .vlls make?    $rev$

  (a) A grinding wheel nxc 9+5bild-p 0.35 m in diameter rotates at 2500 rpm. Calculate its angular velocit+l9-dbn 5cipx y in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.0 ydtwktro p(r *t(3 3)w 5ahkc,s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m k5c3d( *(rpykt at,wo t) r3whfrom the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orbit around the Sun 0)srd3ion 62ro ur: ec   $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed ofih6j w kv1rwos0:s5 3z 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 2vqqkx: sp(e-5h xlmu2)qfu-if a particle 7.0 cm from the axis of rotation is to experience an acceleration of2lhqq)qx e(:ukm2- v5p -sxuf 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelej d+b db)i5,w+0iup zcrates uniformly about its center from 130 rpm to 280 rpm in 4.0 s.+d 5ii cubdb p+w)z,j0 Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radius m o;zr;5 z1akl$R_1$ 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 spacecraft put themselvoj pvuup. :v z-8kt.q8es into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accetzqjp v:.k8p o.uu v8-lerated 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. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpm in 220 s. 83pc7n5 pfailThrough how many revolin7cl35fppa 8 utions did it turn in this time?    $rev$

  An automobile engine slows down fropviuw41k,s -z m 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 jets in a whirling0o i2, kr:3db46eeduor e,gvb “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching itsb v:, 46dr2eborie d0 3okg,ue 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 36* :ems9:d lvibh nyaz6u(8c9 s0 rpm in 6.5 s. How far will a point on the edge of the wheel have tr:m9 9dena8 *i:cuys v b6s(lhzaveled in this time?    m

  A cooling fan is turned off when it is running a*4w fb:tmfxj -6ekl fw17gu3qt 850rev/min It turns 1500 revolutfkwfu7 1w 4 l3gj6:fxqt-b*emions 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 air+d1qa)b f g k+r,/d2ijs dw.s the car reduces its speed uniformly from 95km/h to 45km/h The tirbd) gs,2r/iqifjd .+ kda+rw 1es have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed unifo1cbi 9bc5i)rdkhf3 z9rmly from 95km/h to 45km/h The tires have a diameter of 0.80 m 5ibc)r93b9zdfhc1ik .
(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 j dddzj;-5j s0a bike puts all her weight on each pedal when climbing a hill. The pedals rotate in js jdz;5d jd-0a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the 9w6q7xt7m8/ tt/hgjmv jnoz g3 6fic/end of a door 74 cm wide. What is the magnitude of the torqu7x /to /fwzi 7qgtm896vm/jj 6hn3gcte 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 axl 3 fz30cfhqm*7h mx-vle of the wheel shown in Fig. 8–39. Assume that a frictf3hz-7 mq*xf30 hlmvc ion 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 ends of a massless rod which yo7 r71 wkfqr4k (ms6zpivots as shown in Fig. 8–40. Initially the rod is held in the hoz7yk1k (fs7 ro4m6qwrrizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engin.uq-a/r8t rdv;d8vkh/ eh )ace require tightening to a torque of 38haqra.vh;edrt8/-k d 8/ v)cu $m \cdot N$ 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 +9 v2 qmyfvk16:jjprgof radius 0.648 m when the axis of rotation is throm:jfvkv 6 91 jqgrpy2+ugh its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bi2j4dj: kuc: zn,;iogprr o- )scycle 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 ignored2;-: :oi, rodurgzk4s)jnjcp (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a y6g c stlo8mq6o1r 2q7thin, light rod is rotated in a horizontal ciym2lort g 6cq6o87sq1rcle of radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on z/,qb0 4xcm.gf u6w sja potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and the clay is 1.5 N toqfs wmcg/4 j 6zxu.0,btal.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point objects shown in Fig9n) d4t8u p(h; xtbund. 8–43 abou;d8u hub9t4 n(nxtpd)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 two oxygen atoms whose total mass is k : ,4iwfqc.n7-f n8cdh92 lxgiabmd($5.3 \times10^{ -26}$ 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 cylindriln48u+z4ok;izx jh39d iaew( cal satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the sate4d w 4+ ix89e(haikoz;l nujz3llite 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 radiuox9n vypj pww3*d8 ; ryp5j06rs of 8.50 cm and a mass of 0.580 kg. Calcux wdow53rp;j9yy608p rn j*pvlate
(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 from +ye2cm :a:y 90epcpou tqm43qrest 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-+r w,:;9myindbxuq*e j)hf*3in2 wq wgo-round and is able to accelerate it from rest to a frequency of r;n,)wqjq*w9:fw+ e2b i dmnyxh 3*ui15 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 rpm39pdj )3oki4j rvksj3bo y-gegb()4f is shut off and is eventually brought uniformly to rest by a frictional torque of 1.2d-i b(334g))f j3oev9ps4obk jyrj kg $m \cdot N$ 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 3.6-k68 ef a0guirkyvp 6kmc* 1t:o/ij( w;eg 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 thcu/y7 g gu 5/atb6skv2tgf51se action of the forearm, which vt5yt a7c/s /1ggb u5s f6gk2urotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin g) xppqca n6wyktta-.9j01/y *gdjb)rod, as shown in Fig. 8–46 pc-).9bn)0q1jjtaak*ywg gd/yt 6xp.
(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 consisd s/m 6hoc6u2f 6y8;dwhdq ny9ts 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 wicp3 i)z5 d4aohe 7w8)n1cyu ylthin four full turns (revolutions) and reyce1l)) y38hu74p 5wdac noizleases 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 iner m3vbye7f fk+2tia 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 dek:bec,5uaa8by2we qjj(8 : qz velops 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 7.3 kg and radius 9.0 cm rolls without slipping dow8mk2w9er +6e ul: p7vo zmdlky)z(genxf+k4)n a lane amfe9xk7u4w +() e :np)6geokvdm r2lly+ kzz8 t 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth wico2x6(v g, ycvth respect to the Sun as the sum of two termsg, v6ovyc( cx2,
(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 mal(1d1 jydh6ui ;guemiy 9y43b ss of 1640 kg and a radius of 7.50 m. How much net work is required to accelerate uy34yyed d9u;j1i(gb m lh1i6it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts 2q/: omstcol 8bbq ,h21v p/d9dm-ge yfrom rest and rolls without slmdgve/ o1s2bh l2b, o-9d pmy8cqq/t: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 bayo:vu zx w c8u*t8io1;eex3zvt9z- ,ylanced vertically on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end t3 x*w:8eyz,zvoixu ue1zc8 -; y9otvof the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentumhb )5 kmimilv.g5e ;u6 of a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 mv6k.i ;)mlh5b5gmie u at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8xiu* fqt ur.8 )p8+,1xvydn3otes c( w-kg uniform cylindrical grinding wheel of radius 18 cm when rotating )wd iqn3r.f+* x81x ytpeusco,uv8(tat 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 rotvl zw(0x :min7sf 5i9jating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48,wj s mz(nlfvx9:075i i 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 b(lpz0r2 yv ;uai 1 k+15fa bonz-.icbty a factor a1vb;y +oltkani zf ( rupc5-210i zb.of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck 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 uv/. 3edtni2j6nb h2 man initial rate of 1.0 rev every 2.0 s tn .ijb/ et2u6d 32hmvno a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis throug:/p wz68enr:qsh cx *cd:fm. jc8wg9rh 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 ofrgh9/s.fwn c* 6 edm:j xz:qrcwc: 8p8 the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinning atl4d gdw.3yrk : ,6zesp 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 momentuxguyz8t saajk83lf,lh9 1-)j( y( btk m 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 ofg9 jz) bo-slqo3d d9l1 inertia I is dropped onto an identica9j d13o- od 9l)gqblszl disk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at pbiy)k4 .:txz*,tovf2.4 $rev/s$ 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 of a rotat +ww t3l8of*greb1ls*ing merry-go-round platform of radius 3.0 m and momwlo*w * b s13fe8ltg+rent 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 merc ixjv*.k ,jsjqhz;3jszq30 8 ry-go-round is rotating freely with an angular velocity of 0.8 ji 3,qcjs 830z. hjjszkqxv*;$rad/s$ 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 eventuaxbd bhxz) .:.b9g u+grlly collapses into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest bb ux .+hxr.g:zdg9b )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 winds in excess of 120ge3d dm 4ih;3a $km/h$ 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 zyvij gslka*brv( ) c13up. ;*$ 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 r4ek-rolm3cb ) n3nvp +otate freely without friction. The moment of inertia of the person pll+-p bn3em 3)4nc ovkrus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diameter merry-go-round turvbjjq)kq 7h54af,acqvn3 rv (+em(q 3ntable that is mounted on frictionless bearings and has a moment o vv47mkfj q3(ch)a ebnq,qqj5( v+3arf 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 lying on l2 2m.mm,t9 :jra 3rdt2azx ovthe 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 oam2x:2zm3o . d raltt9r 2m,jvf rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth suchzgmg 2a:r9 po6 that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat th2ozag 9: gm6rpe Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from re38(9 (y:qnbidandq ymst at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a unifo2ryajdit5 z ,+rm cylinder of radius 0.20 m acquires a rotational r 2dy r5j+zta,iate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two i(;o)y ldpp:b6zqf r0solid cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub oi q; rpdzfb 0ly6:o)(pf 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 reo. y h:87sc2xwhc6e rdar 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 s r1ipjkf; k t/,u;1yxvri18zhize of our Sun, but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters 1yfk j uiitpx ;1;8,vrk 1rhz/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 isqq,; (mty 1dlytg:d+9arz3 2jmjw(ws 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 cylindriclgt z w1jyr sq(dm,2(+:;3 amqwy9d jtal flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates aoacgn5 )+uba.m; ,yzdn $H_2O$ 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 rtq-tbm.x m 74folling 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 canl;afo-t: at, /mo sr9ijp-2bu pivot freely (i.e., we ignore friction) ab- o;io2b ltuf9,sa/: mjra-pt out a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on pbpd0gv3qg;)v/ oe* r ji h6h*the floor, and we want to exert a horizontal force F6 p/goib hr pve;j *gvq3hd)*0 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 h.4qkj huw d2/pn(6jxa flat road is making a turn with a radius The forces acting on the cyclist and cycle are the normal forc /.hu(qx4hdpw j n6jk2e $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ 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 whm+8 3vjyxh s)kirling the rock in a nearly horizontal circle above his head, acceleratj8v h3xym+k)s ing it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solf9h7zgn zj 0+v i ;n6kb tqe8,+qyydq,id cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of tz+ f,qhi6k bytzgqy,087; nedvn q+j 9he angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch az: vm, mryot4-ssembly which consists of two cylindrical plates, of : oymm-v,4 ztrmass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 ;8 bzd* 0cy 9li*qxf)s.cfj gyradius r rolls along the looped rough track of Fig. 8–58. What is th*i. ydybf cfq 9;xszl8*c0j g)e minimum value of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but rk5 /0gq s/lyrdo not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed un* htan1j x22hs c2bzc9iformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each tirec2cbhh a z2s19nxj* t2? $\approx$    $rad/s^2$(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