A bicycle odometer (which measures distance traveled) is attached near the whpar.0j 0/ei cgeayd)pr 1u5 q.eel hub and is designed for 27-inch wheels. What happens if you use it on a b0.p)j1y .ua er0apergc5id q/icycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocitxb4j)s q -uyn2y. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the po ysb)u42q-nx jint have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a q9ijh7 r(7kic d;jmgq/nyt3, single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greate4m12mt,5: dqkkwq xbdr 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 with your hands behind your ;xy-ycx p9h.ffuc e:f1+ qhvb2/,it p head than when your arms are stretched out in front of yop -+ye2.f y, f1b9qt cip/fuvhcxx h;:u? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets atcfz /t4qdt7 (;, sfxxag 3iy1o the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why?f37qzi 4y oc,;f(g1d/xt sax t 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 runl 8mmx/31oa e+lhwkw. fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribut 1a8/xoekh lmml+.ww3ion of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carpfq78q cr h;8g yrp.,gv mruk+rs :m1+ry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero? If /ub sz0ujbgq.gsi751hl4 *xythe net torque on a system is zubg70yi szqxbg h4 /s* u1j5l.ero, is the net force zero?

Two inclines have the same height but make different anglet + tx 6,)nesq:di z*kqqkx1:ys with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? sk,k 6) qt :nt xqzdyieq*x1:+Explain.

Two solid spheres simultaneously start rolling (from rest) down an incline. O d ozh9pfgr(:48 ))us/oqxtnr ne sphere has twice the radius and tf uz8d gt r)x)(q:n9r/o ho4pswice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius ant7bu lrl6 qh+ mr+z7r p2ka4psz)k3o+q9e*mq d the same mass. They start from rest at the top of an incline. Which reaches the bottom first? W*bztor k s+h7796 q)pl zuaq3qrl +km 4m2pe+rhich has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angula) b1) l)rn jpmmh0ec-wr momentum are conserved. Yet most moving or rotating objects eventually slow down and stop. Explain.)nl1h0mm-pjw)b r c)e

If there were a great migration of people toward the Earth’s equator, how d5qj jrgs w-6ackzd4 p .b7c+bg5).thwould this affect the length of dr7g h6g5db5at jw-+cq.p4kcb. j z)sthe day?

Can the diver of Fig. 8–29 do a somersaulti4a* 6ym hi9zy without having any initial rotation when she lyy6 9ia iz*4mheaves the board?

The moment of inertia of a rotating sol4 7u/ ggq.ycq3w/jpbvid disk about an axis through its center of mass yug.74qvgbq /wj3p c/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 aq-*3nh /7yqr -igwpse rotating stool holding a 2-kg mass in each outstretched hanhsg73-q p-q/n ew*iry d. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the8bve y4xo5hsy5)o e l( e1+6n/rvyk nh same mass. However, one is hollow and the other is kh5)ve ylh6b/8 (esvn5oy4 nex+ry1 o solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizs /f a3n,2f ly1ekcqi.ontal axle points west. In what direction is the linear velocity of a point on the top of the wheel?el kif ns1f2,ca. 3y/q If 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 edge of a z4 vlpq p7p/culh s4hs3916pg large freely rotating turntable. What hap1p p v4h7pluhg/slq34c9spz6 pens if you walk toward the center?

A shortstop may leap into thf55k )+nrf 8: ujpnamve air to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that hiva:pun 55 n +fr)8fjkms hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discuss why a h; trh*d j8lwbsf6d6.welicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicoptr8wlw6t 6*;hbfd j. dser stable.

  Express the following angles in radzg6v-cvp8k+2d *psf 5wi he/ / u-fld .jim0veians: (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. Cal2(s zx:,or zytculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun an(ozty x2 r:,szd the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beam divect,u b 8 ufd+byq)ym4qhe (+z7s5af c)rges at an +ebhua 7qyfu) f+,b5cc z mtd84) q(syangle $\theta$ (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 r a l 3g+rulfe8(+mzkt:ate of 6500 rpm. When the motor is turned off during opg:lkz (+a3+ruet m f8leration, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the ball zry8 -q5b wtgc o63 znkk8-ga(makes 15.0 re( zty w3g 8kz-k6qa ngc-rb8o5volutions, what is its diameter?    m

  A bicycle with tires 683.b b+w;vsqsb ,c y9gv cm in diameter travels 8.0 km. How many revolutions do the wheels mabb ,cs bv3ysg9 ;v.q+wke?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates a,0vhi,)yopawff*j + rn uk7:vt 2500 rpm. Calculate its angular veloca7 kvjf:ynp,,f )hiwor u+v 0*ity 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;,a ri q-g35nuviwbph . 62lkp.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1. iu3bwp-hk.,26qvi;5p gr la n2 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 ( * s4b1zawqjjEarth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a po:y0jtfjtb9(rgx 4 7a561k q ettz:vldint
(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 rotate0 k aplc354-( bz:ujtzm jpf9n if a particle 7.0 cm from the axis of rotation is to experienc t(4fz n3lju59kbaj0cppz:m- e an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel acceiv mh,t(yx( w-y6oy u(u-7k, axjyq a2mty,4c lerates uniformly about its center from 130 rpm to 280 r7(i-uy ,wx tqkhyavj6t( aomy y,y(m24c-x u,pm in 4.0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radiu2glg1bu/ig, hs $R_1$ 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 Moob/ig *o5q)kigu,snn tb8 r(ai-yv2)tn, astronauts aboard 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 thoub-g u( )osv,aq)n*52 yb kt/nr git8iight 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 acceler9t 5j t d1(guey*ho3oae 2mj-kates uniformly from rest to 15,000 rpm in 220 s. Through how many re1jgt -edy( 3 *omh eoj92u5tkavolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 sh,d(qt6o eg ;t. 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 highspeely2*;7ixu j 38fnwty8zj ke4xd jets in a whirling “human centrifuge,” which ta4nw8 fj 3e8z7j kl;txx2u yi*ykes 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 3knd q4f/3 1iob60 rpm in 6.5 s. Hoq/ndi4kf3 o b1w far will a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is rumuh ,); gi2 r0+kawolqnning at 850rev/min It turns 1500 revolutions before ir) +ak0uw;om2lg qh,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 uniformly foln. wdz6k taw)6)l.nrom 95km/h to 45km/h The tires have aww. )6ok zn6dntll.) 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 i21)bvw qgl-q65 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires hagvql2iq )b- 1wve 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 bike puts all her weight xm :1+ z/ak2zaotam;6dt)gpv on each pedal when climbing a hill. The pedals rotate in a circle16: ttoga)+ak;zzxap vmm /2d of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cm wide. What is the maf). v wrnqku,sp3e-4doo k9 a8gnitude of the torque if the forcvqud-nwrfpo., s9a8k)3 e 4oke 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. 8–39. Assume 4f4:i+xskh- j ffs8sl that a -4 +skfs 4ljihsffx8: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 f+4.yzo 8vg xw ltz/y/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 the magnitude and direction of they./zgt/y +v zowf4l8x net torque on this system.

  The bolts on the cylinder head of an engine require tightening to a tou88 l i:f,+u4 pjgb-hsbozl2 lrque of 38 , sbi z84ugl blo:8fhu+2p-jl $m \cdot N$ 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 radiusxf*.aygu 1o8n( .kjty 0.648 m when the axis of rotation is through its cejkoay tf1n*.(yuxg.8 nter.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. The rim zmlj1, v l40le(oqy tyn 9n:6f.*ful zand tire have a combined mass of 1.25 kg. The mass of th. v*l,fzmy: u6 ely9lo1 0fnzn4l(qtje hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end 7/v2vywcclk3pg 93 zegtkjdq.ix -9vw-g(y/ of a thin, light rod is rotated in a horizontal cir v2ezgjwkq3c(gt //v d-icp 7vw.9l3g y9kyx-cle of radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating at constant ak xv);xqwu, i2ots q07ngular speed (Fig. 8–42).,xu; kxswi0o 72 )qvqt The friction 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 objecjrsu5)b0 e nk tt):p5wts shown in Fig. 8–43 abou0r 5te b )up)5w:njtskt
(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 tiyr;nr6m0 xk:g 1kxr+wo 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 satellite3.b5m5 wr n9gv, u5qadt0s flc 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 d q,35l.tmr nguw5avb0c5f s9 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 radius of 8.50 7 sh*o/5c30td. o hpohy5yhoicm and a mass of 0.580 kg. Calculate5 oy/p*otoh dco 7shihy35h.0
(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, *vjhww +nb) aiwi1v.4 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-driv.pez uqj7n62( ym:o :6nissr4n x qd4ken 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 mass of 25 kg) sit opposite each other on tqr4:7n xpk(zsu6jo d62.n4inqe m s y:he 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 10,3x6,.sx p gl:y0(kd tke00 rpm is shut off and is eventually brought uniform(l: ,k 6xgps.0tkd xyely to rest by a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a 3./j,kk 8f5 rzci6-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-k,)t 4oz4ymzi.egl x0e0f 6*utn)qbcrg ball is thrown solely 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 accelerated unif e*z. fqm ))gty4ur0ltze0i6x n4,bcoormly 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 be1/*- fn c0qmys;brmkp considered a long thin rod, as shown in Fig. 8–4 n1;0/bym *pqrf-ms ck6.
(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 massemqbe;/jj(;cb s, $m_1$ 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 fro t j+pou7ln );a-pbphoq5k7vb yzt874 m rest within four full turns (revolutions) and releases it at a speebnvathptqy7 ; 7)+- u8bjk7p4lz5p o od 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 ht9el5ftk m i:vq(,cl 5o+,fil as a moment of 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 4x d8bw7ti1p1:f oot/fnpe 0 odevelops a torque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg */shog/d zn ,gtri996he m(peand radius 9.0 cm rolls without slipping down a lane at 3.36n/r* met 9dp9/zh gi( hoseg, $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth witlpcfotm,ax4( o* v0f 0h respect to the Sun as the sum of two pxc4 o0t*,0fvfao( mlterms,
(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 7.50 m. How much neh,haz2pxi k lf9y :1k2t work is required to accelerate it from rest to akx9h :2,pf zkhy2il1a 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 res6j:g ihsh(ikd yw7zkw8980 lat and rolls without slippin(8wi8d97ls0 hakhwki z 6:g jyg 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 verticall erfh+h. tz4nj,gw: t3y on its tip. It starts to fall and its lower end does not slip. What will be the speed h. ejg ,4w+ftrh:z 3ntof the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular mom:2ik6(i j fn6di keum6entum of a 0.210-kg ball rotating on the end of a thin string in a circle of radinkdmuik2:i j6e 6f6i (us 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 uniformqi ,ujd 1:l(w dqn6s1w us;7zj cylindrical grinding wheel of radius 18 cm when rotaulu;j:w w1,q71jz6did ( nssqting 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 -,s;egecgaw.is1 he3 is rotating at a rate of 1.3rev/s If he rse3-wca , sh;i.eegg1aises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce her moment of9)t i: lqks6ni b./tez,.vuz l inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuc9tb6en ls. zuv zit,.iq l:k/)k position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her sp4:, .weyha *t8brk t 0t9uayolin rotation rate from an initial rate of 1.0 rev h,:a08y lt* t.w oak9 rtey4buevery 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 rotatt bhxb+xc81,g ing around a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, 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 b8,g xbxc+1thafter the clay sticks to it?    $rev/s$

  (a) What is the angular f82p7 4c wcrkzmomentum of a figure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular z+i-qir hw9xp kv8ut 7yc a795momentum 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 ide2eu)ai0q .gvr u:t *wnntical disq.ng:w0r *a2ueuvi t)k rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2)1;6 bikhy rcl6+xz xmu(,gym.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 atw3rz fc1 m0u:d the center of a rotating merry-go-round platform of radius 3.0 m and moment of inertia 9c ufdm r10:w3z20 $kg \cdot m^2$ 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 velocitq07halh s d+;fhvhv/9.:6wvsvyc * z gy of 0.8h dahv fh0:vlc9/qz+s;.wh76 vvg*y s $rad/s$ 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 cd 8l7ngyriayla gi 1f* .h,h)sffh+q,y59b4dollapses into a white dwarf, losing about half its mass in the process, and winding up with a r ,* n+lh1fi9fdylh7ag5.as4y d g)b,qrf y hi8adius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest 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 e1b-nkvhuv wu6q74 p( 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 q:l i1z/3 spyx$ 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, initiallymzwnf0hsu):h gw6 8 +t nbk::x at rest, that can rotate freely without friction. The monn0+:8xk) wf usg m:z 6b:twhhment 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 edyktu 16 es-i2mge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertet-ys u 6ik12mia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of the roe a-vq/ hzw+jz 0e *7lv:dajh0pe lying on the top edge of the spool. A person grabs the end of the rope and walks a0eehl:7 / wajvv+jad h-q*z0z distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earyerv,(: wv4aj9c1b hwth such that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, tr9 je(bahc4v1yvww,:r eat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rey rjgy-3k:k1*,vml 5 uhi6ldkst 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 0yh-9qfr;st yu hh fm*)w ,9)(ckl lrq7.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by thyywqt) fr) ll-(9u ,hc7shm *qk9; frhe motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, eaco *zj*gc-ui w .rd9l4 r2kgzp+h 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 congrkw2.jp*u d9 ozz+i *-l rg4cservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of the reqdohya*+ .5sy wv 3c6zar 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 timy,ww hd22z ah- yv8hk d/c7uo; ;xkiz.es as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravidai;uwc, xk.h/2d w;zhy 8z -o2hkv 7ytational 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 n -ib:3kq:; mxtw;f zla low-pollution 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 should b :fm-x:w 3;ibqlnz k;te 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 illustrasu466c cv r/aktes an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal surface at svfrkh g:t5n ,4peed 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 ignor3ad5s6wylk s ;3 q48vjv p3tsae 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, aaywkj5q6s;lst 334 vv3as8p d nd (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically o ( xq13.v;; rp t.xozpxjlq3ejn the floor, and we want to 3jxoel;q( 3 ;rpj 1xvx..qtpzexert 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 sn)+g cbdh sv3vh772) gf5u dpvpeed v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist v72sdbdn 3g +p vch7u v)5)fghand 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 cwtc-7ry0kf. length 1.5 m and begins 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 torque required to achieve this fe- ft7wk 0c.yrcat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cygcei o,1cu9j3nn gb.h3wp*l+)8weq z linder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speedsj.gp +)h *wc nzg3l,cq ibno81we e9u3 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 cylindrical plateacihflcvhz+wx ;69.ro1 k: 3 ps, of mash.1lr 36za:whfv k; c9c poi+xs $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 o 8fe9qf+h*.fhnqz b8 xf Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop f zeq+fq .bhf 8*hx8n9if 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 (z9eq 77kxl ysdo not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed unifro:ccn.510 ro:y5vnl e:e vjnof6 ov 5ormly from 90km/h to 60km/h The tires have a diameter of 015ej: oe0vrv6n5 fnr5: yoln:ov occ ..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