A bicycle odometer (which measures dic9w; bo 50c1 fhd*fnjistance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use ic1f;j 9dnibof5 c0 w*ht on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a poi2 szg9xj blseoy,,g lxt+0e0,nt 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 acceleration jgt+sl,ge0xsy,bxol0 z 2,e 9 change?

Could a nonrigid body be described by a single value of th co*tb x69(tipe angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a le l z:uc0ehfg4b+7x4v/il6,ccq5u nm arger 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 zhft v3f.w *+ka 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 answerfzk*3.+f ht wv this.

A 21-speed bicycle hasjh9pq 1u1ev+d seven sprockets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rearj9+pqheuv1 1d sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend ot-/9.wvvzjv lb l5.gfn n-*ce n;lzv1 .b+q tfn being able to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, e;en5.tnvqgfjvvw. bb lv -tz19*l-cl+/z fn. xplain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a )9l5 v)9/mtkmprocn5 g y8gxh long, narrow beam?

If the net force on a system is zero, is the net torque also zero? If the netaqpfqk,s0f w:(ej(n yjj8) .p d7u/yz torque on a q,pu0 zp/fk f): ye (s anjj(yw7jqd8.system is zero, is the net force zero?

Two inclines have the same height but make dib ue ncw.(.0mrj siijw+00mw*fferent angles with the horizontal. The same steel ball is rolled.jw+0i wb(ni .r wucs0j 0mm*e down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (frods9vwflh l m)4d/ s4p0m rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Wm9lhfd vlw s0s4)/ p4dhich has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. They start fromfk/e ,h0.0ds7ravot xki ;g f8 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? Whichda .v r0t 7; ,0kxohfes8igk/f has the greater rotational KE?

We claim that moment l,xtv+4 2ym dn8gk6bium and angular momentum are conserved. Yet most moving or rotating objects eventu6lx2,ymn gt ib+4k8dvally slow down and stop. Explain.

If there were a great migration o1jrov1/ +ejdrch aw85 w(k(eff people toward the Earth’s equator, how would this w(favrjhdk+8 r1/5(e1ej wcoaffect the length of the day?

Can the diver of Fig. 8–29 dowyb 86 bnf3fb/ a somersault without having any initial rotation when she le 6b3fywb/ nbf8aves the board?

The moment of inertia of a rotating solid disk about an axis through itsrw:3)g tsd-sq center osrq:w)-d3ts gf mass is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stoo.yh vuza ;i h bpugu* 5j4*ogi2cm*8;sl holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will yoipggicjh*oyau ;5b uh* 82;v 4um* z.sur angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and ha+o fkkz*98b c m,h:dclve the same mass. However, one is hollow and the other is solid. Describe an experiment to determine which is l,f m8 h: cdbckz9+k*owhich.

The angular velocity of a wheel rotatiwepa/( sc, 6a-8e,t27th z-yn fzauae ng on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? /wu,(2y7 -peatace, tsfzaanh-6z 8 eIf the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the ai3 7ky0ule-whe4.j*zwx cr)-s ak,bedge of a large freely rotating turntable. What happens if you walk toward thjeckew-ay. wb k r7s0,3)*luzax-ih4 e center?

A shortstop may leap into the air to catch a ball and throw it quickly. As 4/p3c bbb m4*t/tnu)jzqa 8ushe throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Expuq )/b8*u/bm z t44sctjpan b3lain.

On the basis of the law of conservati4.b//ft ncjt8wzh( gpqy.ek t qt.(i;on of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can oztq/q .y8e; (/w (kg.ct.j4t hfbnptiperate to keep the helicopter stable.

  Express the following angles in radians: (8 v:odd gxkr y8,83bqea) 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 ama rgit3e.wmq+bz/h8t ( zing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (iim teqw zh3t8/.gr(b+n radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed atje*u*/a;u a ch the Moon, 380,000 km from Earth. The beam diverges at an angle u h;/*ae*a juc$\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. When the moto-zh8*jxez bzq 2+ 45glh7thsxr is turned off during operation, the blades slow to rest in 3.0 s. What is 2xqz*zz4+-8jl hx7heh g 5bst the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on ad333 msgps9 vlrw: lb8 level floor 3.5 m to another child. If the ball makes 15.0 revolutd38v lb3rpslmw9:g s3ions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How manxi5i*at-t 3kqy revolutions do tk5q it-*3ta xihe wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rou1, 0qhp z7zcgtates at 2500 rpm. Calculate its angular vhu1qz z gc,07pelocity 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 make1j8 9gf ;rmz2krmjf*ys one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seamm ;yj*kz 8fj29r1gfr ted 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 around the Sun h-1*we54kv iir r)a27sqat nh    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a po nlux.1)(vqtbitvi 31 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 rotate if a particle 7.0 cm from 8)k5dp.*p p4okwhx w.nfkm)dthe axis of roh.5kxk8.dk )wpm4 *p d) pnfowtation is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its c(y2aop f6,y pvl 4p*s;tir :ekenter from 130 rpm to 280 rpm in 4.0 s. Determipo *(p a;sk4:eliy2,vyrf6pt ne
(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+ e 6jqtr1k9:(frkvaw $R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboard the Apollo spacecraft put thea(oma p0oq0.n ug6jm0 mselves 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 .ugjap6(o0m0mn0a o q 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 uniform7vozs,w+ qf b9 mu*sk,4rfn;gly from rest to 15,000 rpm in 220 s. Through how many revolut4;rz9 ,m sugwnkofsq+ f*, v7bions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Ca(o pz( t8t5( vb*qn qtb1;qyiklculate
(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 highspee ze0ee)...wvmeea v qwj-o(4p d jets in a whirling “human centrifuge,” which takmqea e w4(e.p0-)vvew e .ozj.es 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 from 240 rpm to 3600r l p0u jqtxc)ygv,(0 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled 00pg0ul(t,q) x vcjr yin this time?    m

  A cooling fan is turned off when it is running at s;p:qt4)9frfh l j,na850rev/min It turns 1500 revolutions before it comes tnrsfqj); fl9: ,h4ap 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 cxh/o t el20 pjhd3d.(h ,0fum, y+- wdzmxkr)car reduces its speed uniformly from 95km/h to 45km/h The tires have ) ofdm hxy0z,uhw x,+k./clmrd- d(3 ep0hj2ta 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 reduo0ber 2;s9 s ,js;uh ztjb 0m(gl7-mijces its speed uniformly from 95km/h to 45km/h The tires have a diame7ru jeh;20bsmz0smjjt( 9 bos i;l ,g-ter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each pedal when climbing a hr1dta8ra4l. tf k0 9 u;yxff5cill. The pedals rotate in xytrl;f f5 a4acr98 0.1ftudka circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force o a)r*gn;)xee4 z6cdw sf 55 N on the end of a door 74 cm wide. What is the magnitude of the torqu) 6sxen r )gac4z*we;de 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 tor4(b macmkspqupru . ndh/d d jtlz)2+j11816 cque about the axle of the wheel shown in Fig. 8–39. Assume that a frict1mqzul6 r t+mb.s2j1hj)kpp c8nu 41dad(c/d ion torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are1(l aqvk( 8jd ,xxv7ap 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 p xvlvj(7a(,k 8qa1 dxthe magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require ti ad87rmw6)g:/ czo:dtxw 7nlca 6(zthghtening to a torque of 38 (6n wdt) x7hc7a/aztm6 8:dl gzcrw:o$m \cdot N$ 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 - y 1mvn xh1dcej7r/ 8fk*gsp.of inertia of a 10.8-kg sphere of radius 0.648 m when the axis of rotation is through its s7x1mcy.*h/d8ef-jn g rk v1p center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm igj+wg,ahc j le.l o+4/n3akd,n diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignoredl/ ++4caaeogkw3dgj n,.j, l h (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thinu *7i1rt.3 gfn)higqc, light rod is rotated in a horizontal circle of rad i * 1u)7irght.qgn3cfius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowmb0fnolu9 3e4 p(iuy x. 2a)zcl on a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force betwe3u9e)uami 4c z nofy(2pl b0.xen 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 sh p1wf(.2zye .gbo/rh -tj6l/mkr)bdi own in Fig. 8–43 2f6hbj wdtpl/z.mo1/ry-g)i(e .brk about
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen atoms whose total mass is 7ul3+wp f)ug:u rl8qu$5.3 \times10^{ -26}$ 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 cylindrical2lh q 9as3c9/hmk1 gk:csx/o k satellite spinning 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 radius of 4.0 m, what is the required steady force of each rocket if the satellith1c99xk :kls2kao/ g3cq h /mse is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 cm and a mapd/bm s09lul3ss of 0.580 0pld 39bmusl/ kg. 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 on+6b55;t8 qu shnsvq 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 tangentialov+ )okhj) sl93 2;zbwdzggf 4ly on a small hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a unil z)g4 bv so3dkfg2hzoj+9;w)form disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is even,o6cfkbsa 34 otually brought uniformly to rest by a frictional torque of sakc,boof 6431.2 $m \cdot N$ 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-kg ball atpny 1b-002 s1ckui exev(d3i p 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 soblrb yuf :mbslw--6 ( kj+7q+rlely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerate:(u- jkysbrlwl 7fm-bqbr6 + +d 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 t2u)yw(o-qnww:mcr0b 8 yn b8rod, as shown in Fig. 8–46wbt88(2wmu-r 0 :nbcqyyw)on.
(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 l )z/aq7o4cbd(o ipw6 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 hammer from rest within fou l;ty6fvxc9o+e u- 0obr full turns (revolutions) and l9x- 6ufo byteoc 0;v+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 inertia ofez.b9q+h 8a* t3hbz:9xdk8 6m l nmfcf $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque of 280*j)wn *(gx ko5kzi7je $m \cdot N$ 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 downrkn7l u /xi n1mmadu63h:w ucj/l8r fvp7c* 82 a la ln6vxhu/ruccwf:jru 1/*dap 38mnl7m ki8 72 ne at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy od. /x4m2fqvduri+ h i1f the Earth with respect to the Sun as the sum of tu2dv i+4.x i rhmf1/qdwo 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 kiexjd31 j --93 tt jq:2i5f4yzo lqsflg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation -42o3dj 9ts5qzl1ijqfixlf:3jt e-y rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.jg 3qp3ff3l/s0 cm and mass 1.80 kg starts from rest and rolls without slipping dow3g3 j/3ls fqpfn 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 i d-;6yifpon 0js balanced vertically on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the jyn0 f;p6-ido pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular moment14u.m0 w g*oh v cs cdci6*3wyar.hy)cum of a 0.210-kg ball rotating on the end of a thin string in a circle o u*.rhdam y)o*.3g16c cvsh0 ycc wi4wf 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 gri6 jg v*3drw *vwz q1geao*t-1pnding wheel of radius 18 cm wha g *d1vj-*pgwv3 r w6oeq*z1ten 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 rd. 0bsu ialr5;7ydk6(jvj,ejsryh ;/otating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48/lydu; y(vk,e. j05 rdbrs hij7a6s;j, 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 sc,k/qn:7oh pmkt .co)hown in Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changingc:/mkchp nto ).qko,7 from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial r-m009 e*skj5khnxsruyr 4.t d ate of 1.0 rev every 2.0 s to aj -49kdhrun.xrmyks 0e0*5st 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 rotatio bv dmkq1pgvtv+2oe:b- )*y ;ng 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 cy-+* ;bv1o:eqbtvok2 dvm )gplay, 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 skatv,ac.fn 4xeg 7x7 3v-p kyzng(er 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 momentum of the /w* ah9t 4wbvsiwey 05Earth
(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 ide4(oipe+:x h 9lldd6lk 1tuoc+ ntical disk rotating at angular sp u+dp4d:x+ 9h k ioeclol61t(leed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns att ih(: uqgo5pypum7wf1 5 ay+; 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 hg:m:8 yhj tk4at the center of a rotating merry-go-round platform of radius 3.0 m and mom hk:48 mhjyt:gent 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 angularv20ft (ozzwf 2 velocity of 0.8 t o(0 wfz2zf2v$rad/s$ 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 whi0 l(yhidnj f/*te 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 a/0yn *hjl (dfiway 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 winds in exc,up0uth4sy f, ess 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 99fw wle8gqk d6- wb4t$ 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 - su:gc.ekmz:jhry/u*a9e y in- g4) ccan rotate freely without friction. The moment of inertia of the person plus the platform is ah/*r-:cc iyg4nmu)y9g . :-j ueeskz$I_P$ 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 tu k1 2dvk,spqg7 /w:*ts 6dfjbff:zx5urntable that is mounted on frictionless bearings and has a moment of ineu2617 *fvjgw5fxk,dd zftpb qk /:ss:rtia 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 r3, c9jwyx7bzsc y: kg*olls on the ground 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. 8–50. The spool rolls behind the person without slisgkcxb9y: yj,c7wz *3 pping. 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 th twqs fo6s2:*ec7sn :83lvc 1 p9ulmeme 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 Moon as a particle orbiting the Eartheq2 m6ll18o sp:nsc 9m fe*73:cstvuw .)    $\times10^{ -6}$

  A cyclist accelerates from .*bpax;r*gh.*ero h trest at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of radius 0.20 m acmgy mw 5h3d/3trpy)22iz3 zbkquires a rotational rate of from rest over a 6.0-s interval at constant angular acce2py /5ib 2zm 3tmd gz3hrywk)3leration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two rvb+;-)7or w9 4fkcglyq6vef solid cylindrical 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 ofce+vbyqw r;v lgofk-9 f67r4) 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 th7 tdtkega; guy:qg.x o ,*-i.ne 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, bumq2 3.7ozdwgu t with mass 8.0 times as great, were rotating at a spdq7u.og2w3m zeed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollubxe/es, m2wwrc-j12 ution automobile 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 shou wj2 sr1/exbeu2-,mw cld 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 -+dosnc:hxrlr a /9unt.l8 6x 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 (hoc4b0 at+k *caxfp/fz o)ec-f)op) is 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 pivyev3zy f;mon /:p /d )el)*wmbot 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 accel medynp/;: mef) 3zy/w)o vlb*eration 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 n4jk1y- pfn hx nl :lzyy **6+ogmrq8oz-nn*;on the floor, and we want to exert a horizontal force F at its axle so that it will-lhy 1*: ro*p jg8zmq+x4fk yzoy;nnn n6-nl* 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 ae8tduv ;/fxiyh *b pw4 co-,y+ flat road is making a turn with a rdyeho 8*-y;/ fi,twu4cp+ xvb adius 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 srsr wx+hxkyw ,6,)0noswam+ 3c 8da+eling of length 1.5 m and begins whirling th,0rm 6 wdr8,cxa3+hxsk)+sww+ yen oae 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 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 rods, 1nf)u gchrz *g72cu hhh1g+5nmaking reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and wczcggu+2 5hnnhhr*fh 11g7 )u ith arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cyli,. qzee peql04ndrical plates, ofeqzp. 04qe e,l mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the looped rough track of Fig. 8–50s(ew*aflnp; im(9j/)kn4j g+ytu ft 8. What is the minimum value of the vertical height h that the marble musj(+ iy(aw0nmlf tn)4f ktge*9;/jpsu t 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, buau.:au,4.n l0 w dnbyh/7qmb pt do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduc4n4uox rd:/s-h5rwis es its speed uniformly from 90km/h to 60km/h The tires have a srwn4hs /udior4:5-xdiameter of 0.90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s