A bicycle odometer (which measures distance traveled) is attachedynqcso 5 w,yr- lb+( /zk+rdy. near the wheel hub and is designed for 27-in n+r(y.bywqd/lks o+c,5 -ryzch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on the rim hmnyd5nf wahn kk(i.9 *:31dikkag5g uua( g34ave radigk5ka m n:9h4ii3gadnay1k uf(3 *kn.gw( 5dual 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 change?

Could a nonrigid body be described by a sin 7qwzbwbt +709qz1jj 2oj,gir gle value of the angular velocity $\omega$ Explain.

Can a small force ever exert-p /fkx -c( gjls5mm2k a greater 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 behind8d)ed b4f . kv*zxib9x your head than when your arms are stretched out in front of you? A diagram may help you to answer this.x idx vk94.*deb8 fbz)

A 21-speed bicycle has seven sprockets at the rear whe2r8 ous)rbqm: jj,0be el and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large :0ujbr mq sb,ore 8)2jfront sprocket? Why?

Mammals that depend on being able to run fast have slender lowb;*afg5 mhj2xer legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advanta j52x*bfgah;mgeous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow beao9v g h uxenp8q3ze:2rr5j9+* f fttr(m?

If the net force on a system is zero, is the net torque also zero? If the net lg 2h/ydm- gsr b6o;:q.yoi3otor/rq-6 .smio dogblho:2g;yy3 que on a system is zero, is the net force zero?

Two inclines have the same height but make ds2w.m g;uj7k ha59 oh. qdmqx we6g0d.ifferent angles with the horizontal. The same steel balld.sw. xwqqm 760hg jmg9 oe5k2dha .;u is rolled down each incline. 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 inclit1upozp(mm - (ne. One sphere has twicempt (mou1z(-p 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 same mass. They start from /) jqd zjuzk/+rest at the top of an incline. Whj ) k/udjqz/+zich reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are con5il3 n v35ujquserved. Yet most moving or rotating objects eventually slow dow5l353 qv inujun and stop. Explain.

If there were a great migration of people toward the Earth’s equator, how wo;vu9hl8 tfxps qze:.. uld this affect the lxtz.p8 h 9s l.fu;vqe:ength of the day?

Can the diver of Fig. u9jcz2r m ;e-a3t-uw nq3x0n i8–29 do a somersault without having any initial rotation when she u3cx93 r i njqnt--zum ;20wealeaves the board?

The moment of inertia of a rotating solid disk about an axis 9c( xeizm9k*de c9,0rjnlkqua4 +o1u through its center 4i xozdklq(9j0 9n cramc u +9e*u1,keof 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 stool holding a 2-kg mass in eas7 hb goo-08rg/x m5xhi). vknch outstretched hand. If you sudden.5 oo)sxh-/x8hnmgvb0 kr7gily drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and havel:1vo s2moljc( a+h 14pj8 2n otrvgt6 the same mass. However, one is hollow and the other is solid. Describe an experiment to determine which is6oc vl2vaj:2(m hps ttgno+4o1lrj8 1 which.

The angular velocity of a wheel rotating on a hor8sg6-ea l5 s n/mic4tjizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acc8ntgsj46i l5 eam- /sceleration 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 large freely rotating turntabk+lvu ryp9-ya*bk hp +;, lasw(lcm (+le. What happens if you walk toward the cenc hublpa(lps -9lmykaw+;+rk(y *,+v ter?

A shortstop may leap into the air to catch a ballq8(+dr )1ncmw tiaj m- and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and d-c1+nmj)q 8mr aw(ti legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angulapc7 l12wkk y+7cu86lxlsb,eed5 l-d mr momentum, discuss why a helicopter must have more than one rotor (or propeller)e 2pd-w5se+c kudlcml x71l 6,yk87bl. Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: ( zo u 50lb8lt) aytmb5-)ur1ota) 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. Calculat0,4i auymh *ove, using the information inside thaiy o,4 mv0uh*e Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed amn wiqbxl ot/p4v-;myn .l0012x p)ddt the Moon, 380,000 km from Earth. The beam diverges at an anm 42w 1mlq-b)o dpx;.it/vx0ld ny0pngle $\theta$ (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 m 6e10 nrf 3sjcu0ah9rt**tp qrotor is turned off during operation, the blades slow to rest in 3.0 s. What is the ang0r e6ut3c*j sa9rtfq0r hpn 1*ular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to anothe9 63kinfptvw:. xk5yer child. If the ball makes 15.0 revolutions, what is its diameter? 5n9p6k:e y3 tkiw. vfx   m

  A bicycle with tires 68 cm ijz(6lo+ sbye8 n diameter travels 8.0 km. How many revolutions do the esl86b(ojy +zwheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates8icwv1ny/b4w at 2500 rpm. Calculate its angular velocitybv1c/ 4w8 iywn 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 it(i4a2 b3sw y+xp4sdpy2)vctn 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m42is4atpt 3)cd2yvwxy(+ b sp 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(7teba+ ind xprzxv ea4-,(u 5 its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed o 9ae 6ygua:rev,fg8z 3f a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a 41 oogshzqa5 4centrifuge rotate if a particle 7.0 cm from the axis of ro1qho4 5gzo4as tation is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheelgy 6fdg lj)om .b b-n8va7g;;i accelerates uniformly about its center from 130 rpm to 280 rpm; -b g;mg6i)8j ba.vfodgyn7l 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 radiusre4cby:k(h (tnj6 :pb $R_1$ 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 pubxp/bfab;;nbbi7g r6j 7i33 ct themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated 37bbrbf;ijan37gb b/;xpic6 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 rpmd rbw8bo2( e7q in 220 s. Through how many revolutions did it turn in thiowbe2b (r7qd8s time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Caebg 3(vdvej1 l 2kf*vlj;2u9jlculate
(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 jeevfz 9ww3,( nxts in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete ,e3zf9vxwnw ( 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 diamete1 j-zr )nhz,k:pzonr 0y1 rua5r accelerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a poiryn o1puzrz r:z-kn01hja), 5nt on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It t9 4ntqux/kr xm .ejvj3)o90dvx+wg ; kurns 1500 revolutionvoxn./tevxg)m9 k u;+kx3 j94jrq 0dws 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 red,x+pzvko:(co a*yu-j uces its speed uniformly from 95km/h to 45km/h The tires have a jvycupoa-*:+zx ko (,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 revoluti9+me6zhy q9eh)bu kjy(; syy z 0.2zxvons as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diams b9+z(z h.xyeyheyq)90vm y 6j;zk2ueter 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 ridiny8v1l5ltf.ut j/:skj g a bike puts all her weight on each pedal when climbing a hill. The pedals rotate in a cirusykv/ tj8t 5f. :l1jlcle 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 32f+kx.ljz(aj pr l(ra door 74 cm wide. What is the magnitude of the torque ikfjx(a3r ( + lrz.2ljpf the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fig. 8–39. Assume w,oj/4xzpkkf h t:w +8th k ox 8fjt/zw,k4p:+what 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, avt)gq; (xbig+re attached to the ends of a massless rod which pivots as shown x+ vg;g( )bitqin Fig. 8–40. Initially the rod is held in the horizontal position 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 torque of l;g8k9ogsngnx +r2w ie te:xn8wp5- 4 38 n2 8: 9nieww-n8;4g egxos5+xtglrp k $m \cdot N$ 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-kl y(cgv h361ptg sphere of radius 0.648 m when the axis of rpgv(c h316yltotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 c6ibbtphyu16y 1.t ls u(+1dlfm in diameter. The rim ant h(yb6+l.u16u ty fpdli1s1bd tire have 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 caivb.05e/hh0 d9 pkw the end of a thin, light rod is rotated in a horizontal circle of radius 1.2 m. Calhpdive/a5 bkh0w9 .c 0culate
(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 athn(jqg 8 ecgkb( ets;:kp 100r constant angular speed (Fig. 8–42). The frinjt8 k s1qh0;pbegc:gr k(e0 (ction force between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point objects samao 3 jmg6iushr280(hown in Fig. 8–43 a 2om6s (huir3ja8am 0gbout
(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 2zutd5bp6dmq(byzw wqm(.71 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 posaog 0)ej65 the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satp)j aeg os5o60ellite 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 unh w + -puhhc1vv(70zzlh +f 8qi;s*abmiform cylinder with a radius of 8.50 cm and a mass of 0.580 8 fzc+v7 q+0haw(mp;bz1i v*hhs- ulh 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, acceleracil2l-z ppq a.0+k6mi ting 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-driven merry-go-round and is abcz2h3hisne7; 0pw*ydvnzr9 ,le 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 v0 ,chwie3;n*dn y9zz spr72h(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 1salo 7* f:xektlg4gq ):ti 7b40,300 rpm is shut off and is eventually brought uniform* l)tfsg : 7ogt7l4bixqak4:ely 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 3.6-kg ball at 7qud-0p3: fmbcw* jw zwq *:/ly $m/s^2$ 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 /lh;nwy(98wy y* n vqlof the forearm, which hl ;vw q*/n 8yywy9n(lrotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, as shown in Fig. 8–4bzzb :rn2 ge gxv71t-)xz1bez7vb-r t):g n g26.
(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 twoc2t6n 0 ysnag8 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 fromp*egpip +mg kno;an1r 5xk6sz :2 ,d6t d1hl*o rest within four full turns (revolutions) and releases it an**dtkn6 lmsi:,k2g 1dz;hpe1p+ orxop ag65 t 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 3i30j aiztj* sof $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque of s gsmdjha7wj8 ,qe5g8q n:11 s280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and rawo m,t- rr:rw9jkeg: .dius 9.0 cm rolls without slipping down a lane at 3.3 .wke-ojrr9:g,mwtr: $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to thjlu/pqjvu lg0q1 ;97ie Sun as the sum of two terqpiu j17q0ug 9l/jv;lms,
(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 b3wi ypk3)5x pkg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation3xk) 35piwbp y 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 fro)) c9zlnaex;4 2veby skh.6*wvnm ,dmm rest and rolls without slipping down a 30 am*;e ndc e.)yhmk6vnl,24 ) wxvzs9b.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

  Two masses, $m_1$ = 18 kg and $m_2$ = 26.5 kg are connected by a rope that hangs over a pulley (as in Fig. 8–47). The pulley is a uniform cylinder of radius 0.260 m and mass 7.50 kg. Initially, is on the ground and $m_2$ rests 3.00 m above the ground. If the system is now released, use conservation of energy to determine the speed of $m_2$ just before it strikes the ground. Assume the pulley is frictionless.    $m/s$

  A 2.30-m-long pole is balanced vertically on 4h88fg(4t , t95fgdgumvto s gits tip. It starts to fall and its lower end does not slip. What will gh t4v fm8,f(4tg u 8os5gg9tdbe the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end of.lku ur naik;6i*9/ af a thin string in a circle of radius 1.10 m at an angularia/;n9 u rlfiuk*.a6k speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grinding wheel vo8zda,6,)f 0fmu qfz of radius 18 odf,u6m qaf) zfv 8,z0cm 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, hands at his side, on a platform that is rotating at a rate o5 il7*zfvv lq - zpx):c.w-g xxcdr6e2f 1.3rev/s If he raises his arms to a*vpc6: .)5-q lx-d27v z wizxxlrgefc 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 of inertia 2tg+,iv r k bw2-:-stp9 ydpgiby a factor of about 3.5 when changing from the straight position to the tuck position. If she m pt r- 9i+wg-g:t2 ,pbsi2ykdvakes 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 initia0+ rygllsnnw-k*7 f:j/ 3akpb 80flnhk4 :z rdl rate of 1.0 rev every 4ynb7:kn0k3+ k:0r-gl8nw pafsh l/ r*zlj fd2.0 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 rmh0 r0i98a yumotating 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 i8am0mr9 huy0wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figuredrl;d,q 2 q0ud 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 m3-0x +7j1m gdzu z),sljvsikgomentum 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 Itxcyz4 df8c* 7 is dropped onto an identical disk rotating at yz78 c4f*tcdxangular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2;dbtcc+ t;.pjia51ud .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 rotating merry-go-round platfdaxw kwk/t-4lv4q78m 5gq yn*( g7muv orm of radiuq7w-nl48gmd wykqvm/a4gkv7t 5 ux(* s 3.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an angular velocx 6bn(eda/7xl )6za ltw9r9(myncq*dc rcy(6 ity of 0.8n(/9a(cy96b e axy6d6lntx7z* rr) wc(mcld q $rad/s$ 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 q0 ;i3 5ftu -:pz9jyttcf0hgzixy)1kabout half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass )3y5 -qf j0hftpzt;g0 9itcu 1i:zkxy 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 wi2w/zl i f5hb7cnds 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 bc)jfxorx/h ,: sx)g e;w:5ac$ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, that can rotate freeqv:rt( 8oa6 p 5gxjf;fly without friction. The moment of inertia of the person plus the f(t8qaj 6o ;fvr5px :gplatform 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 thet:tx *2a)h1y eq+eayw edge of a 6.5-m diameter merry-go-round turntable that is mounted ona t:)y*+ex 1wqyte a2h frictionless bearings 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 rope rolls on the ground with the hd a/ im zb:,y;qul-g3end 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 slipping. What length of rope unwinds from the spool? Ho lymq-g; :u3hzbdi ,/aw far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same sideegiw *)(la-kzwe yl 74 always faces the Earth. Determine the r7gw*- 4lk ewe(yza il)atio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates fro6v .k jdyuwbo:p9 -ao*m rest at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a unif + l/c6a6aelseorm cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calc6ell6sa /e +aculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.05jm8zaytyjpa(6ey7 e 892 :sq3psz*xt 0 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches theq:se y ja9z ext*yyajppsz(t382m876 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 angu9 5qgus)q kh8slar speed 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 ouboy.wn hdo4 /1ex c4-sr Sun, but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Asdeob .44-/y1 nwxcso hsume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution autom1 e n,o*cook/ ,ew:uebobile 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.,eoc*n o ,eok uw/be:150 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates an qw9v;1yg slt rp ix1+8$H_2O$ 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 (ynyxhb)f/ ipx j/j8- 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 masiubx (z*ey07gc u(na 0s M and 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 li0*u0 cz(egyin(b7x a unear 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. 2uk7 wmgq28qi+h+abq/zp xr * hdxa:-It is standing vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it xhdwqmp7ub-+* gza +kx2ar 2q h8:/iqrests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making .e *sxpdk a)k9a turn with a radius The forces acting on the cyclist and cycld k*pas9x).ke e 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 cm- sqk,kyr w jmy 25- sb2;:epebq0;xof 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 feat, and where does the torque com2,m 2ebqwx-j0 rsqpsekcm :yyb5;- ;ke from?    $m \cdot N$

  Model a figure skater’s body as a9(w b4az l n ojjsfbs)tg5, :ga:4o9ga solid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretchg(9gj l:a9z4n, ) bsjfg4too bsw:5aaed arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindricaoocwk)6au u7va23 m/cl plates, of ma2ou3w 7cao a)vu/6 mkcss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 rkl-t2. ;24rc uwbg fatadius r rolls along the looped rough track of Fig. 8–58. W2 4 -rg tkaf.c2wblu;that 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, but do n6xfh) hu r.+io d:zr(yot assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutj6l )t cz-dm-lions as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular l -zmdj6lct)-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