A bicycle odometer (which measures distance traveled) i vnwp .1rsrc9;s attached near the wheel hn 9csprwvr.;1ub 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 angular velocity. Does a point ony38 w esxp+2xr the rim have radial and/or tangential acceleration? If the disk’sepxr sw+ 28xy3 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 bodsp9qe gutnn33n ecu.h(,p gz56lf;j* y be described by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert e1*.6icby j jna greater 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 g y qdmbzv-+3 , cd)sijcg 0cofj/0-6gsit-up with your hands behind your head than when your arms are stretched out in front of you? Aso,gj)0-bjfvc z 3-cddiq0 m+ggc y/6 diagram may help you to answer this.

A 21-speed bicycle has seven sprocket*5yh.d o)bkx vs at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a5ko.bd v*y)xh small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on bej ngmk8yx3 -0ding able to run fast have slender lower legs with flesh and muscle concentrated high, close todgk3j0 yxnm-8 the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–39 lx bsydle*e) /1bbyo 2 6qb,qxfz:c55) carry a long, narrow beam?

If the net force on a system is zero, is the vw uj9 m)r*+7 rl9hnxj k;+ qb*1vnhwrnet torque also zero? If the net torque on a system is zero,vn*7+;w uq 9j 1+r*m x9khn)j rlbrvhw is the net force zero?

Two inclines have the same op s0oia( 9w-hheight 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 greater? Explain. ho(i0os-w pa9

Two solid spheres simultaneously start rolling (frojelq/-t (-8n j-et ondm rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Wlnne8jt e o j-q-t(/-dhich 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. They start from 5mrz. f u:g bbyjg0dk0y3h1j :rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at kg j bz uygy:30d:r05j1hm .fbthe bottom? Which has the greater rotational KE?

We claim that momentum and angular un +:h4;)xk; wjnkzhomomentum are conserved. Yet most moving or rotating objects eventually slow down anj)zkw ;:h+kn;un ohx 4d stop. Explain.

If there were a great migration of people tu2o x)x xe/9faoward the Earth’s equator, how would this affect the lengthax2) x9xuefo/ of the day?

Can the diver of Fig. 8–29 do a somersault without having any initial rotation we( :mn z )6lxehzz * os8ws7fadz6/zj3hen she lea hxlsz nwf3zj)ze 6*dzo:8s/ zm7e a6(ves the board?

The moment of inertia of a rotating solid dipr2 /fu7b ;s7 j(s6yp4tbqqpnsk about an axis through its center of massr/p2 4fb7qys(nq6 7 ;tjbpus p 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 af-l(el/0dilxww + sc / 2-kg mass in each outstretchedwl-lif x +de(c/s0l/w hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and haxgm)b.+ud 0: grp qzl7ve the same mass. However, one is hollow and the other is solid. Describe an experiment to determine which is whig.pmu0l+dgx z 7bq: )rch.

The angular velocity of a wheel rotating on a horizoiu -qf-tyh5uk nbs-q/e4mqk( +s9 w5u 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 east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speunt y -qm--(bqe5uw 4 +/5ks9ifhu qsked increasing or decreasing?

Suppose you are standing on the e,d 1 2yogd vwxkho.o.,dge of a large freely rotating turntable. What happens wgdd x,hov 1ky..,oo2if you walk toward the center?

A shortstop may leap into therwd +97cu yr86fxmrx/ air 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 tfx wd9x ury+m/8r76rc he opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation oq :) xbvosul-2mn qr.-f angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate t b-q l)n rvx.-smquo2:o keep the helicopter stable.

  Express the following angle9nd ,o/xuvs ,tof7na0s 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 Earthi22nhm*0(u)tqc9l: j5oie jmo ctr2 m because of an amazing coincidence. Calculate, using the ih ()itreq i*jl2 cm ocmo2j05mn: ut29nformation inside 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 thih 9+ lmz v5j+1oxs)woe Moon, 380,000 km from Earth. The beam diverges at an anso5 l im++ov9jh)zxw 1gle $\theta$ (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 mhv-..st r9x hxotor is turned off during operation, the bladt. 9. rs-vxhxhes slow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another childe() navy ;62jkaskbbo-x (jz: . If the ball makes 15.0 revolutions, what is it(2j)j x; obayb(n:kvs6 aez- ks diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revoj 5m3m8wg er8nnd1 dl7lutions do the wjn5d1 g73mdlr m 8w8neheels make?    $rev$

  (a) A grinding wheel 0.35yh8 p 4dch0ug)d9yn:c(wnz:k m in diameter rotates at 2500 rpm. Calculate its angula y(:d:dn w90u4gckhnyh8)zp cr 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 rikxe5+e gv;+x(ssocr ua0d3 : evolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of ax+aek+ r vui(x ;dge03oc5 :ss child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orbit arbqy ;iw;;aw5i s+d8mq- j q f3lbc8mk;ound the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a k*8br)z4e.itm *tuh4qer rt ; 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 amd/t :wo:1*cy5 doapz,jx9cm centrifuge rotate if a particle 7.0 cm from the axis of rotation is to experienpjt/1 m,xc*mwoz 9dc y: a5:odce an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheellxo chqjeqzk4. r7:)i:l(p :g accelerates uniformly about its center from 130 rpm to 280 rpm in 4g7j i):: c(l. q4qpkh:roxlez.0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radius yk935) ve0ysz jwdd6m$R_1$ 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, a, ;nz ,ivenw(wstronauts aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Snni,z,eww v(; un’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 158zif0x 8lxicgdz,qje +- fo rpm in 220 s. Through how many revolutgff 18 oxi -qj5d0ic 8zxl,z+eions did it turn in this time?    $rev$

  An automobile engine slows down from 2gl.yzst6n iz6b205*xa t tpu 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 highspe 9cia -kh7d:cofrv5m g:pj ;t7ed jets in a whirling “human centrifuge,” which takes 1.0 min to tpdh:-j itcg75:c kr;of am79vurn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 360 rpm in 6.5 y2:jc apypzpr3b:mea3tz-i ++1qc -f s. How far will a point on the edge of the y jria-p ap :b+tyf+32-zpmzc1 q3c:e wheel have traveled in this time?    m

  A cooling fan is turned di58(dp-l(k lx ls9wdoff when it is running at 850rev/min It turns 1500 revolutions beforiws8l( l9(ld-dd kx 5pe 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 i ejxmz8a l8 ,v0ps:bt, 5s4vfots speed uniformly from 95km/h to 45km/h The txs,p84m0lv ts abz,v o5e:fj8ires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions a3;:7fhiaokck2fg+:tkvik w /qx ;2hh1 h3 atxs the car reduces its speed uniformly from 95km/h to 45km/h The tires have a do;/3:i1 hf3kcf2 wtk72 ghit+kahqx a v;k :hxiameter 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 put/ zodz w)yq5f ;+2cd mau/+nbrs all her weight on each pedal when climbing a hill. The pedals rotate inf/+5)ddwb;n yz/acu+m r2qoz 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 forh. 6x1yl l)nna lwscp9+xo8mg/k 7 z7pce of 55 N on the end of a door 74 cm wide. What is the magnk/7.1c 96a)x hlws xlon7n+8 plz mygpitude of 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 torque about the axbyhzv+ xq n av i(r;/; hz/agdvt18h1(le of the wheel shown in Fig. 8–39. Assume th v8z1 g/ +v ;bq(d1yv/rh xa;iha(nzthat a friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, a*avls, s krni;x-*c,qre attached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then rele,,ar vs-isn *l;qkx*cased. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engin( 8rn nq4v) fun/x5afke require tightening to a torque of 38 nk)r5f/48nqun(xa v f$m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg spherexa:va: vhpb.0 of radius 0.648 m when the axis of rotation is through i:a :bp.0ah xvvts center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diam+zwny wi;:7re : o-tvxeter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub :r+i:vo-y x zw7;tewncan be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light b:;e4 p;110ysxg ghwbir c/s prod is rotated in a horizontal circle of radius 1.2 m. Csyep1x ph/0wbc rsg ;41g;:bialculate
(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 cons + a;vo1qw*1py gszmk rz:36ybtant angular speed (Fig. 8–42). The friction forcr:1 1vs+om;z3 wazb 6 y*gqykpe between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point objects shown in Fig. 8–f*+ ismd( ;zq09kiciz 43 aki idf; (sczm0z iq+*9bout
(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 bm vtgffck08pf0, * 5owhose 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 spinnh-a.np tgcj 1(v9n,pfing at the correct rate, engineers 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 isj-nft v,9ph(1ng ap .c to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylindks e7gnd 7kdm6.v e+(ner with a radius of 8.50 cm and a mass of 0.580 kg. Calcdd+. k v(7ne6ns7ke mgulate
(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 swinq c;n-tw3wd4ok0 x y-ml;nqe pmo;9p8gs 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 small cc s9jtoig ov+ol:+q0e,9s;c+x( oflhand-driven merry-go-round and is able to accelerate it from lfqov0;+sx l io9,+ 9tcg+ jeoc:sc( orest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor ro;/j v0bgc.vd7u zqje5 zr ita4ily,5/tating at 10,300 rpm is shut off and is eventually brought uniformly to rest by a frictionaiqi/zjcj5 7lt4/;zbv .,50e gad yuv rl torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a 3t2: tav6,r xoa2enb(x77+m kpmapb 3a.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 bal cr) a.sh+vol*):w+ hc6z6kq oxwir g(l is thrown solely by the action of the forearm, which rotates about the elbow joio)acl)z(+ x q: wchs 6r*6wkhrg vi.o+nt 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. qpkgpi,t.w-j +dyr ut5*:ka.8–46.kjkpi+.y5wptau :d*,r -gt q.
(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 consizs qat90uupyl45h-n ::i d:j zsts 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 tu)phm0u v5n 1,8inbw2w ph k7m( ttld(crns (revolutions) and releases it at a speed of 2hu5(npw7l,p20m wtb )mi c 1dk ht8nv(8 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a moment of inertia of/h(:n t y)( fcj2honhm $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torq7 3cw xtzgqm64ue 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 z+emv/vq (to55bg , oyradius 9.0 cm rolls without slipping down a lane at 3.3 +v oq,mzy (vo5eb5g/t$m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun aazqh -k 8y.irqas8. y3p-f6uas the sum of two teq3kyy886p isafuarq h-..z a- rms,
(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 w4.9t3oejz i*qf rmxqm 7,ang p-bh4*eork is required to acieb-m jn3p9**4 f 4tqeh 7. xmqaozg,rcelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm gt. k adov.7;cand mass 1.80 kg starts from rest and rolls without slipping down a 3vc7d;toga. .k0.0 $^{\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 balann7ehjmdfo5fidb f jo .g6 0y7dc:m088ced vertically 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 o0dm7byf mfe6.gh8 0id5c fj d8 7jn:opole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of ag 1fz. fsq:yd 4r(xa9a 0.210-kg ball rotating on the end of a thin string in a cifa4ya. qzfxg: sd r91(rcle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-k3se, h n3mhjw0e,f9 g:0iymgog uniform cylindrical grinding wheel o 0wg09,:mio , mg3hnejhsyef3 f radius 18 cm when rotating 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 platform that is romkqm(dtfpxu06 :+ .yj tating at a rate of 1.3rev/s If he raises his arms to a horqmj:kxdtu +0 (mpf6.yizontal 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 onaip fd7 ;fx1 .eb(o2xje shown in Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when cf;1e(jaxx .bp io2fd7 hanging from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial rate of 1.:0fx)hjp,pm)9cd tn-l t gd. a0 rev every 2.0 s to a final rate ft)laht:p jx9mc-g)0.dndp , 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 axis throudk2e7ril*g2c5g 0 m 4+e0zaufk ftfg1gh 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, approximate ei05mr1 * f24edzuggf 0kc27atfglk+ly 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 momentum of a figuccufih a2, r;okc6wak*lu7x,1p-z j 0re 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 fdxm5 g/l52gg6 lppj1 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 ineyv1 epn(:4d lfrtia I is dropped onto an identical disk rotating at angulv1p enyd:4l(f ar speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns atfgac c7 3k;;- erejof: 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 center of a rotating merry-go-round pla9-9udbcx*adc( rp yvqa :*x, itform of radius 3.0 m and mo:9a*acvd rq9bdyx , pc(xi-*ument 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 with an angul*x.vyh *lxwo):hb d /wo1*dwcar velocity of 0.8* xlw hvy):w1dwd./*hob oxc* $rad/s$ 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 col-:e(;x xm 3a(lxefa trlapses into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assumi3re aa e-;l(:(tfmxxx ng the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in excess of 127+zw3rdrms (.cgcw2jn ,av1d;zjo8 l me 8an.0 $km/h$ 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 bic6ayx .( ru+$ 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 platfo*00nkdd; anm:v1ff g trm, initially at rest, that can rotate freely without friction. The moment of inertia of the person plus the pf; m*gkvf:t0dda n01n latform 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 afumsipu ;-rn 4,4nbgpk6*p +stands at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on u6i4- *4,bfpsk ;rn+gunpa pmfrictionless 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 grocj : f.xg cswneghe1 +l4ptz t(pgn8- ,l+d17zund 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 ron7cjce-:nsph1z x8 g tlggw4l zt1, e+d +(f.plls 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 an cqj4c.z8f)d, n 4h7q p/xfqthe 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 paxqc4q87)hzfndc qpaj/n, 4. frticle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 1 l(ybabcp k1,p 8g)z0dm/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of radius 0.20 mwra,n+3nat7a ozm 8ib2s .d2x acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by tm8z 7a,wxs a2b.+tno2ian3rd he motor.    $m \cdot N$

  (a) A yo-yo is made of twqb8 ztq2o26s fo solid cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrioqf6 bqz22s8tcal 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, h4 w*nleu o+z( 5k3v6yt ivsk0pow is the angular speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times as great,7ut1(ecjcv x1 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 17vue1t cxj1c(1 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobildyve 3fk :.2/db7vov(wwm t48n gjv3ve 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 uniform cylindrical flywvy:ew4nmwg 8/7 j.v2tvd vk (od33 fvbheel 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 illustratey0hl:4n-d y2 nxhybn8 s 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 surao x5l 545j1i9nunix5e lue n*face 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 pivot freely (i.e., we ignorg31q4mcpui-l js :8 iie friction) about a hinge attach31p mgi8 ulqjs4-ci: ied 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. 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. dis 6tuk)(;x,da h k0gIt is standing vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a stepua)( d0, i;h sgdk6ktx 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 making a turnx 7pz/4k1*(zsypd/ p 72sbz o,efslmw with a radius The forces acting on the cyclist and cycle are the normal force wd, zk f47* ss( oz21yp7zm bslep/px/$\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ 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 am3onux0/ w*qu 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 whewmxn*o/u03 u qre does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thin rodsgi3 ev6ygnt( wy 6l9c ku(ig8;, making reasonable estimates for the dimensions. Then calculate the ratio of the a 38(wng6iu 6;elyytc gki v9g(ngular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which con ofpef84h 1c0hsists of two cylindrical plates, ofhf 8c0h41pof e mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the looped rough track of Fig. 8–p h).5fka3zi zb ko yd./3u5tx58. 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 without leaving 5i.b)z kh/yzu53a3k .p fxot dthe track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but d)sd 9cmy1w/pgi-a um:+m rd,no not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revol :n/xbycmmkb 8/y. ,iqut)5gtutions as the car reduces its speed uniformly from ./t5) umq,m ygnxy:/bk 8ibtc90km/h 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