A bicycle odometer (which measures distance traveled) is attached ne: c)j5b b4x- ao1bwdgtr 5i1 2cbutmp(ar the wheel hub and is designed for 27-inch wheels. What happens if you use it on g12j bi(ua5t x-w)15p4bbmobr:cd cta bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velo77ajxfjaq 75 xcity. Does a point on the rim have radial and/or tangential acceler75aqxjfa j7 x7ation? 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 change?

Could a nonrigid body be described by a single value of the angular velociu ;y+ 2g;f 2n fv*ce(y-jehnggsd -;qdty $\omega$ Explain.

Can a small force ever exert a greater torque than a la k ju)q(suq. i3;c8sjs/ rl.uqrger 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 yo1c5yhdvb l597g 5x35mmyf ke iur head than when your arm55 m vm1ky c3g97bhfy5diel5xs are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has se2l sqcqh go dginch22ix(9 ::*ven sprockets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rxnhic *l9(gdq q2si h::2g 2coear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower le,;d gc ,va7j9:ty woxv5lu(jigs with flesh and muscle concentrated high, close to the body (vj( w7g;,a5v io9d,xl tjuc:yFig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–3 s7um pvkmn7ky u0573v5) carry a long, narrow beam?

If the net force on a sys9ldn()mu .khtrb)py6as45 gj b/nj s 1tem is zero, is the net torque also zero? If the net torque on a system is zero, is the nehy)tl9s64 b.r1 /ndsm)ua b5jg p(jknt force zero?

Two inclines have thl j *49b(2z-6x0zps u-pf/dgi vahbep e same height but make different angles with the horizontal. The same steel ball is rolled down each i *zlge bipu(/vp 6jz2fa 9-4bh-d0pxsncline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an is4: fg lv9pq -:m/evk7z,pi izncline. One sphere h glsvz z9v4/pm-qp,i7e f:k:ias twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the 9hbey sj1ab70same 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? Wh9sb7ae1h b0yj ich has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most moving or rooy r8g2qn9 rt2bzdyy1:l 71wft r1w lyqy 9:ofntb7rd8yg22 1zating objects eventually slow down and stop. Explain.

If there were a great i14v(xt q*/ kfme;yx: 7asaarmigration of people toward the Earth’s equator, how would this afa/i*xr;m 4vka (qy :et7 xfas1fect the length of the day?

Can the diver of Fig. 8–29 do a somersault without having any initial rotatiodwzwkio2:l8u mr 8ll8p ;r5(an when she leaves the i 58ka:l;wpwzlr 28udrl(8omboard?

The moment of inertia of a rotating solid disk about an axis through itstk5ruf nk4y8( center of m 4n5fk( 8ktryuass 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 sittinu5a03+rqdv9de t. ry gx z rt0.iy;(yqg on a rotating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop t xy0. y.q5 tzqey0rui;t3+a(v 9rrdgd he masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identica l7gdo g zolb-xptm:403i4ww- l and have the same mass. However, one is hollow and the otheo34:0-p dolwm wgt b4ig7-lx zr is solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a 0+tt;jux;p+.ma f k* zissi(ghorizontal axle points west. In what direction is the linear velocity of a point on the top of the w +p0a(z*xsk;.t +siuj t; mfigheel? 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 standi+9vwinb6 z; )jf: jtusq,qesi 1o a-(eng on the edge of a large freely rotating turntable. What happens if you walk tos9oae1b q,6q+ jiiv tw-sz e;( n)u:fjward the center?

A shortstop may leap into the air to catch a ball and throw 0qsnrrvz 7r;)it quickly. As he throws the ball, the upper part of his body rotates. If you look qun0rq v7r)r;sz ickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, bu+( a yad:h+t a6yh o7lg2k0kdiscuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can gd ou y7(a+kb0+ htyhlk26:a aoperate to keep the helicopter stable.

  Express the following angl y)pey *zr0xhlm at7on9dz)1- es in radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincidence. Calculate, usimkfj7-hy2x / ung the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Ear u/k7ym fh-2xjth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,0.) :9x2sbx:;ylng7f2iy ;jd ojq rnxg00 km from Earth. The beam diverges at an ang 7ysojg :2ilx;.;)x2ndb9xgnyf:rj qle $\theta$ (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 theo- ,dsqw92 5uytuwyk1 motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blade 1tky5 u,o s2-wwyudq9s slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.54//60ha jyiy0 9gggp aszlvb4 m to another child. If the ball makes 15.0 revolutions, what is its dil a9gsjza /gh/bpy6g y44v00i ameter?    m

  A bicycle with tires 68 cm in d( r6x7s lc ,oivkf.3aniameter travels 8.0 km. How many revolutions do the wheels7ak ocinlrf3vx (s.6, make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpx-1 6)ydu m) 710hlipisthchx550m+cou tex sm. Calculate its angular velocity ine 1pxh i)mu5ctic x tms5)06 s+hl7xd1-you0h $rad/s$ $\omega$ =    $rad/sec$
(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-rouneej69 4s4txf yd makes one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linsx9fe44 jtey 6ear speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orbit arojz 9;m rzahghu9(7ym 9und the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of ho.kj070 s laea 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 particlfhryutsvdrk1v h ;+qo)p 5 x 090oxg3/e 7.0 cm from the axis of rotation is to experience an acceleratyh;rt xgsf3o vor0k5 )hvqp xud1+ /90ion of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpm to o)7xehu in7ti .s2 l(i280 rpm in 4.0 s. Dete s.n7hxio7()iuet2 il rmine
(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 radiuu9 :ys8gmtdwd6:szh(lh ln;58wvp u +s $R_1$ 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 spacduowf/ ; sf io tj1h1qq1h7*/mecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they acc1*;hwh7oqo jtq/f i1ud sf1m/elerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpm in 220 s. gnm bit6c;xu,(i 6tb: Through how many revolutions did itmb 6bn(c i6ix;,:tg ut turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200se7af2:lr y - auzn:odns*h j,q 6-cc2kf k4i5 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeed jets in a whi*;grm,x4ls8bul l;lk *l ,a mdrling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its finalrl;*lblu4mkgs, ,la ld ;*m 8x 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 nb f .5.nsunkic: slng,f.g8,uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edgn .s nn8uf: n,kfgic 5b,s.lg.e of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It tkqru q6-acv6. urns 1500 revolutioka-.v 6qq6 curns before it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed uniformly f02c r3tbzzla)l 6mg54ist+xhrom 95km/h to 45km/h The tires have a dz lm2c3h0t+zgar x)tib645 ls iameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces its k/z/3 rzfl.8dvit i.q 52 vdgsspeed uniformly from 95km/h to 45km/h The tires have a diam5.z2/ rlvd.i8 i gsqzfdk/3vteter 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 ,i6g( 4.x bwkdfhil5cx5 oof(jz-*log57 r bba bike puts all her weight on each pedal when climbing a hill. The pedals rotate in a circle of x- jb56b4r wfdzg og 5o(ckillo7bx(fi h ,.5*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 magrxk +m0-rs hh*nitude of the*xr0s- mkrhh + torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torqfe86 ; ia6nn027h u eakcs)sdmue about the axle of the wheel shown in Fig. 8–39. Assume that8hfne6 sa2muk 7a) dei;6s nc0 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,15tew2vr8il,:yzh v k are attached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal posl ezrk it821:,5whvvy ition and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening to a to: ya/ze ajxb:*rque of 38 jb:: /aaeyz *x$m \cdot N$ 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./+yvk)esww.i; qlsm *5ez f6k8-kg sphere of radius 0.648 m when the axis of rotation is through its cent+sqlkwf5 m sek)v*ez.ywi6 ;/er.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66i g 4s/ lg8m dve5p:dopb sc3p-4:h y8oudm+,d.7 cm in diameter. The rim and tire hab5im h8 scgyd, v8 op:udmlp:d4/d-gsop3+4eve a combined 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 of a thin, light rod/ z5 f(qmo 1d-fl-rpujojr)o / is rotated in a horizontal circle of radius 1.2 or)jd(-f-/juml 1z/ po5 fr qom. 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 rotati r jf+4,tsyew1ng at constant angular speed (Fig. 8–42). The friction force bw t+,fser jy41etween 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 poibfhvv gt2d l-sc-.u2y4,o bi.nt objects shown in Fig. 8–43 aboudgt4o-2b2 vfc. u -slv,.iybht
(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 coa; 31xmudowpe +3v fh.nsists of two oxygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the correct ra9hhabuji7d c,6 c x(w4te, engineers fire four tangential rockets as shown in Fig. 8–44. If the satell9b hhuja7wdic, c6(4x ite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius o 1pqc.m wv0yg3f 8.50 cm and a mass of 0.58003pmvg wy 1.qc 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 a bat, accelerating it frowo6;ryk 0nkpa-/nevq3rc u,. m rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven qk x:dgq0h7 *(2gkpkwcts .q :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.pdtg*2 qkx ksq: .gk w0(h:q7c5 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 eventu8*mo2 dj pmy44rpq(q sally brought uniform4 * p2jy4(sdpqr 8mmoqly 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. 8–4 oz*nulp( ep;2ww-rveb ( 4ol65 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 thrown solely by the action of the forearm, which nmwy /g:1czrk:0x d(u.,hic3 alx8xw rotates abou ncaxr:/ciywkz1l ,(xwxm h8.:gu3d0t 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– nvtxm e2)urs8he2cd3f9b6* w4v 9ex6*mdbt 3cnr8e2)fuhsw 26.
(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 co (arwl6p+h05e emwropd+) y( mnsists 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 four fufrjl:07f 4sqoll turns (revoluqjfs : 4f0r7oltions) and 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 of2,pt izw8qzvyl0j .,,.3w dumr cj0yc $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine dzyu+ xl3h-xe 6evelops 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 kq )vgs)*(s,v5kk tl qqg and radius 9.0 cm rolls without slipping down a lane at 3.tk)vqss(lv q) kg q*,53 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of 1e)o8sk,(rh)tl3t 8 n 2it ec:rwbmi eno-hu9the Earth with respect to the Sun as the sum of two term8i3o9h1-te:tkcs i8rb(wnmh2) u,erel on) t s,
(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 164;fvb4fp, bs0g 1nva: z0 kg and a radius of 7.50 m. How much net work is required to accelerat fn :a04vs;gfvbb ,zp1e it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls withj/h8y4csofosu5 fvk3 5e -7vmout slipping down a 30. /v 4-sfv3 8ycose 5jomuh7f5k0 $^{\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 vertical8 1cbije:/w9lp4q mz gly 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 pole just before it hits the ground? [Hint: U4gwz/cli9j1 mpe :q b8se conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the qrgm ) fr/k30bend of a thin string inrf) /r0bkg3qm a circle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grinding wc m1q -ic6(wy sxc34coheel oiy1c cwm s c6c-o4x(q3f radius 18 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, handga 9b,)r.1rhyurk , v.g hcj+fs at his side, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases to 9uh)a,r h.krcby vg.rf+ 1g,j0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in6e m/b oom+6os Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing fromo6e6o +obsm /m 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 s a2uz(ujnvbv k;jww.9q 8t eb:o9oh*8pin rotation rate from an initial rate of 1.0 rev every 2.0 s to a final rat(znbtjbo9w9a;e j wq 28h*ukvu v o.:8e 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 through its cen 7v3gkqmtv2r -ter 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 shapet 7mk-gqrv32v d 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 npjuz on8c.00y+ eu*fluw/k8 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 momentum ofwy0.8f 1ou tgf 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 8)(crqsrtdgbr 3sy- z0uct3 5gpsp.5I is dropped onto an identical gzd5 tr 8cr trsuqp)-b3sp(ygc035.s disk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2(i slakqs :f091i: maaxa-)ho.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotatiwu4wx+i9mtlo; p+ 5hgks 02;q ldv*ecng merry-go-round platform of radius 3.0 m and moment of inertia g;msx2* l4tdc o elhkqw+uw v +0;9p5i920 $kg \cdot m^2$ 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 angug ymm(n8/ wak qg*l,3olar velocity of 0.8 gqm8wy nk 3*lo/,a mg($rad/s$ 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 about halftb nddfa w .whg9h*c8w8* ,a0q 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?(r9af*c80gbt h *a,hd.ww dqnw 8ound 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 wfl0hdbgj w8/ s 5m(7tpinds 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:9vsl /pywbovo 5/b78 n4p cbdjrzce,ek:8: $ 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, initiale uf*pht70u*yly at rest, that can rotate freely without friction. The momeuey*p u7f*h0tnt 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 t4-p:k omndv x.he edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless be.vn kmp:xo4-d arings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope6cr+hjtfb-a2 ;;p gzq4vh5 ur rolls 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 slrac4b; -h jtug h26+vzfp r;5qipping. 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. Detewh08;.59osfu7mm fpi y0gco brmine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moof5fsiyhg97 mow8 0bm; ucpo. 0n as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of7/*tkkc7,4 a5n ob fz8ggufy,0 gn dcr 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 cylinde+jug (,vui-vyhh n7bgj d9 :a3r of radius 0.20 m acquires a rotational rate of from rest j, +7n d- iuhjga:u3bgvvy9h(over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disx*cen;u1m ovf/peh .;ks, 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 v;/f e;* .h eouxnpc1mof 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 skad94n;bgf 8up.d kib gf/55ipeed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 t-ra2- elpgj1x nb2o4+ gcecw-fc-;q h-g(qjo imes 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 unif fqcgqcpr- o-n-e lj2 og- ;4jwgb (-ea+1hc2xorm sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-poljqmz .wc 5ces:2o,q n5lution 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 k j:2.5z c qq,cnemsow5g, 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 ad8 jkmns h:bp )cddz5911q)pn 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 ; f xnh.+cp7q rr9xbaec3o .t0horizontal 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 and44chuinez ( l* 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 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. 8–4hl( uzni4*ce 21g.]

A wheel of mass M has radius R. It is standing vertically on the r6)6 t q+v1wfh5 fbtwufloor, and we want to exert a hovh6 qff1)r56w utb+ wtrizontal 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 k pe8/0smpkkgy+ 3fd)uf5.hs is making a turn with a radius The forces acting on the cyclist and cycle are the normm8y.)k/hf p +f0ukssd e3g5k pal 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 ro55a gp a6gx)mfck into a sling of 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 axg6 aapg)55fmchieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and hy,2m jh1vubz x 9u22zzer arms as thin rods, making reasoh12 b yzu,x9z2j2vmuznable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly -z x)knlvkw88z 7h4hw7.u1 w qssic3 lwhich consists of two cylindrical plates, of mass kq w sh7 8 k8-.szlxuhnww7v4i3 cz)1l$M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 aloncj- ck*)c:tc ug the looped rough track of Fig. 8–58. Wh:)*-ccct juckat is 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, butb . jkp(w03i9ghv:vit do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its sp*vmw1: :meobp /x8e qheed uniformly from 90km/h to 60km/h The tires have a diamet *om/::8vqpweh m1bexer 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