A bicycle odometer (which measures distance traveled) is attached near tiiy1+g. v6ps s1vo:owhe wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with+vyi6posv1s .i wg o:1 24-inch wheels?

Suppose a disk rotates at constant angular velocijv-)f8l(v8tq /pv 4pheks zz4i ): gvxty. Does a point on the rim have radial and/or tangenq- zpl4/h8vzvs iegpjt)4(: 8vkf)v xtial acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be desc9znh7xxan0-x(7 mzu;.qo g4gdd s,s)vwcz j - ribed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a gread(6z)eq3b6 jar eb6 elter torque than a larger force? Explain.

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

Why is it more difficult to do a sit-up with your hka2)g+lyw k +lands behind your head than when k a2glwk+)+ ylyour arms are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the reauxoef/f e j:5:kvfn6 8r 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 l8ek ufvfn6ef5 /x :jo:arge front sprocket? Why?

Mammals that depend on bex-pl+ / wvxen -niu3a5ing able to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain wi/-5 3unnav exx-wpl+hy this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8,vp -8 kt,.vio 58msp8icfjf f–35) carry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero? If the ne3gl40zjfj qt ;p36w6 exf1jsnt torque on a system is zero, is the net force zero?n6l sf3t zgw6jpe0; qj13jxf4

Two inclines have the same height but make differentl+(xknub4 f,tm*vamnu;ip1c. b q ;;p 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 greaterf iq;u;bnc l+mx*pbv,(k;4 np1.u tam? Explain.

Two solid spheres simultaneously start r -:.fmprgd;j(kuylx- + nta, wolling (from rest) down an incline. One sphere has twi;.w dkgtl:xr, nj(-auy-+ pfm ce the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius ao8cw34sqi 8tgeby g 4-nd the same mass. They start from reb qie 8-twg 83gs4coy4st 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 greater rotational KE?

We claim that momentum and angularnp)d l +r f+0n*igt)ew diht0+ momentum are conserved. Yet most moving or rotating objects eventually slohdri+)+ nlw*0de0 )ift gtnp +w down and stop. Explain.

If there were a great migration of people toward ther 5( oeh1b rs)o-sc8yn (aoxz8 Earth’s equator, how would this afrh885bzc1-(n yx s)r a oeoo(sfect the length of the day?

Can the diver of Fig. 8–29 do a *ft5fo/o/(n df hy2p osomersault without having any initial rotation when she on2t5/ fphffyod/(*oleaves the board?

The moment of inertia ofn-fd +xi)- wu xell4y1 a rotating solid disk about an axis through its center of m +- e) llwnfy-x14diuxass 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 hold-dhqizv62 uf0n5 wkqw.p6b i1ing a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angula0z kdpfinhubqi65 6. qww21-vr velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However,vk.jo t l8 s*u,wrf(v bth4/9b one is hollow and thvfstv4 / (bjlkuohwr *89,.bte other is solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal a cn8fu+;(ai:t7mx )p)c/oeq t: gsasv xle points 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 of this poin8:e+p:/oas()ttcsaf)n qmg cuiv x7;t at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rot1 thq(xei*ryu fk+myy i;0:1 rating turntable. What happens if you walk toward the qe0yu h;im+yr x:i 1fyr1(k*t center?

A shortstop may leap into the aiqa9pp*)d.e ; -gnaxso r to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explaoad*pas p 9en;.gx)- qin.

On the basis of the law of conservation of angular momentum, discuss why a v98teu0xle/ xpe q(cc5 ; 6jqihelicopter must have more than one rotor (or propeller). Discuss one or more ways the second pr eq98e/iel;vcuqt0 c5j(6 xpxopeller can operate to keep the helicopter stable.

  Express the following angles ing5d)6- kph9 li*zm zlj 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 amazing coincidence. Calcu + obz*dd2puq197itrclate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen oodp*qz7r+db ti2 u1c9 n Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed lm5+*lnkj z6bb 6n x1bat the Moon, 380,000 km from Earth. The beam diverges at an angzbnm6b +xln1l5k *6jb le $\theta$ (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 rlz) lk+njo(e :ate of 6500 rpm. When the motor is turned off during operation, the bladese zk)+j:(lonl slow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a llm j)/dpi2 a *d.q tax)(z,ouwevel floor 3.5 m to another child. If the ball makes 15.zxomw )/d,qu)ji(*td.aa2 lp 0 revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km.dvhshz 8hdj)*,k:ki( How many revolutions do th*,jh s:hkzidvdh8 k()e wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rnh b6t.z3 v6mu-iq /drpm. Calculate its angulaumnbzd6 ri-/3 h6t.vqr 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 revu(z7qcb) avn5rd-u 6nolution in 4.0 s (Fig. 8–38). (a) Wn au7nbcd)5vu( r -zq6hat is the linear speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (pm/j u rvg0, yvnrj 60+*bi*qfa) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed o; de (wh0j+kifxs)+v uf a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particlesn)) nuq 0un*aw) n*l8a.zba hwkq0 k-:h:ysq 7.0 cm from the axis of rotation sku: wn)ly)u ).*bhnkqsnh0qanq-8a z *:aw0is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheeah2zj*t3eik4b nva*nr f-2l gd0 b0u: l accelerates uniformly about its center from 130 rpm to 280 rpm :30k bja 4lir-* bn*0ut2hzeadg2vnf 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 radiux7 (oxnkib /*ys $R_1$ 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 thww3 i0jm 7f7qle Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energq7lmfj0i3ww 7 y 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 rpm tm* az km(on.e d,q4(ho7qe3win 220 s. Through how many revolutions did it turnh34(kanmq.(,towod q 7eemz* in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calculj)2+*rbtnuo qate
(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 strestbu )x wq+n9m)ses of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutiox)nm)+u t9 qbwns 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 240-nd.b fmw ch:6 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the ww. dnm -b6:hfcheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/mi ysgd5ben:9q0 n It turns 1500 revolutions beforen :e0dbg9 5syq it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed uniformly ny06ne +js ir+from 95km/h to 45km/h The tires n 0jei6rys ++nhave 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 aqx4z: h0 6;/p nhfrql) sxe zgh(8/whss the car reduces its speed uniformly from 95km/h to 45km/h The tires havgh 0hrlne/4h)p ; sq qxwz8z:h(/ x6sfe 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 clipmxqm *2* kqf.mbing a hill. The pedals rotate in a circle of radius 17 cm. m mkfp2*qx*q.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of twn7 e hw*13fe(1 sfo8wqx5cl 55 N on the end of a door 74 cm wide. What is the magnitude ow7 e f*ox1clw(3 n5fw81eqshtf the 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 tor09,w ccg:-ix0 o+ppb t rax9ykque about the axle of the wheel shown in Fig. 8–39. Assume that a friction torcy: i9ocp,bw 0xgrp-9tax k0+que 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 ofmkh8gv9e.a6y z67 s(p- lvzwp a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and the(yvpzwze- 6pvk98m76 .ahl sgn released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engiegqs hma :/q41 pm7km2ne require tightening to a torque of 38h mmkem/1a 7gqp:s4q 2 $m \cdot N$ 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 p r)pw0 e/x8eg0.648 m when the axis of rotation iegr8pe/xw 0 p)s through its center.    $kg \cdot m^2$

  Calculate the moment ofay onnag:s,92 inertia of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass 9a ynns2,og:aof the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in a horw9 -s hgy+g: n; gci-e5bo2qneizontal c ye-gwsebnq9c5: nh ;g og-+2iircle 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 wheel rotating at constant angular sp1 )ro u6h.wnm lc hiqzls4m-(/eed (Fig. 8–42). The friction forcm/hcqzs m- oullh1 i)64. w(rne 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 inert p: c-e/ag(/s:jyhifoia of the array of point objects shown in Fig. 8–43 ase:/-(p :gjiacoyhf/ bout
(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 congj8s1hzo . bq-sists 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 cylindrical satellite spinnxb;y8hlxcc4 qs /c+2lu z*m, oing at the correct rate, engineer,lx;u q82*l4yczs mc+ cho/ xbs fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with, r2qxw1)se8lss4 geb a radius of 8.50 cm and a mass of 0.580 kg.1b s82ewr g,e )4xsslq 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 bi gz0(l-/ zpb16 :jckdf y9awrat, 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 tangecoxwor ,+a: nmt ,6-h0g8zirrntially on a small 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 other on the edge. Calculate the torqr iwrza0 8m-or,o cxhnt :,+g6ue 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 dn1q 1aqj2xir 3sj4x zd:b3j3,300 rpm is shut off and is eventually brought uniformly to rest by a frictional torquq1x4:dx ndq3zi j a2jsr3 1b3je 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 acceloi*wrkk :3*(b-sjf hw erates 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 the b;ae;ww,c 6g caction of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is acceleraaccbww6e;;,g ted 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 bemb:a qbi7 rsnmk+a v q,g 328t:/5hmhh considered a long thin rod, as shown in Fig. 8–46.a+k nqibqhv /:gth s82mmm3 r,5abh7:
(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 8uj:xjdf 9r+n consists 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 within four full turns ( k: 0afj5ac a*n(xc8 c+nazz7xrevolutions) and r :cjnak8fca*50+z 7acxaz( xneleases 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 ht/)w 98unwxr oas a moment 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 develop e)g hxcb26xa5s 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;pd:kar x)( mt without slipping down a lane atd kmx(: pat)r; 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respectosawu9q ; 0k ik7-:s)k9zs)gkec-qdl to the Sun as the sum of two term;)0d sq -lagwk e9ockq) iks:7uz-s9k s,
(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 net p. 5c-gejc q,iwork is required to accelerate it from rest to a rotation rate of 1.00 revolution iqe ,5-jcgp.c per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg start) 6hm.6lhmtyj 8hg,0- kvihfp s from rest and rolls without slipping ,h6h 0pf6. m8yh)l-hjikmv gtdown a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

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

  A 2.30-m-long pole is balanced vertically7exx-x xxbg yts n/3** on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hinx-3xxx/gb xn7 tye*s*t: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end of a thi02r5xk wa 8 4b:m:4qjsr tgnibn string in a circle of radius 1.10 m at an 2 ::t44i0jmsq5xkbrwn8gb ra angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg unifo, jwmsgk 7z24zrm cylindrical grinding wheel of radius 18 cm when 7 zs2mgz w,4jkrotating 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 1i;50v;lf g chokez (cb;r/e wat his side, on a 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 oh;lcwbzo i (1;g 5ce ;kfr0v/ef 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 inert7bs, w32 vpozfqk v/-bia by a factor of about 3.5 when changingkopw,7- b3s/b 2zvfvq 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 izenpemv0q6-4 nitial rate of 1.0 rev every 2.0 n4z e0qe6-pvms 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 arofbeko8qex j11 q 5b6h7ii: zw-und a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk ofb k15e 8zx-6wb:o q7e1i jifqh 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 momentum of a figureequa.sgwtl+ k7n3f6 3 qfd,6j 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 m8t571i aa qodwa*yv,somentum 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 momeh/ *iv, zo7zb9ua1c3ss5w9ig kpu f qha/, 7uxnt of inertia I is dropped onto an identical disk rotating at angular spe vgz9z/ 7uqo7s s,u1*,i b9h/ facxi3wu5akph ed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at ro:vc *2(5;x- au y9rz xn(hfsyjt4xl 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 sz. n cev 5hyvmw,aol6 *ap/ova0kr:9)tands at the center of a rotating merry-go-round platform )vezavc:v l5w.am,ory a 9/0hkp* n6o of radius 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-roq ox;tslj gh/iak0 xr4b.3564ink(tkund is rotating freely with an angular velocity of 0.4k5h6l0g/txn4(bokx ki 3i; a tjqrs.8 $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, losing ig3iglwk4bagczm p h-; as 3mmu(z42nu0x. . .about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carr(43. m3igmm kh4-.axcla. up0wi; gsb2gznuzies away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in excccws .j)p8hxa: mgxf52)i e3hess 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 l3ay 3*6 hyxk hf-0dec$ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, that can rotat20x+ kbylgli ds; i+5he freely without friction. The moment of inertia of the person plus thig 0;xd 2kyli+h+lbs5 e 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 merry9pug.:oja z/ -vy pj*k-go-round turntable that is mounted on frictionless bearings ap:pk.9j ozv-/j*a u gynd 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 b7sr;zah rd;v+t,j+p 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 ondht;+ar;bvsp 7zrj+, to 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 the same side always facesy +bu5 .+pmfjl the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earthy.p +l fmj5ub+.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 1 m(nwp x 6(tnz7eg2lzp8/$s^2$ 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 ri 7ok z; q6l4vj0v/aithe shape of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest olkii4/ravo7j0 ;v6 qzver a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two s 0l /vwaf3 deamg f v)jnip0m:w2:vfw/(3q8 feolid cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservatiinevm3 jf 0v28w/0m :q3)a /f:pfldaeww gv(fon 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 l* wrkspe)8t/is the angular speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times as gril nzjkc18+i *eat, 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 1 zjnii+1 8k*lc1 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 lbrihzwa6 2oli9.pt:.ow-pollution automobile is for it to use energy stored in a heavy rotating69i.lz:ob ithpw2a r. 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 illustra+1no . 5qlmhpqtes 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 rolli u4e:qmc2.q:u -fj;0aa dff dyng on a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivjhg;t ig10acrv 53- zhot 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 acc gi 1;t5crhzg3 0-jhaveleration 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 standin,b hbd 6:v dpnrpk,6b2 4tc3gbg vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a ste, dpnpt66bbgk,chr 3:d2 bv4bp against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=w- oqr0 (s dn+0yls,te4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist and cycle are the r,ysn-elwq0 ( os+t0dnormal 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 o2 beftcz 52x:0.50-kg rock into 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 dobx:eo2zf 5tc2es the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thin rods, mak9 inpavi-0(ba ing reasonable estimates aab( -piin9 0vfor 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 which consists ,xl m:mt*eu x9) 1h k-mt9vsfhof two cylindrical plates, of mass : m1-e,9tvk9hxm xfh)*lstmu$M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 andaqd7 y;lu 9kw- radius r rolls along the looped rough track of Fig. 8–58. What is the minimum vaua9 7y-qk wdl;lue of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do notprfy5k65rkxguzwrdx 1vh79o x9/-)8b*tykf assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 ren)i7sg-yda.;vg oc -bvolutions as the car reduces its speed uniformly from 90km/h gvyo .-s a;-i7bgcd)n to 60km/h The tires have 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