A bicycle odometer (which measures distance traveled) is attached near t9id fgr+v ,t:nhe wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch whet: 9rng, fvd+iels?

Suppose a disk rotates at constant angular vepuki5z sw4unq 52d3e qja: 37wlocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangeqp7akzeww ds3i:q5jn2 5 u3u4ntial acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the9,9kfkt h:yp z;37q n*(ev.mzexpe tg angular velocity $\omega$ Explain.

Can a small force ever exert a g 4ax4 5;hfmaefcos4 f7w2pr/sreater torque than a larger force? Explain.

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

Why is it more difficult to d3u7b ,m g xi:-sio5dlqk8w k+ho 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 t73shdq8ixmw-+g ,u: ok lbi5khis.

A 21-speed bicycle has seven sprockets 5hz9t-2ce rtnqnp d pe 1ly3p;8/yeb*at the rear wheel and three at the pedal cranks. In pt 8n5;bdlc3-ey qpn9 *e 1ey2rz htp/which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able t-u,gtc) d;ma9 .x 5yvq k3;b dpu)aqqfo 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 distribution of mass is ad3 q fmtd9 uq-5aq.yav;,c))udb g;kpxvantageous.

Why do tightrope walkers (Fig. l;o z -uu 1)nbmif0g1(7 ftdnk8–35) carry a long, narrow beam?

If the net force on a system is zero, is the ne9e(6 , 3)t vr pcqyxufg8o +j93aanmkit torque also zero? If the net torque on a system is zero, is t3k g9,8fxc+t)qa p uo(6m nea jr3ivy9he net force zero?

Two inclines have the same height but make different angles with th s- yj 5mfp;i16ox+fcre horizontal. The same steel ball is rolled down i1 r;ox-c6mps y5 jf+feach incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start ro ;7rty0rb.qwmkrvq + -ifd-v:lling (from rest) down an incline. One sphere has twice the rwkvr7t .fr qdvbrqy0 ;-i- :m+adius 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 a6n91-3fncrhoepvh0t +j(bv the same mass. They start from rest at the top of an(6+epnc 9 tj -h0bofv 1a3vnhr 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 angulh7irrj*7d;v kvm /d1o;/qcd7b ;sht w ar momentum are conserved. Yet most moving or rotating objectd ;7 ; r7sd;h/hb1crvi w*dq/mv tjok7s eventually slow down and stop. Explain.

If there were a great migra4mw nckgpv :kj2qe( -+tion of people toward the Earth’s equator, how would this affgk+eq(:-npm4 cjw2vk ect the length of the day?

Can the diver of Fig. 8–29 do a somersault z41q9n. bja br5mw6nf without having any initial rotation when she l.b 4b9q jm5fnr1 6wanzeaves the board?

The moment of inertia of a rotating solid disk about an axis through i12ud0hnvyn:-* ya ue +pb 4mubts center of m +mn nb- bpe4udv01hu2u*: ayyass 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 l v:xo4rc082 bshcwq7stool holding a 2-kg mass in each outstretched hand.7 sx8:whq4cro2cvbl 0 If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, onl;9c0e ls) hume is hollow and the oeu)l ;msl09hcther is solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizb u.z zh61vi3do,,f 0(zace faontal 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 t31dvhei ,fz(aza6, o0zu bcf .his point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large *e oz:qmq+wc. freely rotating turntable. What happens if you walk tomco *q +w.q:ezward the center?

A shortstop may leap into the air to catch a ball and throw +s yd 1v g7j i/ (wvv*cx/bhq(pj5/dddit quickly. As he throws the ball, the upper part of his body rotates. If you look qu d j vcsj vq(vy5p1+//d7w/bgid*d(xh ickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discuss why a*bme ca o( rch+44up/b helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operatepc mo(bheua *cr 4/+4b to keep the helicopter stable.

  Express the following angles in rad 7xvyxjfz:w29jj4f o7ians: (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 g w,f1.g(5;jyqmydo e because of an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, aw ydeg5o.yjf qg1,( ;ms seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beam diverges a :rx 1oxcbhj: x97i0qa3d/r buta/x30od79 b1 rxhqu:ix :jrb c an angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6 / qbnzv8ck2+d500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s/2dv+bkn8z q c. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a lfd5avs63**hl6 mh o cievel floor 3.5 m to another child. If the ball makes 15.0 revolutions,36**am5os vfc6ihl hd what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revoun f4hr/n5 spm h3:+ 9f7iv*znaf 9rqalutions do the 7 pmf9h na+ :zr5rih*an3un/v9fsf q4wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate 6k7ne2(ds;ipkiunk1 3;fsse its angular velocisk s;ki; iep 3(fke1un26d7s nty 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 s (Fig. 8–/:c uzgl 4 3qii)b;mm s;su5tg38). (a) What is the linear speed of a child seated 1.2 m from ;tmu5z /igq ;3s suc4mgl)bi :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) in4pcgk pjp+4h yh0g1 a/ its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed okbv( t5 w6f8y*.ygtwnf a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.0 cm from the axis 3/o3e ;: bc8mysrzg elsg12tiof 2/y1i8mg ;zs3et rl:esoc bg3rotation is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpm to)kt3 rg9 uq; zr4wvqmd u)+o1v 280 rpm vtumu +43)q1gqw9orr ;dk)v zin 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 radius 2jc p-a5 t4nat$R_1$ 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 puih;wtki9;tw/ 51fc mkt themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they a; wkmi9tc / h1wtkf5;iccelerated 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.5k53xlpky 3qs Through how many revolutions did it turn in 35q xspk3 y5lkthis time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm irxde:adz43 n f,;rzyga, )ys(n 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 highsztvpi0(zxrdh* ,rpam2b3z p5 *3iw )w jaa60zpeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before *5 r3m(6w0ai)ijpxp wzp ,aa3 *v z2rh0bzztd reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates unj ito ,1r cyjdr.a;z;,iformly from 240 rpm to 360 rpm in 6.5 s., dzi; tj,yrarcoj. ;1 How far will a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is runnn5*r+v s704zvbq a f g.vz7najing at 850rev/min It turns 1500 revolutions before it comes to arnq vv.j s5+bfzg7v4zan7*0 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 its speed uniformly f : f+)kgsmvkg1rom 95km/h to 45km/h The tireskks:mfg1 gv+) 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 unifoxiwf3 6/t:gbo w0q b;xrmly from 95km/h to 45km/h The tirew:b ;0t6q xwf bgiox/3s 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 ridingwnqh;l e(81i54;3 jt zqxbtl2 ma et0s a bike puts all her weight on each pedal when climbing a hill. The pedals rotate in a circle of radius 17 n4q18emj2l t50(twix;a;q3lb zehst 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 end of a door 74 cm wide. Whwf+ws : :w)nzr8a r5 mxh0w9qpat is the magnitude 9qzn sf +h)mw0x::aww 8wr pr5of the torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about/qv +3yp 5s+*r5qj.ntl h7xagdh pk(c the axle of the wheel shown in Fig. 8–39. Assume that a friction torque of 0.4 7vjkdrt py c*p q+(.s/5qah3h+gx5 nl $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a man4gfsvxksb 53 .ko/1xssless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then/b 5ko14 x. gxf3nkvss released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require 2x *:w/lpwaiutightening to a torque of 38 lx:wu wi*a/p2 $m \cdot N$ 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 1e1ris*i 1c*t.iaz 7u d0.8-kg sphere of radius 0.648 m when the ax zs1 *c.dua*i1rit7ei is of rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. The r,zoi/)*d kg ssim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignoredds g*s,z k/i)o (why?).    $kg \cdot m^2$

  A small 650-gram ball on acwby/8qz9px.n/1v u the end of a thin, light rod is rotated in a horizontal circle of radius 1.2 m. Calculax9 1p vbwqn a/.z/c8yute
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating at constant angular spee u 8,5p:nz)o qsnt srq 59dy;fp2w2umnd (Fig. 8–42). The frictnw,) q;y: p2m 2rfsnpn9 u5uosdz85qtion force between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of 9uc -t*wj-zjb3-i u sbinertia of the array of point objects shown in Fig. 8–43 abs-bcuz9 *wui3tj-j -bout
(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 masy j.,d c,s vz,cyn+ ,3j-wbrpts 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 rate, engin0 you2qq3bfx9wy)4 oygd:jo.eers 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 th uq)oj0 4yyxy29ob f3go. w:qde 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 cm7wij -vy:5ynt and a mass of 0.580 kg. Calcu-y nj:wi75 tvylate
(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 swin.u-t9 qzi bnr;gs a bat, accelerating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven merry-go-round and g.kwskgtl7-4g 1b4 r ot :dll+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. 4k kw1 g7s4d+b:-rtllog lgt frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 1lj09 )e e*zl geund-k+8d 4jcv0,300 rpm is shut off and is eventually brought uniforml98+eeejd luj v-4* kcnz)0 ldgy to rest by 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 3.6-kgta vs9l-6 7rnym*.)*dpfe poi 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 forearp-2a4 fld0aw4+sn bls m, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly fr-a0lba4np f+w 4ssd2l om 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 ro bf6n)iop/ 18va 6q*xsooljondc,d; 1x+hq s9d, as shown in Fig. 8–46v6ol )cibx9o*a6jn+p1sno q1 d,f8shd qo;x /.
(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 consis+ mdgz-o(ct0w4p3 y7avy j fn-ts 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 acceleratx02sx:3zfqozr u216yz7p qbd*. u vzues the hammer from rest within four full turns (revolutions) and releases it at a speed of *u2v bzz1 y uz q3rps.7od6qx2u0zfx: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 om(a:s0e s:z5yi um:mx)i h5: n0esuquf inertia 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 de- uh +c d/enk1;,whxyhvelops 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 down a la vh 4bm8sd65ci1 :rldyne atd4y b61sm5irl8h dcv: 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy .3t3q nlm9 mb+kmgsj( of the Earth with respect to the Sun as the sum of t.9 qlm(k nj3mbsm3+t gwo 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 m0w *gwdq1n 0/n3jynry ass of 1640 kg and a radius of 7.50 m. How much net work is required to accelerate iny0/j1g3 drn0n*yww qt 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 from rest and roll2y/mm h wbdbi;wpz 62,s without slippingmb,myzb w; w6/p di2h2 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 ie d 0a4be6*vyc+jw3nl s balanced vertically on its tip. It starts to fall and its lower end does not slip. What wila yd+eln* 4 cw03jeb6vl 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 end of a thw 9aeq)sx1c39 * e,lgzimwye;in string in a circle of radius 1.10 m at an angular scqxs)i,; 3*gzla1 9 myewewe 9peed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical lua*ll 7iok+4grinding wheel kll4i7+la* ou of radius 18 cm 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 hismg*3ab(ffk 9 a6joiy/yp 9g618xxa mk side, on a platform that is rotating at a rate of 1.3rev/s If he raisea*g(ma 31kb6igk 8jo y/f a9m 6yxp9fxs his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in.p1;cxep1b( cqc(iwg:w, ed nn9p/ xp Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tucppxw ./ep;1nxpcbe1(9w(gqci:n ,c d k 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 an5: f p yt88utgft(nz:q initial rate of 1.0 rev every 2.0 s to a final r:n(u8 5fytqpzf :8ttg ate 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 rota*nax r/l(u5 :gh:nf+tq(hns nting 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, approxift x a s(5:g*nunn+ql/h(:r hnmately 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 angulvy5mr ly g145a30 ydc6fn;z etar momentum 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 momentu6afqyna3i*dsrt c+q 7b0ai : *s:mm/wm 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 id3-5: qdgbao /u0gw5eh ,kbswnertia I is dropped onto an identicalbq 5sog: w/d e kb,h-53awg0du 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 f q7ac7y ew-c-2.4 $rev/s$ 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 rotating merry-gp, x0 evww-2rbo-round platform of radius 3wpeb 2-v0,wxr .0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with a psd4qvr/i 5n5c q:fd*n angular velocity of 0.8 :5ddi/n5q qsfpc* vr 4$rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into an),vp 1tu,w9tafd4 s5vhtq j1 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 wp j1aqd,n 9 )tvusht54f,1wvtould the Sun’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 winds in46) 2 iwmzmvfv 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 b:5rimi6mb, s$ 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 tg 4e(f2 bo,ncplatform, initially at rest, that can rotate freely without friction.b,(ogf c2nt e4 The moment of inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diameter merry-g.i vdbx h m;-iu92;pweo-round turntable that is mounted on frictionless bearings and has a moment of inertia of 1700-ei ;b p;u md.iv2x9wh $kg \cdot m^2$ 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 lyinoign ws k+ x rs do*ei-/(vt26,e,bw2ng on the top edge of the spool. A person grabs the end of the rope and walks a distance L, hol*o/,sb,rw+(gson kt ind wx 6v2e2e -iding 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 E a z p zxz(*v+.d6xxq0e2p,ywkarth such 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 the Mxyp +,pa xevzq60x* z zw(2.kdoon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest a w;xf2v)arc4ot 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 uniky7tgom v6i7c vy 6:no6jgz+ l )p:0tvform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval atv7+z66:tpyl y)0 6vgcn jm: ogkvo 7ti constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of masls-w3f . p 5ra4-tmuj:z z/vbks 0.050 kg and diameter 0.0.sv/b rmf: lj4zptw3-zk- au 575 m, joined by a (concentric) thin solid cylindrical 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(x+prnu3iq,kx2jc / s the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times as great, nj)b c -hcf6p9were 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, whap9h) c b6cn-jft 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 automo( s)e*w8uoqb /3-v-(s.p mfrcv6kgc0 qhzj mxbile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 356cqmh/zq3- f8 vv(wp oe*sx(rm u.b-gjcsk 0)0 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustra;gi.htdz 8 1t*,nkmwutes an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) isn7dbp/kt/ut o9 op d-* rolling 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 a e;wez2xk(am 1nd length L can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, d1ez(2awek ; mxetermine (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 the flo6o0z.6u m:+ adnsh k uas8 (fv03bztyror, and we want to exert a horizontal force F at its axle so that it will climb a stu00 6amst usv6(nz3 a: od+bf r8k.zyhep 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 ron02w eo l3t0dl4( aewg8lq5xcad is making a turn with a radiq(0lle2ew4d08 w la5nxgto3c us The forces acting on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.5 m and begins whirlej9, qzs,r,/k 5wobhting 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 torrjwbh,ko q5/s,9t, zeque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solit9n;mc)t ll5 yoo6+mu vj,- 9ge2bmgs d 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,ng96tl j-)5 2,s9+oo bmmcuvy;gle mt and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cn 9up5bcw- exc-4*/ ur atg)vhylindrical plates, of m h-9cwb*a)n/upv- xu5c r4etgass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the looped rough track of Fig. ;b *oegsyr cux;)f(*i1mo:f/3ewyfx8–58. What is the minimum value of the vertical height h that the ma( 3 ;xcmbx o*fiye*o :fg1fw/;) rsuyerble 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 nott457a 6nx8kokts ag: 05d nttv assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 nvi/5+lud s+ drevolutions 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 angular acceleratiosd5vu nld+/+in 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