A bicycle odometer (which measures distance traveled) is atl427j;b./ajjzy hget tached near the wheel hub and is designed for 27-inch wheels. What haa;jl4 g2/zjeh j7y.tbppens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velswuvu +7 m- 9bbdghsr*4yb; lx5 e75wwocity. Does a point on the rim have radial and/or tangential accelewdbsweb gw; 7y+ 9-uv b5mr5u l*h4sx7ration? 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 descr5vht6u9oj 4p qibed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Exp7 ov1f5ki 2mhvlain.

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 si.m2 t;tb1s kdwt-up with your hands behind your head than when your arms are stretched out in front of you? A diagram wtd;1 ts.k2mb may help you to answer this.

A 21-speed bicycle has seven sprockets at thec :)fd,kwyr v9 rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket?vyd9 wf:rc,)k Why?

Mammals that depend on being able to run fast have slender lower l6lsny;t g slpxg +/xb8 kp-+- zpdy6k4egs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotationa4ls+ppkd+xlyk;xg6s - gb n yp/ -z86tl dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow b* df 1sx*iuwnt4e*g /ieam?

If the net force on a system is ze4pcdqc..8w)- ypddfb ro, is the net torque also zero? If the net torque on a system 8dd y.bpc)cqp 4d-.fwis zero, is the net force zero?

Two inclines have the same heipko.0 s. m*7:trpnu might but make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be great tnsm0..*p7oumik r:per? Explain.

Two solid spheres simultaneous8jh3/zrah 6vyfmovp( 18 cl4w ly 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 /83z8 jahh vml fp wo146(yvcrfirst? 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. They6q c m1l6)ptxe4:lwzl 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 tzx m :1lll)64cweq6 pat the bottom? Which has the greater rotational KE?

We claim that momentum a6nnwchj8r 90 x2wp*:.s xkboao-5gvi nd angular momentum are conserved. Yet most moving or rotating objects eventually slo oikw0.bv5hx o 8- npng cw9x:j26as*rw down and stop. Explain.

If there were a great migration of people toward the Earth’s equator, hoh:tosvdnw ,-2w would this affecv-sh wo2,: ntdt the length of the day?

Can the diver of Fig. 8–29 q8 f0e mr2(tnpp9 xl0wdo a somersault without having any initial rotation when pr9(nx pm2le08wfq0 tshe leaves the board?

The moment of inertia of a rotating solid disk about an ubop 2a,;- --nvzty9 two 4swxaxis through its center of mass-to;wvostyn2ap-b -4x9u w,z 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 rota gkuiqj57fytwz 6*9h8ting stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity incr68fuzj9 *wyki5 7t qghease, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is hollow andmskxrp,u /2-(prfs 3pfig m7* the other is sol-,us7/(ik f m3xpp2 fm*psrrg id. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points west 0 dbn hrolmh2 2dpkncq, gthk,g.+,2v--z6gb. 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 of this point a6-p,v.zk-k,,d 2qdnglgt0 g2 mchr2b+hbno ht the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating turntabl.h k p .bg5qliu/-ulk6wb23aje. What happens i-/gp6lu bq5bji.k aw.2uhkl 3f you walk toward the center?

A shortstop may leap into the air to catch a ball and throw it quickly. As he ta ;x0s7vp21jg k-j uhl7 mauw4hrows the ball, the upper part of his body rotates. If you look quickly you will nou0mjka2-uv gaw lp1x7;4j7h stice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of con)pq nsi c /8yxoc:7:kcservation of angular momentum, discuss why a helicopter must have more than one rotor (or pr ky8is/:n7c:cqxp)co opeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles osb0xgj-hez/8x n20r in radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an q,bz msq, l;t 7+h.,x(wvx1)+ 9k zdmfgfrldbamazing coincidence. Calculate, using the information insibrz(,l)vlz.fq ,wdxmtg+ k h+x;, fq9ms1db7de the 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 at the Moon, 380,;f+ wx8*wdd40jc tgfm p(yp *a000 km from Earth. The beam diverges at an angle*xd;ffj+pw *48waymp(c dt g0 $\theta$ (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 mc4sm gvh a1k74otor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades a41gckms7 v4hslow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.9 p22o0s5je 8nzy sjls5 m to another child. If the ball makes 15.0 revolutions, what is its 2jpsz0y29o8n5js lse diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many gxia i9w8r13 dhu 2svf -j-v-rrevolutions do the wheels m93g-du j riviwrx2f vs-ah81- ake?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calcu 1n; (b,s 9.nohxoavcxlate its angular9no.;n(, 1cov sbahx x velocity 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 j 1(wd9q3-ic)-s m4-dstaf hfh h*xs pone complete revolution in 4.0 s (Fig. 8–38). (a) What is tc*sfp)d9i(j-xs h3 4 -qhw-f1 thadsmhe 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 tboyf6 gh l lhvcjr2)mk)g52 :;he Earth (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 point.g3j(m8dgx0og i ani (
(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 ij 6e,m*)v:14br oglup. acacjf a particle 7.0 cm from the axis of rotation is to experience an accelerrg c6. a:e,cpv lmjubo*aj)41ation of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its 3zx lww8xct5 *center from 130 rpm to 280 rpm in 4.0 s. Determinez8xt* w35 xclw
(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 radiugeq42.jhc;z8 d si: wrs $R_1$ 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, astronautsy oxo2 jol17(gh71gsc;r4zpy aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerate 1j zs 214cgo h7yp7xorogyl(;d from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpm in 220 s. Thrxe43 1 5elm :h.hyp z8nsbem vevow3(1ough how many revoluyeelmm8v.bo13x ew5nsvh z :p1 4(he3tions did it turn in this time?    $rev$

  An automobile engine slows down fra;w xxi. timj(8tc ;8vom 4500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeed jetvn p,v gds0+q xy kt)xhcvtdg (46.8/us in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions befor,.t4(vkd xpctdsn6vyv+8/0gg u xh ) qe 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 diamaf nzfb16zh, (f-bqo8eter accelerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of th f- z6obabq8(z ,fn1fhe wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 15hp.y(5(v w rsx w byd,1ohf69e00 revolutions before it comes ty x5owy( vb 1h9 fw(h,rps.e6do a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed uxm y 3)5imup3wniformly from 95km/h to 45km/h Theuy)33 mipw5xm 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 mak8gpi*e1tf u5 2hhub9 jn67msee 65 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a ep98h5e7*s umfbu1 6i j2tngh 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 ped-l.:ybtvqt;to0jy dva 3 ww2vu b(a.7al when climbing a hill. The pedals rotate in-(bt y 3a 0dwva;y:l7w qj bot.tvu2v. 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 cm wide. What is t tlm2mhuh,0 s o/.avttyzb,v:y4o -y+he magnitude of t + lyohoya.h: -/v,mb,u0vsmt4tzy2 t he torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fig. 8q-n) inh(*h-o+ttzg -up .(q d0m ymkb–39. Assume that a fricti yt.diznhtqm( u-gqb( m)-np+o*hk -0 on 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 ends of a massless rzwmf;fp8 1qz6od which pivots as shown in z; 1mq86fwf zpFig. 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 renk.15iso zhl70to ;g hxvl8f0 quire tightening to a torque of 38 0z h8nhot f7 l5vig0.;sx ok1l$m \cdot N$ 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 wheng (x7t3.ak,- lhynhbfn rd1 y5/tbi 8f the axis of rotation is through its cen i3a5(dnyhf7r y xgfh 1b/.-,b ktn8tlter.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. Thr/djk8rk n(; ce rim and tire have a combined mass of 1.25 kg. The mass of the hub can(; rjkcnkd/ r8 be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thiq+;tnf 5 c(cscxs pwe1w0n au: -(lty-i3 /kftn, light rod is rotated in a horizontal circle of radiufsany5e ((ct+lt0wpiwk-c su:- 1;3c/xtnq f s 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 a0k03yl;.n o 2pdupz fd potter’s wheel rotating at constant angular speed (Fig. 8–42). The frp.d o;y03 pdnf0uzl 2kiction 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 inert4 8ys/wq.c bdoia of the array of point objects shown in Fig. 8–43 abo8bsqdw . c4/oyut
(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-9le / gc ;0 g xab2hkwhe*wtvk.gt5:r mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the correct rate, eng5955/:hgmz e i *uznc0 kyr9p(grmvueineers fire four tangential rockets as shown *ncmeu9yk(05pmeh u r/zzrvi g5 95g: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 witx::xly3o5n l6 f7rbv hs:7ovyh a radius of 8.50 cm and a mass of 0.5vy73n yx6:bhor sx 5 :l:7oflv80 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, acceleratiy:)ce9bhuyj ruw;zkei o;, 5kk m+6q+ng 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 ablea3,u, )nme6m t nl3pge to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) 3e a) ,n3,gm6 tenlmupsit 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 isj e -)xx9cpry) shut off and is eventually brought uniformly ejx-rp xc )y9)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 amevei; . u3g-- vv3lk(.i rycg 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the actionvlvx* :*n jc+a of the forearm, which rotatn*+vvc ajl *:xes 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.4 syh(z2f/ uio 8–46o4fhsyz/2 i( u.
(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 e vf 9 ntxrkkbauk1vr00(d0;; two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelera(fym y9ck)4vfjr9asy 43 ht yq v(k5w7tes the hammer from rest within four full turns (revolutions) at v5j4)39cy(savm9wf yrh yf qk(74yknd releases it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a moment of ineowwi(co1vd2.ov ; n r sto854qrtia 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 torque of wkyn8 (i9a8kx 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 withodj;fxouq :s08ut slipping down a lane at;x0d:f8o qs ju 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of tn /*v -r tkjgr.e3a-xm he*su4he Earth with respect to the Sun as the sum of t.*vngkr3tr eax-*-/ e sju4 hmwo 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 and a radius of 7.50 m. How much netlaigcig rubi/9n:. 1/ work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 nuagg i./b :9lc1iir/ s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starhmtnxi2fn3 /0ts from rest and rolls without slipping down a xmf/tn2 ni 0h330.0 $^{\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 starts to 7wrm (zt dwe8.fall and its lower end does not semz8wrtw( 7d. lip. 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 on the end of a ty3lbpla8 *qjn i*:. akhin string in a circle of radius 1.8pa ynail. :*3b*qljk10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum4 t5hdv2 1tenc of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotant th1cd24e5vting 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 c8ywgj09i go -platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreas 8i0o -g9cjygwes 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 r h fo.i zgh0rc+)8gea4ji8 lo9educe her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck positio848ic ogo0hf+g9 l. jezia)rhn. 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 1.0 u/ail2 xgv 74ocf1ew t-4pwy*ree cxpwy/4fvg iu*o2a7t1 4lw-v every 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 vertical 6z; .tt;k2rp fwqf fl0axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay f.kzf;l06 wrt2fp t ;qsticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinmsjp6 ;.qwi 5 h qc6it:6os(j(n3awufning 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 m,gjljdmt. :.omw6 i, komentum of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I ieqkrr9(:qqxa7dj9 cjtvl*73v,o+ ts s dropped onto an identical disk rotating at angular +vc9*3x7tkj qo sl7e(v:tq,jr rq9 daspeed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turnsf6 efzxmd80-h 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 cenj0mn ( /ggb8z*/dc bxyter of a rotating merry-go-round platform of radius b/mz c/xb08ygj*n gd( 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 wib7r,x 8pqpx9 wa: .cdhth an angular velocity of 0.dxap9xp wc.7h,r:8 bq8 $rad/s$ 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 collbq udsyv8*/ 8wapses into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries aq8wy/ud*vb8 s way 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 *4, 6(/z mr ags8t3lq4u ;bbewax,a;pr glxpywinds 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 va +/h w 5qtz fy8ll)3yuty5m1$ 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, thabden n1+,ze za+a se8 mj3q2jpe/s,g.t can rotate freely without frictio g ,.z+d+/8n2 mqsseebjje, zanae31pn. 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 ths-j.w bi9 ;o vmxxdh.jr06(gfe edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a mome0ow; bxfvdsmj g.-(ix.6 r j9hnt 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 rol qbdf*n88ds2 uma: 0bkls on the ground with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool 8f0 d * q:sbbna8um2dkrolls 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 always faces the Er :;clib)xt6nqrfo o4k;5 )d sarth. Determine the ratio of the Moon’s :5qoki ob4r ;sntcx;lr6) )dfspin 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 from rest at a rate ohu z 0(szs+e(6xmio.(0hzyw gf 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 c,r(:rykt:/ *) ksela4zp rpkshape of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Cker,:k4yt/)* crlpr( : s zapkalculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disr doau7a6x-ku/q2k1-wewr pl6 8uzd. ks, each of mass 0.050 kg and diameter 0.075 m, joinedxkuow6d-./ql e 7rur6- uaz8wda kp 12 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 rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of t+ fp-d2khalz7kiqw mx7d t* cu: n;5b6 id*:zshe 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 wimh -ls6x j4x*tv8p1wtth 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, losinw-l tp mv4hxj8x1*st6g 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 3befql8tv. ba19d2lnautomobile is for it to use energy stored in13n fqe2bv 8l a9b.dlt 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 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 illustraopacuaur4(f +7zt2 f ;rmha 4-tes 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 speede64gl/n hbf 8s.u+jz r 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 lengt,:+yu r rgmt ;wnf)6yvh L can pivot freely (i.e., we ignore friction) about a hinge attached to amn:ygy ,v+;6u rtw)fr wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing ver*bomte bt,vnh *vnxf/0s c, 77tically on the floor, and we want to exert a horizontal fo stv*v77t fo*m/xh , neb0,bncrce F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is ma0p /tc)eghl a,king a turn with a radius The forces acting on the,lch tp0e)g /a 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 m0a suxpn, p9ul+,*ldpn:aa9w;l hrv/ 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, an ax:vl, p/aud9ps,w* 0nh r +u;lp9alnd where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thin rods, maki0lzm q5s.mj( mng reasol. 0smmj5(zm qnable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a ch3wj46l rpp nkf s37b:hu- hfewmm7,(lutch assembly which consists of two cylindrical plates, of m hs6fbl7n hpwukp(7:e mh-3rw,jf m34ass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 ro *3k.l/lyaz2hxkm6wx b +d:x yugh track of Fig. 8–58. What 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 mz/wbl x2d.*ayk6x3h+x ykl: without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do xmd2 a- q+np1dnot assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uqm7-pzjc u sex;xp1-qwivf;4c.pa i79: :xxkniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular accew qsk jfz7-xii7pxc;e4 :1uqx m:p-x.ac9;pv leration 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