A bicycle odometer (which measures distance traveled) is attached nea)t b09,t cgijzr the wheel0z),t ci bgt9j hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant ani1rl: /tlarl* 2tcgsoo9m*; z gular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential aal zt*oc /l1;ts 2rmlgi9*: rocceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the anguaajj,vran4* 6kxa,e h15fn l.lar velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a85-ky9 l uyi.6,jp pnfh9wjng larger force? Explain.

If a force $\vec{F}$ acts on an object such that its lever arm is zero, does it have any effect on the object’s motion? Explain.

Why is it more difficult to do a sit-up with your hands behe *f xuz bx6yat / m7llmwnq6-7vu9x;,ind your head than when your arms are sv xqf- llxnu6zau/yb7mw9;7 6e, *mxttretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel a:;r4mny3b 1e haasx;se 0o+g snd three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a largges;s s 1eran: bm40;+3axyohe rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run lfdu7rc*/rk;fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribut /7rdkufc*l;rion of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, nar-f (: znbz1soxrow beam?

If the net force on a system is zero, is the net torquy5qo wm+c.b-; tagf n-)hj g2ue also zero? If the net torque on a system is zero, is the ofg.2h j5 ty;qabm c-)-g+nuwnet force zero?

Two inclines have the same height but make different angles with the he gttv2q*i :jp*-p2oporizontal. The same steel ball is rolled down each -ottiv *q2pe*g2pjp:incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start a hxi9ed- + ig-1a+ ubj,upr)lrolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches thp9-xi+l)1 jgba + d-eih,rauue 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 radius3gqn;hpm5p:r1p lr l. 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 kn:pm l5qr.;g 1l phr3pinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and af+q sprx1xf3j-3xgj hw .*xq*ngular momentum are conserved. Yet most moving or rotating objects evenx*ff+xq.wpg3rq *1 - xjxs3hjtually slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equa.+9 z 1mpoku7i(zs bhgtor, how would this affeci (uhzgkz7s.91 mpb+ot the length of the day?

Can the diver of Fig. 8–29 do a somersault without having ae:8lnwn*lvb71 mdu +ony initial rotation when she lenvebu +n8dm ll*:w 1o7aves the board?

The moment of inertia of a rotating solid disk about ag:l v bk tfw6nvs3,t1/n axis through its center of mfktt n3sb6,v: wvlg1/ass 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 outt+4ofb 9ym,mkh :nec-stretched hand. If you suddenly drop t4em+kb,fm: toy nhc9- he masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the +r(ym8(n/ e4bck q qahsame mass. However, one is hollow and the other is solid. Describe an experiment to determine which i84 / kqhyqn( cbm+(raes which.

The angular velocity of a wheel rotating on a horizo8ua .9 lec9xdvb,jo u(ntal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points .b,u lovcuad8 e (x99jeast, 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. What 6-6cz(zwdb, n 3u k9dsv o7rb(otb+z5afb/ dfhappens if you walk toward b(uk fbfs dv bzdn,rb5-7(o39t66o w/zdazc+the center?

A shortstop may leap into the air to catch a ball and throw it quickly. As he tek *5nt2o(aa qhrows the ball, the upper part of his body rotates. If you look *kt aqa5n2 eo(quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discuss why j.z(n i0jih*aq(b t( es4n:mua helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep*ji( :(qushneb0 (t.i z4 amnj the helicopter stable.

  Express the following a )/zf;qs vk/zlngles 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 a3 dx- ,/ep ro(1(iblqf4x/ ubvoel8g tn amazing coincidence. Calculate, using the information inside the Front Cover, the angul4d8e-l,t/qr exgofol bi( 1 3xup(b/v ar 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, 380,000 km from Earth. Thee-jwe: pvjde5eq.+x ) beam diverges at an angle exe.+jj v )e:ed-p q5w$\theta$ (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 motor is tu)j wpgy l0mgzg,t5y+ :5p;o;ie jme;orned off during operation, the blades slow to rest in 3.0 s. What is the angular amg zpewej,ij; :;m;5y)gpoty o5l0+g cceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child1 g5wj gq4kd7. ontd)c. If the ball makes 15.0 revolutions, what is its diamd ) 47 gtjnqdw.c1g5oketer?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutions d kctf-3wnh .3co the wheelscfh. wc -k33nt make?    $rev$

  (a) A grinding wheel 0.35 m in do rbea o.z+28gwt99ml+ yr wy(iameter rotates at 2500 rpm. Calculate its angular zo8eltagb+r m.wo9+( 2ry9wy 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 one complete revolutio- x*fl40d zn*jryn c9kn in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seatef-4j *y0kz9d c l*nrnxd 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 eb- wpf9z2tebb+7 .k x7agu)ithe 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 pointsq ptd*d.)pn) (f*dst
(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 frol .,ja*jb rii g.g/0qhm the axis of ro r0.,/hji .biq *gjlagtation is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center fromsagr hrdqhhm/ 1itz: y2*fufg5v 8:4* 130 rpm to 28 m4 :gh *:/8agt fqr1*srdify25zhhu v0 rpm in 4.0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radiuxuc: /ic.w -w x4ls+ous $R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboard the Apollo spacecraft put them xsyua5 ;u b(vm377wfbselves into a slow rotation 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 spacecru yas;53bm( 7w ubfx7vaft 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. Through0x 48cuv/ua88rosj nw how many revolutions did it turn in this c a8u4xrow/nsv u 0j88time?    $rev$

  An automobile engine slows down from kl9hrl .ld.d*c p7 n zxr+hg-44500 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 jets i2:mtw5+sr+b i lvgv5 in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its fin :i+2vb5lm+ sg5irvwt al 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 a td,r+;tj;c. ux8o xtjccelerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled in this .j d;8cjtor +;xt,tuxtime?    m

  A cooling fan is turned off when it is running atv )q b,+wzqshm,p7 bt/ 850rev/min It turns 1500 revolutions before it cometqbv, zh+w qm),b7s /ps to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed uni0ona0e6 lqw zwrzd 185formly from 95k68qr5zzwdol10 nwe 0 am/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

  The tires of a car make 65 hip kd(p/sgh( :3xxs- revolutions as the car reduces its speed uniformly from 95km/h to x-h/pd i (xgh3ssk(p: 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 climbing a hw tt1l)8b )f je,bvzm4ill. The bvlf),m zt1b et )84jwpedals 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 cm widemp sac5o be9dcs7 q:, tw;c-(kadj40w* qhch ;. What is the magnitude of the torq*pdsdc7q4hc5a q(w wj, -m tckc0: ba;ose;9 hue 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+.b: ga3kgyq sd abn)* the axle of the wheel shown in Fig. 8–39. Assume that a friction torqa*an):bgbds.y+ 3 kqgue 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 masslesjd. 0e,p5yq ;m nt /v0lkrjw*ms 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 m;e0jrvk jn/ ytwd0 ml5q*p,.system.

  The bolts on the cylinder head of an engine require tighteningbp,dq(rj of; ) to a torque of 38 (,ojrbp;fq)d $m \cdot N$ 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 inelg67fq249) y4 1ht2 npq3+ j cmfcezlvrp5ixjrtia of a 10.8-kg sphere of radius 0.648 m when the axis of rotation is through its cent4nf1 25jg3 pvcqp4i6 z72 mxy q+rfe9lcl)jhter.    $kg \cdot m^2$

  Calculate the moment of inertia ofq6tugp/kz8hk+o3l 9qc y+ qs; a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignored (w8c/h+qq9;tgoks zu pq+k3l y 6hy?).    $kg \cdot m^2$

  A small 650-gram ball on thuf,7 ;wkxn uj7e end of a thin, light rod is rotated in a horizontal cijk;7nx f,7uwurcle 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 wheely8igmee a.db-9 as9*g rotating at constant angular speed (Fig. 8–42). The friction b ea ia-8ggse*d9y9m.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/xf2tp5tdo2,v,e p xf inertia of the array of point objects shown in Fig. 8–43tx,/ot2ed vxfp5,2f p 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 consi2v j; qhr+tj)9 x8azjusts of two oxygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrit06lao,v0bng e so,*hcal satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the sag,aotb*e6hvl, 0no 0stellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius odu lk:43 zpnxt.bl i5;wefn+.f 8.50 cm and a mass of 0.580 xp+ n4.bf.5:d3ulnw ikt; ze lkg. 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 swingstd4jgl 9*fr4z)f w v7v8ib+t3 tuh0uze f0se0 a bat, 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 smalsu/ l -5tuodn7l hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each t7s/5 -no ulduother 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,rv /yry ,t/yl9wbsl-:cxtizez28 2y -300 rpm is shut off and is eventually brought uniformly to rest bvilw2l yy/ 2ryx,e9zzr:t- cy/ 8ts-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 b9x2jh445u jeleo 2apaccelerates a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by th)4up,xn omc v1qmqq :1e action of the forearm, which rotnq1 u,v1cmxmq) 4o qp:ates 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 blad:gtr x + 7eb;(ixqxmp6e can be considered a long thin rod, as shown in Fig. 8–46.:xm+ ;( re 6qbpg7xxti
(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 cms zarmv(/mbwcr0-1j kkg)- wx w*:: -vb a*vqonsists of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest with9p98w nw ava7hin four full turns (revo99hna a7wwp v8lutions) and releases it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a mo:bqt;2g ql) +m( ezwcohf6p7tment 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 dd ,++ji/;tir3mnjh rrevelops 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 a 8ycfrsqcj .:,oxwot2 0.1luhlanerlt2qw1.u0xc:f8yhc so .o,j at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth wit:7npt (c4aupa h respect to the Sun as the sum of two ter(nt4 u 7apapc:ms,
(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 ofn2x6ga4pw5tetv;rx t cfel/lp6 j+-, 7.50 m. How much net work is required to accelerate it from rest to a rotation rate o l;lnttj+xpf c6t6epe ra4xv-2 ,5wg/f 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from romigo r.+omm *-h/2ecest and rolls without slipping down a 30.0 /mr*.+ooe-mihom2c g $^{\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 fa/x ra ybgb+ 7fbg7q8t5ll and its lower end doe 78fq7gabg+5/rxb tybs not 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 bal3rm2de3q2o lyl rotating on the end of a thin string in a circle of radius 1.10 m at an ang r2lyme332qd oular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cyl+vvti6nkaf7d/ o27 x xindrical grinding wheel of radius 18 cm when rotat2nd7xvo kav67fi+xt/ing 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, onyx 5eud iib403yv( cg71fhyd , a platform that is rotating at a rate of 1.3reyddbcye3( yg0 i5hf1 4,7ivuxv/s If he raises his arms to 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 reduce her moment of ief3t3 r3s,ysqk8v*vqbb l 5f6nertia 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 angulatvlqfv865 e,3 kq by33*sb rsfr speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial rate (zofqo dr*sqo-+b 6+ jof 1.0 rev every 2.0 s to a finaord +zo f(*6+qsqjb-ol 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 rjp.sj6h 1 5jiuotating around a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk o6h sj p.51ijjuf 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 9*c utco gl2(gmomentum 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 thzk(w+.cv(h.itfp 1 - yrt)sk te 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 .*5shz5 ec fbakm9a6t89o dtw of inertia I is dropped onto an identical disk rotating a5b t8w9o6 ezfasmhkt.* 5acd 9t angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2y ;lyb. xu24 (jxumr hp:;2q,u3b36gsr e nery.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotg,gp mm2 s m .xjmww2 ramt+llyi6-4*+ating merry-go-round platform of radius 3.0 m and moment of inertia lag+pm l 2mxg*r-wim4 .,m6 mjty 2ws+920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an angular ve7ck0w )ot nbqduodh1u;hg7v988+ysch y dr 51locity of 00 1ywc h dhhc7 )d81;urn 8ok7bvuo9+dyqst5 g.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 zij,wy7 w+ifrv8, ny6dwarf, 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 away no angular momentum, what would the Sun’s new rotation rate be?(round to the ,iw,7yyj +8 n6vwzifrnearest 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 excess ofqctyp:(kf) 2/gw*w jrz n-ut- 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 u51x 8ct26zfssyf vyo )e9ahx7b .uy)$ 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, temnx7lcmkq-4vj8.yvf2j *- vl shc,b fm,8* initially at rest, that can rotate freely without friction. The moment of inertia ofb.,jxjm 7vye4vf-lcq,fs* nc- kl8t2 8hm*v m 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 the edge of a 6.5-m diameter merryc ,d 8dl9wsia;-go-round turnw,sd8;ldc a9itable 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)rns1 mh0ks+2 fsvzjm1wk7 r: rope lying on the top edge of the spool. A person grabs the ends1 : r20 wjnkmsv17)hsk+ frmz of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that thgpbtbfw ..o 2q*) )pxwe same side always faces the Earth. Determine the ratio of the Moon’s spin ax ) to w.2wp)f*qbp.bgngular 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 -grq(syq ,cx/:de i/ ifrom rest at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape o u(l)jykz h 06 o/dko2dofal;)f a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0jo26 )oko;d( al ) zl0uyfhdk/-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made ozswlnp26*di g)7vh t6:rm4qhf two solid cylindrical disks, each of mass 0.050 kg and diametehd66v4:m2 szw7thlpgi)* qnr r 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 rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of the rear wheel (zdida: ) 7rhi(ma0 w+jg/fi m0$\omega_R$) 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 8.0608ncd2tuv yf p,f wh. times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to t2fy fwp nhv0d,8u c.6a 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-ztffbykb3g :*g, 1va 6;ao7h bpollution automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniforma o kz *,vb6byft: g1h3gfb;7a 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 anmi (3 pg3wm8+im0iids3 o cplmzca-(2 $H_2O$ 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 sn8))294dx s jvzwnw a5b ye2jjp4qb.fpeed 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 length2 dawu 6yq+p+q L can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. A+u w6yq2pq+dassume 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 0f4m(9s j kmz s))bszsR. It is standing vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb m)skbjz9(sfmzs )40s a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.v/ 8ogp 4fkodu :d4nudl:q:j02m/s on a flat road is making a turn with a radius The forces acting on the cud fpo u::: l8o/ng 0vjkqd44dyclist 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 indx u x:,0.-j6hjtitvy to a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come xujh:.0t,6j-v yi tdxfrom?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her ;ic0n u:kc/n bot 71yoarms as thin rods, making reasonable estt;: cknuco/b10y7noiimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly wr3prz+*s ied: hich consists of two cylindrical plates, of rz+i dr3:ep* smass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 track of Fig. 8–5qwi,(0(xdi/ vj dsan/a sa.) i8. What is the minimum value of the vertical height h that the marble must drop qiiiw/,x(das 0a /j )( dv.nasif it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not d2ud3cuhi4ii7 ) kqv) assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces itsvj4z, )2 xdhxc;enw6can** a;fae:v z speed uniformly from 90km/h to 60km/h The tires hc* )efxaxavz z 4;jvdc6,nhw2*a;n e :ave a diameter of 0.90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s