A bicycle odometer (which measures distance t t 6 m:dbloez3c11o6jkraveled) is attached near the wheel hub and is designed for 2j6kl1bdztc o 1eom6:37-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotate;v3zw0 ,*q )jpd joayus at constant angular velocity. Does a point on the rim have radial and/or t*3pv0 j ,yjazod;w)uqangential 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 single value of ttlq8: uurkep-lr.*: zhe angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a. 9mz,s2fvrgn 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 difficnr p ;av ylrk gef,.9814csg*zult to do a sit-up with your hands behind your head than when your arms are stretched out in frrg4f9 1vrl; yan sg8z *.e,kcpont of you? A diagram may help you to answer this.

A 21-speed bicycle has sem yoj11 fg((/oeia(j fqgc,3iven sprockets at the rear wheel and three at the pedal cranks. In which gear is joc3 yjge1q( o1/,mig( (ffi ait 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 front sprocket? Why?

Mammals that depend on being axt5xmw( 5ea:vble to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational evt( wxxm :a55dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig.+w/uug 3e1ihiqkh) ( m 8–35) carry a long, narrow beam?

If the net force on a b5(u*a tq0q mesystem is zero, is the net torque also zero? If the net torque on a system is zero, is the ne u*5emb(ta0 qqt force zero?

Two inclines have the same height but make different fn :*d quvh0 )b7f+zarangles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball ) :7qfrhbavz*+dfu0 nat the bottom be greater? Explain.

Two solid spheres simultaneo8j tya( fz)v;itfsqk 1n,e *:nusly start rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the fq*(ens j fz tn;),i1tv:kay8other. 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. ;mwzvt(g r78tThey 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 vrgtt(8wmz7; at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. f2fc4 yfj9;er7j1n r tYet most moving or rota jff7y ;j9n t24e1rcfrting objects eventually slow down and stop. Explain.

If there were a great migrap4 6 zxq*zru06(dhjyftion of people toward the Earth’s equator, how would this affect the length of the day?*jp fqd0huyr 46 6(xzz

Can the diver of Fig. 8–29 do a somersault without having 6jejkqv:50ut/mn/uhs/l kt + any initial rotation when she leave j5qht/k ve0:6um/ntu+sljk /s the board?

The moment of inertia of a rotating solid disk about an axis through ad lf3wl7.se; its center of mass iswf.el 3 7sl;da $\frac{1}{2}WR^2$ (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 each outsjv6scu dxk) .;tretched hand. If you suddenly d6 x; vj.c)uskdrop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and hd)rj c 7k-io bsdk58a;ave the same mass. However, one is hollow and this8c d k b5r-o7;kj)ade other is solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal ax,dmx mf11 esm*le points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleratfm, exsdm 1*1mion 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 turntable. y q* im(t7 e7uo: spg3rn(5ljzWhat happens if you walk toward the cente iepnlyjgz:rt*7su3qm57o ((r?

A shortstop may leap into the air to catch a ball anae ohq2 7)ya l(3yqf1id 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 legs rotate in the opposite direction (Fig. 8–33 o2) 7qay1f(qylea hi6). Explain.

On the basis of the law of conservation of angul/blb ha, 7au7gar momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operatel7/ub g a7hba, to keep the helicopter stable.

  Express the following angles in radians: d; nb19k v.txcw bd:b:(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 bec io/ibe,2,4kfjnc 8ipause of an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun an p4iej2fb8 on,i /k,icd 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(:8kwhskw9k*tp6*t4zcv q x z diverges at6kxv8 *w p(*qt:twhzzk9 s4ck an angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender r o59 q9(wmsnnuotate at a rate of 6500 rpm. When the motor is turned off during operation, the bla95nowm9 ( nsuqdes 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 t yrs3d m2s )n i(37urzvno:p0dhe ball makes 15.0 revolutions, what is ip02ds7 unrnvz3i(: rdy3om ) sts diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How manut:)v k3ht-h+ ax 2*n*wg+whof0yn lfy revolutions do the whhwtlntf kg w3y+ * un*x-0fav2+o :hh)eels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates35ce(ld yrgx, at 2500 rpm. Calculate its angular velocitc35y(r dgel, xy 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 4.0 s (dv8 4aame2k *cbbs( *fFig. 8–38). (a) Whatcbks *av4e2 a8m (fbd* is the linear 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 arojd8cp )8l-5khion47it c0rc ound the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speedq.gry jtt4r 4; of 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) musfiu- m .(easp4t a centrifuge rotate if a particle 7.0 cm from the axis of rotation is to experience an acmi f-u.s4 ape(celeration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about itn ix*1r3-,umh5pv wss s center from 130 rpm to 280 rpm in 3v,5swx- nmirp* 1ush 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 radius g yu /or5m(um9*wdit 5$R_1$ 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 Moox(-it e/bpwaf +.dp)i n, astronauts aboard the Apollo spacecraft put themselves into a slow rotation)(x+p/. feapit wib -d to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 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 15,000,10kv o3q7 eymxj2:lllpg t-k rpm in 220 s. Through how many ,7kvp ego3l2m:lqxk -lyt j10revolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calculahz7pdf- t9oedp mt 03bh4p 1jk1n xr53te
(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 wed;o5lv 5c cd6c0 / a.i(kg-uimoo:i vhirling “human centrifuge,” which takes 1.0 min tvoedo5 .;-li:(ca0vk/o gi mdc5 i6cuo 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 uniformlyfqy 9 *8q7gogy from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel 7goqqy89g* fy have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/mi-s v3kzr*xng i tt9wf6l-9o8te)og 8kn It turns 1500 revolutions bev 9tl ki9r-k3g -f to8gnzto)* es86xwfore 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 6 *p gh6yyst*3.5jft omc+ +soj5 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0jp t+g yjscf5oymt.h*+6 *3os.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 makqct,x7d/- ; nol/btmq e 65 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0ntqtb-7 x,;od/ ql /mc.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 weit3-0:,gd wcksf pbo0bx wqx-bp ge5o6c(6p9 aght on each pedal when climbing a hill. Th9 ,o bbtkc0g5wxq3bs-6(wgpe-p0a6dfox: p c e pedals rotate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 orncbm7ce (,5ps2q:gap s i81 cm wide. What is the magnitude of the torque ifnpips ,5or1(:c8cgmbqe7 2 sa 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* n4ux3ou0cgm . Assume that a friction torque 0nocugxm 4* u3of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a masslesszw, 8qbyz *s,*fj1hv m rod which pivots as mvf8 *1j y,,zhwbsqz *shown in 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+j 7sfqt9ar1tfhw71:a + 2ikfev,eng a torque of 38 hqgaa ris+7+wej2k9 7ff e1 nttv1:f,$m \cdot N$ 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 10.8req) jes*l63nubgr5 *-kg sphere of radius 0.648 m when the ax6 r)q3*j*er b ugenls5is of rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cmt3p77sr js 6wb k0jr1t in diameter. The rim and tire have a combined mass of 1.25 kg. The mass o0j7p7b6 rtwt1s3r sjkf the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of v:e6z bk)h2rza thin, light rod is rotated in a horizontal c):z2 bh e6kzrvircle 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 wheqz j ;6rvm6tk2,w ydix)mjz5oaqez0o )64n2rel rotating at constant angular speed (Fig. 8–42). T m qeazzdjy q) 0;o64t622irm wvjx)5zn,6k orhe 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 pof(g8idw+-j l 2dkclp- int objects shown in Fig. 8–43ic +f28kddjg-ll p -(w about
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists ofc7.bp7owd --qz8u ppz 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 satellvf--.:2c mr ku/gvosdite spinning at the correct rate, engineers fire fou/.o:u-gdmk cf2svv-r r tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the 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 rad24-g 1 vtffrktius of 8.50 cm and a mass of 0.580 kt1 vffgr2t 4k-g. 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 9 iy0 1ac.xeadbat, 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-driven merry-go-round and so;p n/)l,xfg(k0, 3mxxcbyq is able to accelerate itfql0,smc n)pg x,xo3/ky ;( xb 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 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 p3( bm 0 xz7yx j3kgw :uhs13;vnbu6seand is eventually brought uniformly to rem6jpsbs7;(k0w3:e3gbzv uhxu1 3yx n st 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 ball6 mz9we ef5+fq 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 t *v v8oh(gl*i jkw0xl8he forearm, which rotates about the elbljv hvwo gi(8x8*l0*k ow 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 sb g(htz65 l9 k(t5bjmhhown in Fig. 8–46(5 gt6k(mjhth5 lbz9b.
(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 twbwc buun0uj53;j 2+ ieo 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 withinbv rv7yfai46ey: 0n v6 four full turns (revolutions) and releases it at a speede0nvybr4 a: 6fv 76yvi 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 ofb*u(r5 ;f n)c2eiw y o:zixj6u $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine de;k.a ) cu.zq2 vokg i8aeida(;velops 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 and radius 9.0 cm rolls without slipping down *vn.nc pb0(dq 2,9oolweua:b a lane at 3.3 :2n *b9eaq ,np 0b.dcv(wol uo$m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the sum ofl-n*f,gg uwur :eq(u/ two/gwng *r:u -,lq ue(uf terms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg an/i53du 9 )tdajj1np4joyy5yl d a radius of 7.50 m. How much net work i3 /uioa yj4ynljd5yp9 1d)t5js required to accelerate 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 stag: )ymkst ;wag q4;*)ku) tyfarts from rest and rolls without slipping dowk)*s tw;)aytfgy;umk : a g)q4n a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

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

  A 2.30-m-long pole is balanced vertically on its tip. It startsm e0mxl 56y)c/b) ncgh to fall and its lower end does not slip. What will be the speed of the upper end g6 h c5lme)xn0bmc/)y 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 0d g1e d)(kljqrotating on the end of a thin string in a circle of radius 1.10 m a k)(deq d10jglt an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindr2 d6 z55inubh.* sdm0zv stte;ical grinding wheel of radius 18 cm when rotating atdtt0s6nvmuzz.2 5b i5s;hed* 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 ra) t (1ilidccf5w; tcv9te of 1.3rev/s If he raises his arms to a horizontal posi51ccvi)wf t i (ltd;c9tion, 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 shownz /xdgt pb yq:r,7vx7o.lc87qyht v57 in Fig. 8–29) can reduce her moment of 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 tuck position, what is her anguz7yvp77:8t,r7qo. dqy/5 g hlxcbxvtlar speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation ratpo2+p c zn3x c7sb9q3ye from an initial rate of 1.0 rev cpbp+qsx y3 nz2 o9c73every 2.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a v,p,l 4tsuowiqos --9 uertical 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,-qis9p4-t oul,ows , u 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 x*9boit9g- p qyl2w/ /h)wq yxmomentum 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 momentum of thhe d)*ws 9l(j()8whgm43s +tq fo vxive 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 r p84549l ner) eiyqo kft,s3aI is dropped onto an identical disfrop4a 43trynl ie,s 9)kq5e 8k rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns k9)m5z( xhah ax;(qsx at 2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the c0, ;lhttwmnrm8v ga62 enter of a rotating merry-go-round platform of radius 3.0 m and moment of inerti mh awgnm 026v,tlrt8;a 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 rotdj (cmowwwp5zp ( *8)iating freely with an angular velocity of 0*((w wpwo8icd)5m p zj.8 $rad/s$ 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 aboutfp6(57swan a w half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the l w5 7pasn6f(waost 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 in exceud-n zsxg5uk a* ps*5z4-7n xcss 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 shbuy,i x 7lvu5+h.u($ 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 r0+ *)m/hkhtah ij iny 8w(vu8ootate freely without n 0au8o imhh*j/(k+i8)w vht yfriction. The moment 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 thw:ra dx u 8dvk9y0b72ae edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless byk9x2: da ub70vwd ar8earings 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 end of the rope lrqmr.s ; :zwp(s* u2djying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto ir qsp.r w:sjd*zm(2u;t, 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 Earth such that the same side alwaysxq4+;hxx5k/rhn743y- ox wauwea l2 x faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbi3r-wa4 ;yx/wk752u xxxaxlh 4+ noeh qtal angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a92rzwxl58qb; rd* pgo 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 unifor0 f,c;bp1uaq.vo 7xo gm cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by th;qbgavx70c1f , o.opue motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylef 0d+86 wkljkqh)rey *;kyr3indrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches th 0e jk)r+e qdky; f6l*rhk83ywe 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, h:qbz s yj9)l:yow is the angular 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 our Sun, but with mass 8.0 times as great, were rotlz70p( pq/+ 7xuywvmue)us w.0zm) oxating(ypve0.z /lwuqz7 up))u+0 w mmsxo 7x at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use energy stored inhw8 sn*g1 h/r y,h2boy a heavy rotating flywheel. Suppose such a car has a total mass of 14,1yg*/wbhrn hy h2 o8s00 kg, uses a uniform cylindrical flywheel of diameter 1.50 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 7ydgszc8co c+;i*a:f wz31wcan $H_2O$ 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 cqbapy; ncor.t+f.a v;.8) ntat speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and:m ts, .h hsl1;f7t;cjfo l6xq 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 horizontallymq h ; ;fl:h.j67ctosslt,1x f 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. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on the floo9eptd (. ci5txr, and we want to exert a horizontal for i(e5cxtt 9dp.ce 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 witjdvuit 7t99 0xh speed v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist and cycle are the normal fi 9 uj79v0tdtxorce $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ 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 length 1.5 m :9wu+cr wei*pa0hwefjgy (*8 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 wher*(eyu+8iwaw f0hr pg9wcj*: ee does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and herwm (-w4g;3a x;w4.igy.kydw5cmc rc i arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with ouwigcc ;;r. 5ygm3wwdi-x y4 4m.kc wa(tstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindqzom ;k3+ty)erical plates, ofotqze3y)mk; + mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the looped+j.l ,6ve7a,.oa:o+xgfm kucj d40xdfl z*i m rough track of Fig. 8–58. What is the minimu+0avoz.f 4om dxi,,mg ucl6jdaxe l.7+:jf*km 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 y/ ll p-jw-9yruo*rnbe 4,wms/j -lh2do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uniformlypm4;kvq sx cn4j,6 .sfgard44 from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What w.444n asxj4fscdq6;vpgk, m ras 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