A bicycle odometer (which measures distan)gg*2i o9rpx mce traveled) is attached near the wheel hub and is designed for 27-inch wheels. What hax9mgg )p i2*orppens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Do8 /tirnd.cay4/dhx;wq 6or,q es 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 tangential acceleration? For which cases would the magnitude of either component of linear acceleroyq, hqa8cw./r;6d td irn x/4ation change?

Could a nonrigid body be described by a single value of the angular velo3zrz4rfe )gi -city $\omega$ Explain.

Can a small force ever exert a greater torque thavwe)9hj:1xr a n a larger force? Explain.

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

Why is it more difficult to do a sit-up with yoy bs0b ;y3aox2ur hands behind your head than when your arms are stretched out in front of you? A diagram may help yyb2o yx03sab; ou to answer this.

A 21-speed bicycle has sevenrid3otxz 7w*) sprockets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a smw7* di3 t)oxzrall 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 to ruga0sef1 2vz-tn fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explefv0 zg -2a1stain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. o - np;dax*h z::8xh0ahar/i a8–35) carry a long, narrow beam?

If the net force on a system is zero, is the n4i 8op:j(2c/a rn tk2al7kex/ - tmmgnet torque also zero? If the net torque on a system is zero, is the net force m(kx72nlo a r4/it8t/ekgpmn ac: j-2zero?

Two inclines have the same dfl :wh3zz/el504,( cyevic pheight but make different angles with the horizontal. The same steel ball is rolled down each inclizev w/: d0lhzi3c (c,5yl fe4pne. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an incl9fw29bnov o. wine. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the ibww. 9no9v2 foncline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the sa1 vbqlmj,8q0 zlqnyp7a 6(tg5me radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater spee6q,1tajm pyv7n5ql z8b lq(0gd at the bottom? 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 ihw6hrju t u3x,lp8xd9q9p/ 8or rotating objects eventually slow down d6lr qtu hx3pxj/,h988wiup9 and stop. Explain.

If there were a great ;ixw7za ooyv 6ek9-x8)q) fzgmigration of people toward the Earth’s equator, how would this af-) 6 aoq)e8xkxwzvg9y;io 7f zfect the length of the day?

Can the diver of Fig. 8–29 do a somersault without having any ijk3x39a3e n 3b rb hq4fwhii67a-6x grnitial rotation when she leaves the boardhr x6 3-iwbjag i34 fek3hnba637rq9x?

The moment of inertia of a rotat(yp s0;t wwvv zm(+3sring solid disk about an axis through its center of ma0vts+m(wz3 ; yws(pvrss 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 3-k6sxd h vnp22-kg mass in each outstretched hand. If you suddenly drop the masses, will your angunx23s d- 6kvhplar velocity increase, decrease, or stay the same? Explain.

Two spheres look identica1irm3ulp)7 x al and have the same mass. However, one is hollow and the other is mxr1ai7)ul3p solid. Describe an experiment to determine which is which.

The angular velocity of a wgd2 j3vfl 6h s3a-k*usheel 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 points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed incrf kg *adl6-u v23s3sjheasing or decreasing?

Suppose you are standing nf0bflldn,y 1. xh9i7 on the edge of a large freely rotating turntable. What happenb 1i,.nffnh7x 0l9ld ys if you walk toward the center?

A shortstop may leap into tre4l9vol*b xv9 yri9y)2 4orohe air to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. rvo4* oiol2 x 9e9v4 blyrry9)If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the lawq(r 5mrr xxedyk p1as6: 4(6rn of conservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or morrnk1(dqy4ps:xx r( 6re6m5a re ways the second propeller can operate to keep the helicopter stable.

  Express the following an)3)x+8h64f rci7agb cdbcjw) qko 2kigles 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. Calculate, c )h rbkf6s(g.0npoii8dm60 zusing the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, 86pmfi0 kb)ihzo0g.scd( n r 6as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km frc;mf6yy2 wobd. mpol*/ l1:psom Earth. The beam diverges at an o ;2/16:mpo .cml fpdy*ywbslangle $\theta$ (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. Wh* co2(aiugi. s2(am,mu u;htc en the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the bs (.t ommgc(u*c2i,2aa;uiuh lades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the ball myb35x1 tsl kt6p9*/6*sjfbfjtv ( nvmakes 15.0 revolu6bl(n1fx/9 6 * vfsjyvtst5 k3*t pmjbtions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels vkdkq ,gj 3 unl.vf3gqb1b7(38.0 km. How many revolutions do the whd,gfvv(3 nlgqqjkk 7b 133bu.eels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter v)r5mq y,k8d lrotates at 2500 rpm. Calculate its angular velocity iqrk5,m vy) ld8n $rad/s$ $\omega$ =    $rad/sec$
(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-r; pa4z blfm.-8vuc5yu ound makes one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a c5. b 4l-8ac;fzpuyvmuhild 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 orbit arolzau:rn8 8wf (und the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a po6t+5o.ass 3elxa0(xn*uh o:kt +qy dd int
(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 rotat2e syw dt g)cm 2.i, v7k;yhir9x+2take if a particle 7.0 cm from the axis of rotatis .x2iy+i) cvka k wd72 9egrm2htyt;,on is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniforml5mozpo1cpajjp ozg48 mtk0) 1*,2 qqny about its center from 130 rpm to 280 rpm in 4.0 s. Determinez1 *)g4o,2 m paooj0 ckm p8ptjnz1qq5
(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 radiuspmfu ml1-4g-j $R_1$ 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 acnre3(.rr b51/1r4 uq nk deqhboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of arc11derer.5n3 uqb4n/ r h k(qs 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 acceleratvtmo 0hnl-w* )es uniformly from rest to 15,000 rpm in 220 s. Through how m0n*v)mw tlho-any revolutions did it turn in this time?    $rev$

  An automobile engine slows down fro5jt-nd bf): ram 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*-l x,kl :405pc gaip1 zfn p+nsnn.dt jets in a whirling “human centrifuge,” which takes 1.0 min to turn tg- lkd5safxi: .pz n+nl,ct1n4np0* phrough 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from)4s3e t 7qv,ugliogz ;ci 5yad8;f3cr 240 rpm to 360 rpm in 6.5 s. How far will a poiiz;ydg3c7egi4va8sru l, f35 )oct;qnt on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min Itu)og*7rc7nua.fxgb/(* pp zs :wu8 kr turns 1500 revolut7g.kou *(w b8*pgxafzcu)rpu nsr /: 7ions 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 its speed t.i4hiyj8iajqlm4kmzk/i b4. ,9y*e uniformly from 95km/h to 45km/h i,.4y.m /lb mi i8ktk4y *jqa9izh4ejThe tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces its speeaj++u :;bmvn9u ihv x/d uniformly from 95kb9nv/x ;jai+hu+mvu : m/h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a b0p1(e d 8ywb4ln:kab p 3hi0ck44zglmike puts all her weight on each pedal when climbing a hill. The pedals rotate in a circle o(l4z k 4b0 napb3 w8im:e1dh0l4pcy gkf 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 54 w p5q)vdzo;p5 N on the end of a door 74 cm wide. What is the magnitude of the torque if the p)5op;qw 4zvd force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fig.4*vx.uxm/v3os2 fjit, /e;vf,l qnnv 8–39. Assume that a friction torq/*vx,; qn3j 2 xvonf/l,.vueim4f vst ue of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, arcu0.ogre: n4wkimm2:z w)h x) 2 bqnr;fx48bhe attached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate wrw ;:0rcem hx2mb44g bh.q)n ox82:iuz kfn) the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an;l3d9x;ofpba 0sbob- engine require tightening to a torque of 38 b9; b3lspf -oo;dx a0b$m \cdot N$ 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 radiuef+ ph:6 wua16 thasfo :h:ih+s 0.648 m when the axis of rotation is throughai h6 u+owt6:fh:a+hsh1ep:f its center.    $kg \cdot m^2$

  Calculate the moment of inertia on5av9 u4 qs-za s4k4bof a bicycle wheel 66.7 cm in diameter. The rim and tire have a cov4zk4 sb5-a uaqo4s9n mbined mass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the b 40lplfce xp4:1ny5w end of a thin, light rod is rotated in a horizontal circle of radius 1.2 m. Calc50xlpnc 4 fb1e l:pw4yulate
(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 bow2d+gk,ud6xji 4:*mski x )vr v7kah6ul on a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force betwes) uk4 m6d:xuiv hkk*xv6a+g,jr72dien 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 inertx:q srljro(2 v hf6etc6q,e45 ia of the array of point objects shown in Fig. 8–43 aboflx5 r6qvq(,c2 to:j4 rhsee6 ut
(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 consisqivgy bf0 s+*8ts of two oxygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the correct rdc/kl*9g hsu7,7)vz3.qrg jwd)r ks8l p )pk fate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kfrk3v lszpsd wh)lrj.g, k k8q7u9)p7gc/ d)* g 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 ra5fzc st;7t:g)cp1 n(io( w ghmdius of 8.50 cm and a mass of 0.580 kg. Calcula5w);:1(cgc( nf is zht otpm7gte
(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 p bb 6t4bgaj;;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-roudr0: 6cq)gpuq,9piza nd 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 r qipa0)uc:d9, 6 gqzp760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is eventuniysemiz ) .2 4zmdb46 pm6g1dally brought uniformlyesbm6mim). pd4 iz1z4nyg26 d 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 ac -)ykqvxp*c6cx9/ 3(nqqn thvtk; -rpcelerates a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the action of the forearmjy35;.t /wf teaawm5jpa1* wh , which rotates about the elhatemja 35/ p5wyw1j*;.fwat bow 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 rod, as shown in Fig. 8a) w1vjnsse( glg /h v+hj+e34–46v +easljew+g3(j g n)4/sh1vh.
(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 2+jwshbo 4qaiji;m* 9jf x- u)consists 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 hammer7b:7g vdj( rco from rest within four full turns (revolutions) an(d jgo77 c:brvd releases it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a moment of e01,zd;qc 9fknk (uri 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 develops a torque of 280 o9zst 4h dobgu)qn4d+1 -znq3$m \cdot N$ 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 s3bog 7lz2/ pz q(zfel)lipping down a laz/)gz b e(l3 zlo2f7pqne at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun asd;l x(m)3t89azrr7,bmv hnwu5civ * w the sum of tw(rwvcrt93a*,mlxv5z bn 8whm);du i7o terms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a rad2egeu/erdx6l 8p+p8j ius of 7.50 m. How much net work is required to accelerate it from dlge6eu e/+xp8rj28prest 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 rolls witb bc7m 4xx ,,7:lxyvdkhout slippinbk7x bxxd vcl,4 ,:7ymg 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 vertically on its tip. It starts to fall and its bdkhd+u3em,ah /.8 awlow3/uwdha+d.k bhe8am,er end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentq,4+bvd f h) jb4fkxdx;6 5efjum of a 0.210-kg ball rotating on the end of a thin string in vfjqjk h5xb;4,b+)d6fx edf4a circle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grinding wheeatj f z)ld q,9z0:xm,dl of radius 18 cm when rotating z m,ja,0 9 ddqfz):lxtat 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 is5 j gh- j97 rga.r5nxt.1*evj+zc nulr rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases to 0.8 g 9.5zt* vjj+r 57rnlhe.raj nu g-xc1$rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fi2mkxag 7 .b(bjg. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck positionx gk(. 2jmb7ab. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial rate oflkz9:o a;kbywn 2)m6v 1.0 rev kl9 yom:a6kw 2vb;z)nevery 2.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertica/z;nl8-*b 3+ lyio4useobyiil 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 fry ib- +;iibz/u o43l ne8*oyslequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skate7p ,/l-nc ;wsdew :zjtr 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 +vgv 3y yg ;woe3mec5-h),bpc momentum of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is dropped onto c 8q-zut5hy3mxgwe 65an identical disk ruh5z x -c8 3wt65mgqyeotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns a9j* n0r wxf*1yrtq -bwt 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 stangjpe cy-zre vab,d.844fad5- ,g az5tds at the center of a rotating merry-go-round platform of radius 3.0 m az-dr5-zbj e.ydpea,a 8g,c a454vg ftnd 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 an angular velocityevi 21 6/jnmyc of 0.8i2/ym vj61 cen $rad/s$ 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 eventa7k evjl-+h f:w ,d.ikually 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 cl- .,vf ik+hkew7:adj arries away no angular momentum, what would 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::av tdg0fj3 e ig:6ncc*jg0l winds 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 7e 00njm ti5bt$ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, that can rotate freely w4nfx f(2 a+vaoithout fricova4(+ 2nfxf ation. 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 6.5sr7ei35 1dea-x ub (tj-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a momentuidrst(7 b je53 ex-1a 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 jk 8(,md-8lovh2vmd jground with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8 (mjm-8dv, 2do vhljk8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces the E xv1sb*t4tux 0arth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital ang0t xt*4bvsxu 1ular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at ajy28cbfqw/n1-x0uz3f xhk -w 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 uniform cylinder of raxrf;i vvq( nf 4u,a*u3dius 0.20 m acquires a rot i3uu 4;rxfafqv(n*,vational rate 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 solid cylindri-c/hn y,, xgqn b2(fdscal disks, each of mass 0.050 kg and diameter 0.075 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 shq x-f ,,ybnn2cd/g(rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of the reart 1kqhya ;;cmn:s1dr, 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 tis;4fpbxj tu 7 (o :napem/o5l-mes 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? Assulsx ote47 omp(-/n uj 5:;bfpame that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to u:p+ urq k5hr4yse energy stored in a heavy rotating flywheel. Suppose such a car has a tr4+p u yrh5:qkotal mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without 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 an )lcg0 t7n1kn-ild0fy+n j)e y $H_2O$ 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) id0gv 4)w uxq6us 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 and length L can pivot freely (i.e., we ignore fri.6m ;zc-v mhtection) about a hinge attached to a wall, as in Fig. 8–54. The rod is helv;e mth-mz.c 6d 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. ,s:0s sll-wgd9 n5acu;6tq wyIt is standing vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against whicn qy l;u-9wsgwa dt56l0scs:, h 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 makipkm2r0i z3 ,heng a turn with a radius The forces acting on the cyclist and cycle are the n krp3ez2,ih 0mormal 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 b4ysjkzz9 cw4:a 886pp0rfzwg egins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the to868krpjy 04z9wwzapcsf 4g :zrque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid a4hvyko60cu 7fz 37rm cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculatemc7f k3r70haz oyv64u the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylw:y40v5+gn+ rbjrcr qindrical plates, of mas:rv gc rqy4n+05w jrb+s $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 royom2rdy 0g5v viq m*-7ugh track of Fig. 8–58. What is the minimum value of-ygd205v iy rqv*7mmo 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 do not a/e bqf u(d2r:vssume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speeds* iwjjv kg2 :qz1cf31 uniformly from 90km/h to 60km/h The tires have a diameter of 0.91cikqg 3*zf 2ws1j:jv0 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