A bicycle odometer (which me5c 5ov zaf sdfbvp(cfo+l)/*+asures distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicyclcpov*+ zfl/od5b+) asfvfc ( 5e with 24-inch wheels?

Suppose a disk rotates ykb*a3 idfd9dc i(-1-mkw5ea at constant angular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity afmd(y- 3ia*5dde -k9 cb1ikw increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be qaytyq*tb/gn.x21vdg26 r- kdescribed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a gz)fvdmq8acscxvzr+ k() 06 uw,6j d2 freater 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 do a sit-up witpbsr2vp;c7 / d8rkz( idxkc44h your hands behind your head than when your arms are stretched out in front of you? A diagram may help you to answerzs(k44p8i rcr vd2b /;kd7p xc this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the pa 5z(ghd1 fz.wbar, zw:*fgs;edal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rearg1,zzsbhwz* :ff 5a(w;ar.d g 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 run fast have slender lower legs with fles1* 3z3dp/j adxfzfy- yh and muscle con a33dd/1zy- xy*p zjffcentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry 8 agl5n/p vp (amad2:za long, narrow beam?

If the net force on a system is zero, is the net torque also zero? If the net*e+8mhfnz x/ k torque on a systex8en m k/f+*zhm is zero, is the net force zero?

Two inclines have the same height but make dqoqsrwp +(jnq 6lgjg/6e6fp412te f;ifferent angles with the horizontal. The same steel ball is rolled down each incline. O rq6e; 1lof66qtqswjj( 2egg+ pnf p4/n which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously s-jjw4wdh(d cjj du8ow.my -k ml/6*2xtart rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which8--m d jkuywl4*wj(o m2.d/ j j6wdxhc 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 zo 5t6re ok7zxr1+s +j:,x*abk+o utgradius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greo*+kt +6z+ :bxo,rx7t s rk5a u1ojezgater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most moving ocbyga,.1j ;c zjna /c6r rotating objects eventually slow down and stop. Explain.cy6c.gaj /1; , jczbna

If there were a great migration of people toward the Earth’s equator, o,yyw :1rmwck1doh *5 how would this affecd:y1r * woh k15yo,cmwt the length of the day?

Can the diver of Fig. 8–29 do a somersault without having rw6w7 :j, 0 5ndktweuvany initial rotation when she leaves 6w5,:dveu jwtk0w7rn the board?

The moment of inertia of a rotatinjy j(cvlgn ),+g solid disk about an axis through its center of mass i)+ lyjcj,(n gvs $\frac{1}{2}WR^2$ (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 g 8j:/m kcmc.st ,u+-qzqr(ajrotating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increas+.g/8cj,qz at jm-cmrs q(u:k e, decrease, or stay the same? Explain.

Two spheres look identical and have sh7;ph-wq ns**oa9z4sm( wmlthe same mass. However, one is hollow and the other is solid. Describe 7s n*m(swomla wzh*;9ps4 -qhan experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle povxx0e xbuw7 p+s4:hg /ints west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration poipue:x0x +v4g7/sh bxwnts 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 e2pao-6sd/c v edge of a large freely rotating turntable. What happens if you walkv-/p 6ceo2asd toward the center?

A shortstop may leap into the air to catch a ball-ckq2(ap+ dj z and throw it quickly. As he throw2c (za kqpd-+js the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum,- zxskb-s6su: discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keekxsz ssu 6-:-bp the helicopter stable.

  Express the following angles in rad 2o8n 7mtoex:sse,w*oians: (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 cosy8lo-gao 1qihm(gh5 hp6v93incidence. Calculate, using the information inside the Front Cov365h a hs(9q g-logi1 hmoyp8ver, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at t,agp843 jrww iurxt.r uct51- he Moon, 380,000 km from Earth. The beam diverges at an anglerxcw ,arui5u tp3-w 81g tr.4j $\theta$ (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 ar6b 5idjglzobj +7xl )bjs8iv67p; e.t a rate of 6500 rpm. When the motor is turned off l7dgj z 7.)jv+ l8 poxbbis65bj;6 ierduring 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 3tw 3mxs-vc9) hd28jui.5 m to another child. If the ball makes 15.0 revolutions, what is its diameter? umcsd9 )v-w2j x8ih 3t   m

  A bicycle with tires 68 cm in diameter travels 8.0 p(hbnx-3nvv:km. How many revolutions do the wheels3h nnp bvx(:-v make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate itv5w xfg .rb3t qv-mfd5i,w3 x4wt+ct.s angularwm43t5bft5r3,gd. wx+vq v wt c-.ixf velocity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.05s up:h7a t;w5tot-t n s (Fig. 8–38). (a) What is the linear speed of a child s5t hu5n:7swotp;ta-t eated 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 r.e39 r ho-pkdits orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed o/:yfye3;mn on f 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 parr9zif 47ycy:)0rz kq aticle 7.0 cm from the axis of rotation is to expe7if )krarq z49c:z0yyrience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel acceleraeem 8os (t-aqb z8),lttes uniformly about its center from 130 rpm to 280 rpm in 4.0 s. ,qz8(lot- )bm8taeseDetermine
(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 vykp3rquebt m0 7 5d 7.oakz:8$R_1$ 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 sp;g/bwd oxjnh *(4.w7e3j fyg-oy0zbx acecraft 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 as a cylinder with a diameter of 8.5 jh-4nx ; b.ygz0bwdf* yoo7e ( wgx/j3m. 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. Thr nz9icmtr14d0cch- h,ough howc1dh cz ,rc-i49tmn 0h many revolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500r 3+ho mav tczp..94sy 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 highspeedalf/(iot3 i ion:v -;u jets in a whirling “human cent3n f-:uov(ilai o;i t/rifuge,” which takes 1.0 min to turn through 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 frd9 ifd-i w;t+ ow;hx:qom 240 rpm to 360 rpm in 6.5 s. How far will thd;-:;9q+doi wxi wfa point 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 It turns 1500 revolipgbrej179hv qxq.wsx6;:yud7j h70 utions before it comes to a i e0jh.7ux xqv6 hjspd7r1y;7wg:9qbstop.
(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 unifpli;/sm:a.g h 6wbwf-ormly from 95km/h to 45km saw;.l6-:hfimbw/ gp/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as fp hd;i62m m9ethe car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.80 m96mfp em2dih; .
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each pedal when climbing b-qiulmr9hd .; ))yig a hill. The pedals rotate in a circle of radius 17q9.y;l g) dh-ibium r) 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 or,hs lq93: rvbqn: w.dddsez,i n-d25n the end of a door 74 cm wide. What is the magnitude of the torqdz: qr2li ndv,ss,bd eq .rn3- :5w9hdue 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 abouq/oqj +w-/q:fkb dr,j )+ scjzt the axle of the wheel shown in Fig. 8–39. Assume thabj)/rcq:o,zq +s+qd- /jkfj wt a friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a masslessb25qwzlq3uh:vo k/c n e/s4+x rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then rele/2 xbsqveq5 4nwz oulkc/ h:3+ased. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of a1cmb) w .l ; :yu-k-dc0k /ipghv(eoayn engine require tightening to a torque of 38g1k yhcmoc b :lwdkvp)iu(0 y-.e-/a; $m \cdot N$ 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 +*knhp:lmz/06 ga c uha 10.8-kg sphere of radius 0.648 m when the axisug+ /n0m ha:lph6 kc*z 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 g-)w;vaxl0j1g fc m:frim and tire have a combinejgl:xw-v 10g f f)mac;d 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 end l6wv,zenl(t9 8g.85skqomrc 7 lhvp ;of a thin, light rod is rotated in a horizontal circle of radius 1 ;7g. l856se(hq, zrovl nlkcmt8w p9v.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 a potter’s wheel rotating at constk*kphx:bf wawu8/,a( j3j-q 3zsk h6bant angular speed (Fig. 8–42). The frxfku 6 3h(ab3/w8k -: sk*whb,japzjqiction 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 inertia of the array of point objects shown in Ficac; 2 rt4fn -/j o)1b;qsemaxg. 8–43 abcx4j e /1omr;fqa;a)s-t2 cbn out
(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 ato3qvfxpi)o ;; h3/ bwd hii 6)fjcyi/:6eholf-ms 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 spinningn2z2hyn 2 k(hvtz927 pyvxtfds9mt7 2 at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a hn2zyttf2s9k (y2mpxhv729 vnz d t7 2radius 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 uni 3sczhcg:*m 2vform cylinder with a radius of 8.50 cm and a mass of 0.580 k:smh2 cz*vgc 3g. Calculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings az9 ute 7yj+h r2or8xd- 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-dri x:c /: 497l; zuinrleatpexe7ven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a au9xez:4l7i7x e;e lnctp / r: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 tw;4 yedqljr sd9+*)f rpm is shut off and is eventually brought uniformly to +yfj) r*; dsqe4d9tlwrest 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. 0bezb 0 z:k ,ryz/bqn).h nn;9ifz h2a8–45 accelerates 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 t0y1 anz7zblfr-g(y (xhrown solely by the action of the forearm, which rof- (7a yg1(nrzy z0lxbtates 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 rod, as shown in Fig. 8–ugy6i0 dff(6r f6b zxoa)7y fx w703rk4dfigofz(f063 xfryxyu 06b7r7akw )66.
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine consists of two masses, 40a tbyviy 0y5dx*p8,x o:gt y$m_1$ 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 withiykw i0q h-n3y-n four full turns (revolutions) and-y-iw0 khqn y3 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 xmift r i3)( .;hth t+jvrck;7inertia 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 28t a- 1 y9ydpmbcnjjk-x:0t;:x7 k9toq0 $m \cdot N$ 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 rolag)(bm k3xvel2 fm( .kls without slipping down a lameb.xag(3 ( mv fk2l)kne at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the sum of tqd1p4 q:p+f+8lmaase,z 8yt awo terms:1s4,qlapt zqefam8 8 ad +py+,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius of 7.j+h/ -fn ;opvy50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8;o-n+ fjy /hpv.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 2gzzc(mdf hoel); m,hs; 8ud4)0.0 cm and mass 1.80 kg starts from rest and rolls without slippzh u8d) gm,mz4);scldfe( ; ohing 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 i+un es-jrw 1m.1m w/gtts tip. It starts to fall and its lowers.ww1r/1 nt j meg+m-u 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 mom9w,m+/sl5 f s i19p is.kwxrcventum of a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an angular 5issw x 9l,irkmc.1sf+9 wp/vspeed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum ofa1q44k+mzkcye 7r uv, a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when cme4vyuk,qz147 ra+k rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is x0rvb v)1re(( p xzwo-rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreax ve r((wxvb0)z-p1r oses 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(q0u1 fc cyj;ehs073sen8 bf)uxhlq6 moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck pcx31eu8nhb ;70 leusy 0h6sj(fcf ) qqosition. 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 initz8v 0qd1spv0 8fcdy ,hial rate of 1.0 re8f 0d,1hvc8d 0syvqpzv every 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 vertical axis throqx )eu-2knj 7ough its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk ofe7qju-k)no 2x mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinningi pkm0d,3x6t2nvci/-hi-d zzre1 a7s 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 momedlruzp3z4w xf-ts- 8( ntum of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is dropped onto an z, (fzq*izqkx/ a-n6 tidentical dizf- (niqtzx6,k/ a *zqsk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns a4ogiuu, q c,m8vx(1z ht 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 cent v4:i/scx ;t *h+ekgorer of a rotating merry-go-round platform of radius 3.0 m and moment of inertia 920 exsv+r hcko/4 g;*t: i$kg \cdot m^2$ 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 wit1m a gk(px/v ce(1uk.dh an angular velocity of 0. a.ck/xv ((gdukemp118 $rad/s$ 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 a white dwarf, losing oj b2;ro*p6 4n..epeh6x 6:n(etej*mujfsg k 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?(rje(6no.2rns.o :e46 6;u tjhgpekemb px *fj*ound 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 in excess of 129qy5iu n fi(.2xrqn4t0 $km/h$ 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 v7o ;ui.e(/cj sze:eg8-x9 tngfypdcv- hc )-$ 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 rotard *5 nkuu9 w9)og ip7nm3pic8te freely without friction. The moment of inertia of the person plus the platf m)dr*3ki79u o59cn8unp wigporm 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 lwlo8e95l 4v j.5n(ta7ae,whk2td dx turntable that is mounted on frictionless bearings and has a moment of inertia of wl(5k4, tjn2 lt 5d8ax9wavoeh 7.l de 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 ft ua7;gdni9)xkw -a 4the 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. 8–50. The spool rolls beh-i7gkn tda 9fw )au4;xind 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 Earth. Deterrz7x f5nl ss )4q8+cudmine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.) d7)cqufls z58nx +rs4   $\times10^{ -6}$

  A cyclist accelerates fn, 0xmfe,x gpd::kqw 9rom rest 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 shap(n.kj 3u . f .v(enwwppaj7 iaj5cno:tj4b (z4e of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calcul4jpvb:7.i a4 k(n.po3a.5njn uf ejjtww (c(z ate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass h3i nmhrf-y-k7 9lw2s0.050 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 iy-klh9r 7minshw-2f3s 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 rear whee00/gwdne 4gqs7 kp3 oll ($\omega_R$) 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, rgk4lwg j5b9fy,n a1x0i z1/obut 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 of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and thatr9ln f kz5,wa4i1g0 ob j/gx1y the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is 0sz4n+xr:bye) rsudf wd*r 4hjr -49j 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-ynd )rusferj4s:4w xbz09r*h dj+ 4r, 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 a:78pv idw ybx+n $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal s aqkjpuu-io3w+.+ 7t5s iel4u zhf;1jurface 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 jda/ruc m: m2/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, determine (a) the angular acceleration of the rod, and (b) the linear acceleratjam/2r:d c m/uion 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 id. uqgar),// j3ykqwR. It is standing vertically on the floor, and we want to e/)ya w/rjqi,k gu3d .qxert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making hxm1m 8*ku.px( p mvo/qjv8. za turn with a radius The forces acting on the cycxok m. mvm(xz8 8pp*h q.j/uv1list 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 lengthq14kmk g37;cmb p6 igj 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, a;mcj7kbi1pg 4mkq6 g3ccelerating 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 solid cylinder and her arms as thin bvoxjnwywba46 a;: (toon:3 ( fc;m n6rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and withx;o:o6o;(at3m nw nyb( jv:w nf 4bc6a arms held tightly against her body.   

  You are designing a clutch assembly which consisba s8hw+ r2;jets of two cylindrical plates, of w;e8shj2 +abrmass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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.skm 7uz8:p dt r rolls along the looped rough track of Fig. 8–58. What is t8.pzukmds:7 the 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 do not assumgnk68zwsa1 jk2n0u1b5mk e ny*n la-6e $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutih/ g (5xs6a /+0b-ucxqecxupsons as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 u6cg(sbux /h sxep -a50x/qc+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