A bicycle odometer (which measures distance traveled) is atta9 s 4wpkh 93.t-jmognpched near the wheel hub and is designed for 27-in4tk3 -.wo pgpns h99mjch 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 havxmax;1v. ki 6ye radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential6.1k xayxm;i v acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body b55+;+y 7ri+pjkb eynz lpf wg5e described by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Exo89 z w+s1yz/daw hz:1v*+z l kmby+sge,z 1bdplain.

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 l4r;y7qq9r 9ufy ./kn,e;zs4pgztzd do a sit-up with your hands behind your head than when your arms are stretched out in front of you? A di 7,4uqtse.zl;ngkpyyf 9zzq; r9d r4/agram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and thr*ti/i) +a 8gd el w(zrml4mn8dee 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 peg+dt*lma/88e wmiz4nl d(r)i dal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to rcsqswg999 o,2*ux yc,x5q icvun fast have slender lower legs with flesh and muscle c29 y5 wc,*vxcsq,u9ig9 sx coqoncentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8jcfj:b9-r b 2ax 9rre) wkt8h.–35) carry a long, narrow beam?

If the net force on a system is zero, p 7nx*w5+bz todn)v ,ris the net torque also zero? If the net torque on a system is zero, is the net force ze57ptdnxz +wnr )bvo, *ro?

Two inclines have the same height but make diffeqf81c96pqd lurent angles with the horizontal. The same steel ball is rolled down each inclin1d fc6uqp98 lqe. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simulya ge(0v4d/ rntaneously start rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has t4dn 0 v(egyar/he greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have 2w svlm2p-ix9wbe1*kthe same radius and the same 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? Which has the greatem-wxlk ev29wp *b12i sr rotational KE?

We claim that momentum and angular momentumawzf)nm (i -+g are conserved. Yet most moving or rotnmi-a(g) fw+zating objects eventually slow down and stop. Explain.

If there were a great xhu3hxo)rz(e: l8 7efmigration of people toward the Earth’s equator, how would this affect the length of (ef loz rh:8ux7e xh3)the day?

Can the diver of Fig. 8–29 do a somersault without having any ini/u4)up 7waide.mx4 cj tial rotation when shej7 ie/p4au.ucd x)4mw leaves the board?

The moment of inertia of a rotating solid disk about mxr3ic- 3: 1 sc5 exo( geffdxn*/iw2pan axis through its center ofpc m 51(f33i esx xew *nr:-/ofgcix2d 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 each outstre2thaqee8;b0f r7cp9 ctched hand. If you suddenly drop the masses, will your angular velocity increase, chpqb0ar 7; e9cf te28decrease, or stay the same? Explain.

Two spheres look identical and have they9 obxl0za oh,; e5;zs same mass. However, one is hollow and the other is solid. Describe an expe zobayz xh,o;; 590lesriment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle pointc vhfxd469)jh4. :uqt i7i fxbs west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration ofxc9vb h4:ff 6h xitd i47.)qju 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 rotaglh7ub 2q(j:qi 1 bll4:q1bpy gp;j2s ting turntable. What happens if y qbl1l1ib2sl;pj h(uq2::jggqp b74 you walk toward the center?

A shortstop may leap into the air to catch a ball an3, 0igachs hh3d throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that 3i 0a3g,chshhhis hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conserv cmuc .om:mo 68qkf78vation of angular momentum, discuss why a helicopter must have more than omvqk7o.8 f6ucco m:m8ne rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (a) 30 c83x-h,u*s e/hso nzr$^{\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, *cu9 jo.gnt(qusing the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Eart*t ng(oj 9uqc.h.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the M/q9byqy8usem 0hp(j) xg,w0 soon, 380,000 km from Earth. The beam diverges at xeywq,mu0g0 shj(b p 9ysq/ 8)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 rotate at a rate of 6500 rpm. When the m,rdqv0dfz .+-wg.obk otor is turned off during operation, the blades slow to rest in 3.0 s. What is the anv,zdq d.w ob0r kfg.-+gular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on z2g uo o7evoc k/je*ven1d2,;a level floor 3.5 m to another child. If the ball makes 12 on7oevjv1e,/zc e ;kgodu2* 5.0 revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in58lsjzvw(enu2 * w- pcc,rk,s diameter travels 8.0 km. How many revolutions do the wheels jvw 2c*,uknew8c ( szr-l5s,p make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Ckuaxh+9n 7 vz:alculate its angular velocity inuv 9n7ah k+:zx $rad/s$ $\omega$ =    $rad/sec$
(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 (Fig. 8–38). (dku2 0q3 c.1qhhqoqv5a) What is the linear speed of a child seated 1.2 m fromhq .qh10v5o23kq udc q 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 E (n x lwt9-3+uhn:wuhcarth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a poiq t3l8s:1uhr int
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.0 cm from thet t1tus.o mfr0xau;/w24s8 ey axis of rotation is to experienc r 8st;0y2/t txufsueo.mwa4 1e an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly a*2ce l f1b: 6up.vuu.gfdyq 4tbout its center from 130 rpm to 280 rpm in 4.0 s. Determif du 6le c.bp2tf.qv:*uug 1y4ne
(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 radiu3,+d tqc )ckces $R_1$ 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, astronautm8 pt+pn0b ik/s aboard the Apollo spacecraft put themselves into a slow rotation to distrpki tn8p 0/+bmibute 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 rp mb(t*baoh-;+ci(opqbbna0 5 m in 220 s. Through how many revolutions did it tum50n ab ((bhoa q*pit;c+ obb-rn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200s nf/xnjc5-qj* l8 -xz 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 highspee67dbdz yu lw5,d jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20wzd7 uly65,db complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from oid+0gn*,fx//c6i 6 ab*pcycpg/ x y,m,htdj 240 rpm to 360 rpm in 6.5 s. How far will a point ,ibj+tdxd xacn **fm,o6y//ig cc,h 6gypp 0/on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned offlc d d4le t40,a1t/vpw when it is running at 850rev/min It turns 1500 revolutions before it comelw1p, l4ec0/4ttd advs 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 thkoe8e:l9-h mq eob f8 ;/lo2ltp4t ,ide car reduces its speed uniformly from 95km/h tleqptllt;o8,4 i 9k bo-he/8mo e :2fdo 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces itsnaye a/*l+x138 vq fvz speed uniformly from z1aea 3n8*x/vlvq +yf 95km/h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each pedal when clim7op ztcp6z++ pbing a hill. The pedalsz6+tppo7 zc +p 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 d-a5e; bwb*ms of a door 74 cm wide. What is the magnitude of the torque bdema *sw5b-; 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 torque about the axle oh)l wslr (td56f the wheel shown in Fig. 8–39. Assume t shlrt 6d)w5l(hat 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, are attached to the m+)m:ut +oj quxay;y. 1kbmg( ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this sya1 )+my+x u qom(g:;. btujkymstem.

  The bolts on the cylinder head of an engine require tightening to a torque of ozod 794nmf1 4y9mq t8uaqfr* 38 o z 9d 1nfyfma9tm4u 8rq*o47q$m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of radius 0.648 m when the xzm;zpolo u6h2 o /*h4axis of ro/; mo2zo6hz uolx4*p htation is through its center.    $kg \cdot m^2$

  Calculate the moment o+2ottmwsww) kv 2l 40nf inertia of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of thew0lon4m+wt w)22k svt hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, ; 7zpf5xye)j klight rod is rotated in a horizontal circle pxe)f5k;zyj7 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 pzt)y jg*h85zu w d+u:a bowl on a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between 5 + utw:ugpd)yzjh8*zher 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 shown i0er:ia5bmf -6cgo kp.n Fig. 8–43 about e - 5mrfgb.i:a0c kp6o
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen atoms whose total mas, f0q;mhyoc y,.fk6kjs 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 spinninge bc/igj e23r(adf(v8 at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 362( j (dgbc vai8er/3ef00 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 radi63f rlzmt) 5;jebbx. bus of 8.50 cm and a mass of 0.580 kg. Calculat 6r x.35beflm)jb; bzte
(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 frog9q+2mkt i qg cg4rhd60.) opom 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 mew8 7c c7xbgysrzak-p ) wjfa7,;r3i.1ji 9 ifvrry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two childcrpi) ybks ;a9fx- 73i .77va iwgj,81rjczfw ren (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 ro ;3p+ymir 5riwkvioely 4ro8.v( pr,:tating at 10,300 rpm is shut off and is eventually brought uniformly to rest byy( oi;mp8er 5:,ori+r.4iwlvr vp3 ky 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 mph9y m+yzyz3u .4/w f1gu)+r,g zpdx3.6-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 actgju3 8m6u;/ v /oryhfiion of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly fvmfi3u6y;j/o 8/hrg urom 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 j,g jld ,qoo3/a long thin rod, as shown in Fig. 8–46g j,, j/l3dqoo.
(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 tn6.m /i dbeva2wo 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 full turns (rev)waq5 fsdl0r.rr7 d2molutions) and releases it at a speed of 28 w.)d q s 0drr7lmra25f$m/s$ 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 bh1d.mi(frgpkq*i6 )6 cuy* bmoment of inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torb;/v.: uhpjtj que 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 slipf q; mldkayl 8ws:gu6k8y,:(zping down a lane ayugl::;q kl8mfya8(6dsk wz, t 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with resp/d ghg1-ldi labm2m5u2q u5 ;hect to the Sun as the sum of two te;52miqmd/ 1gh5 h-ldla g2buurms,
(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+c5y xwwo8 q;v mass of 1640 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to c5vxw yowq8 +;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 stalkvz ct ;i2z:kjoj(/ ;ag m.c5rts from rest and rolls without slipping (/t;g2m 5:kv ; jczicazjkl.odown 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 bala,do d*he::7o9sjvu 6 l1 eg).vtaoda pnced vertically on its tip. It starts to fall and its lower end does no1se 6vvdoup,)hj: oegota:7dd* . la9t slip. What will be 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 2k4 su2htb0svbh ;xegkfo:)5 on the end of a thin string in a circle of radius 1.10 m at an angu)k5okgs:;tfsh 24 e b vhuxb02lar speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.croc3nz 4 o 6:x*rqm7y8-kg uniform cylindrical grinding wheel of radius 18 cm when rotating at 1500 rpm? cq6n 7yrr3 zxm:o 4*oc   $kg \cdot m^2$
(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 thq nw y 56y78o;ln/usogat is rotating at a rate of 1.3rev/s If he raises his arms toyy/8nouo5l7w ng 6 ;sq a horizontal position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduc4 lrdz6ur/pn(e her moment of inertia by auzrdl4n (r/ 6p 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 angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial rate of .kkn 9zr xubhrik+--4 1.0 rev every 2.0 s to xh9k -k bruz-i4rnk.+ 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 p.o 6+qzctbv ) iwnq(3a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a vbt3qwnioz ()cp6q.+ uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momed7o:yynj;2s)cn/; w mn5ukom ntum 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 ofg)2z(dsz tq , i enu1aa:4go*x 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 inerti(n*, 8vutlhqkgs7n /sa I is dropped onto an identical disk rotating at angular speed s,kh(t g/ *vn7u8qsl n$\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns atn oef n0)0jbv.9rcf1u 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 syv )(5u9.dpnm; zwdvvx g j.s kku(w(/tands at the center of a rotating merry-go-round platform of radius 3.0 m and moment of inertia 92xsv9k((yz d;m /5.vnd wkg)(uuwjv p .0 $kg \cdot m^2$ 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 velocitdjt*3+b mc yh.y of 0.8 *by tcjdh+ m.3$rad/s$ 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, lm i e) a,rxfkm(fm;-e)osing about half 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?(round to the nearest inte))fmare;imkf mx- e,( ger)$\approx$    $rad/s$ (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 inzfq-+fsz7(7sj+ys(h vk/( czo fvsp 7 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 w pj1 qw;bjm hvoe-+pd1p2n+.$ 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, iq3iim0z4 rsv) nitially at rest, that can rotate freely without fric)sr4zmviqi 03 tion. 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 thw1c 9ywh c1ly4e edge of a 6.5-m diameter merry-go-round turntable w4l19yhcycw1 that is mounted on 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 end of the rope lw yl8a(. ot+r 23-medf*1bszs+s9 vsrhu t.eeying 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. Wub et*a(rsese2s+9w3l d 8yt1ofh.m r-+v .zshat length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the0y1qs 8m dv( i (jmar:henre5* same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own aer5rsm 1(v0iaye d jhn8*:qm(xis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of eqye-e- gb1qqc..- nh 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of radius 0.20 m acquire(40 cf1yo78aqitbgzup :l1 r ws a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by tzr0(74:1iwft8olpg qycba 1u he motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mj*fc8yxysm23:j-w5cpy 2 et 9s7y ly 0by6 zagass 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 the end of its 1.0-m-long string, if it is released from restye 223zcljf068-7y9yg:s w xcypybytm*ja 5s.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is tgb)yo 7n kl9s7he 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 rotay3(pat7 : tcjxting at a speed of 1.0 revolution every 12 days. If it were to undergo gravitatio37( t jtx:ycapnal 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 store*2k sfkiok ;trjn+ cb:bv*n 9-d in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travejvn-29o n kk*+i kbc:*; ftsbrl 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 illustrate9z w08sfu(j 7qtjj8avs an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal surface at sperys3sink7t a; 8;a al.ed 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 and3f/lii. g f. kkxp ka;x(gwlw2f7 7b2j 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 o alwk3f(xb2 /2. gx7lj ik .ifwgkf;7pf 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 floor, and we (n 88jl zu/vczwant to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The l /zzuc 8j8n(vstep has height h, where h

  A bicyclist traveling with speed v=4.2m/s on f*q,)7w4l rvd4buh x) fm vvj*a flat road is making a turn with a radiufum)vh w*xvf l 4q,dj*7b)4rvs The forces acting on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.5 6d3iem(y */ u*odri3exuj2gym and begins whirling the rock in a nearly horizontal cix *ui i2(jmyeu*36gody3er/drcle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms4y0n:hqcsvk* g lj03h as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, andqj3hlv *shng yk00:c4 with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylind7zf)3mge31s wkjd8r; 8dps/vpw51de ebr 1jm rical plates, ofs8 dk7vdp;m 1wjp em 331s z1/r b8de)wjfrg5e mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the looped rough trackc fgbx+i6-4he of Fig. 8–58. What is the minimum value of the vertical heighf c4hei6- +bgxt 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, but0sl 2hk xc ozdv/5-:5svske7n do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uniformlylsz.65nezf s8qkk*67 +fikw + ba ,pzm from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular accem ks5zzfl.fk i +wnq a6s8kpbz,*+76eleration 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