A bicycle odometer (which measures distance fgq 9nja*2.+yo2gj vttraveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle wijvn.ty*g29q2 jf + agoth 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a poinst+ cearh9zo5w1s81zi6v89,sarab z :p :k q wt 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 acceleration? For which cases would t1i 5z8 v18c9a9:sz: qz +rskerosp bt,ah6wawhe magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the angular velocv 3 ls3zyr ddjn)0n*wp8/)cnrity $\omega$ Explain.

Can a small force ever exqwr2lnsz0p-z p .j9z1ert a 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 s o3mo4ki (jtwul3k.a i/,esa4it-up with your hands behind your head than when owis33 (k.et4joilak,aum /4 your arms are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sp71hq,x pyeqy+4rg b7 k+b,okgrockets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large7y g+oq,k, 1 ryxbg bh+qk4p7e 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 fast ha:w x m*6rpg0save 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 6*aw :gr msp0xdistribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) ca/ehold)g h gtmn7*8;3/.n nam:pmxa y rry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero? If the ne0s( li,(c 0e a+ )oyicqeqqm6jt tor jc +i o(yq(60sl0)q,maceqeique on a system is zero, is the net force zero?

Two inclines have the same height bu a0l y +obo svphx;d/+n8as3m6t make different angles with the horizontal. Theb30 nolom+ as /6+h8vy;psdax same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneousc sne w 2/b+k grtxo4u,9wo)r0ly start rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches thk g0xt9 erocu 4rwn,wo)/b+2se 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 and the same mass. They start from /v oktb;-n8ys rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bots-bvt /;o8n kytom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are aytf,6j /eb2pfauxb9e2ewlc80uw90 conserved. Yet most moving or rotating objects ev2j e/6 leyfe20txuu0a 9wp,c8 f9wbbaentually slow down and stop. Explain.

If there were a grea4yy8 bk9 3hm ) ks3o60ymqr spxtu/c(nt migration of people toward the Earth’s equator, how would this affect the lengthb933cs4ophyky) r /muy mtnsqxk0(86 of the day?

Can the diver of Fig. 8–29 do a somersa68;o/ dlb-x 7dxojsjwult without having any initial rotation when she l db6s odw;/-xjxj l8o7eaves the board?

The moment of inertia of a rotuee 2tekc x x+3el9g60ating solid disk about an axis through its center ofe2t xeku3e+x0g6l e9 c mass is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2-kg mas-p(9 o9izgw4m k*gwths in each outstretched hand. If you suddenly drop the masses, will your angular velog ihpk4gwt- (wz *m9o9city increase, decrease, or stay the same? Explain.

Two spheres look identical and have the (n evc htg3u87same mass. However, one is hollow and the other is solid. Describe an exp7c(n8ueh3v gteriment to determine which is which.

The angular velocity of a wheel rotating on a horiz 9z lsqeimh4ah+)usvnd80gq)d 0 *ri,ontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? d r+zli8*90 g )snveqam q,4h)hd0u isIf the angular acceleration points east, 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 ro(yylue)4 22hmqn d/elj p2wp y)i/9zv tating turntable. What happens if you walk towarde yhz/im )ypv2ye4j /ndll2)u( 9w2qp the center?

A shortstop may leap into the air to catch a ball and throw it quick( mql5ic-.x3 uqoh+ puly. As he throws the ball, th. i+uc5lo(hq xqmu-p 3e 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). Explain.

On the basis of the law of conservation ofs;b8z2 cw+ upub4h (,fs )bz *filx.ff angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or morewfips h(sf2 b+ ;.4l8zzb f)cuf,*bux ways the second propeller can operate to keep the helicopter stable.

  Express the following anglesjuh6+ dkp7i*b 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 becadbho- fza1i8 /use of an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (inhob i/zfa-1 8d radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed s6l,blp7-;ugi: cqp i at the Moon, 380,000 km from Earth. The beam diverges at an ang ulsipgi7 l6 pc-bq,;: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 rate of 650zqi 38u/ yxqkj/ebm-*0 rpm. When the motor is turned off during operation, the blades slow to rest in q*ykqb8z/ j3x/u-em i3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor a50tu3 uqk6i b3.5 m to another child. If the ball makes 15.0 revolutions, what is its ib3 kut0q 6ua5diameter?    m

  A bicycle with tires 68 cm in divseiod(m+ b q 6t79 7s7cehji6ameter travels 8.0 km. How many revolutions do the wh 76ioh(smv9cibed7te+ 6qjs 7 eels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter ro 9 d5idlucd5nx(v4u x(3h*fof,dqt2gna7ws )tates at 2500 rpm. Calculate its angula(ix)tw7h5dv d3ucn4 g f*o d2uds (9qa x,fnl5r 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 revolution i qykua7 h,7hasc sp)w8syz)6 (n 4.0 s (Fig. 8–38). (a) What is th7) pu6sh(yya)7sshcakwz q,8 e 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 (a) in its orbizr.5phj(7d s zu epuk x.efw :a00as40t around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear spmr/kv1bzd ( a32 wcb2peed of 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) musi(7vnmztg q, 2t a centrifuge rotate if a particle 7.0 cm from the axis of rotation ig 2t7n qzv(,ims to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accel5 guvlrw3.sw ,erates uniformly about its center from 130 rpm to 280 rpm in 4.0 s. Determ u5g sw3,rlv.wine
(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 radiux2o4wk jzkwh4:v6t1bs $R_1$ 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, astron:(mhm:/d ckmkauts aboard the Apollo spacecraft put themselves 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 spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Detemm(c:mh/d k :krmine
(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 15,0 wc,6 fvs6y(shua*hs f q68h ,o5/fgrw00 rpm in 220 s. Through how many revolutf*sfs,qow/y6u66h5afh gv (rh,8 wscions did it turn in this time?    $rev$

  An automobile engine j :fvrlhpy()dt( wj4: slows down from 4500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeed jets in a w/cqf xgm trk) (l, sl4* z.z/h5x0voskhirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before)x.z4s0(k* slqrmlv g tx//5,h ckofz 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 apom)v/f o, yx:ccelerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the whe/,xo f)vm:p yoel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev*.puz p4 4,gyfbu2z jr/min It turns 1500 revolutions before it comes to a stfz2ugu 4z4 *ry.,ppjbop.
(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 s s.i.ryod 7q4xpeed uniformly from 95km/h to 45km/h The tires have a diameter of 0.80 m.7 .o4 .xsirdqy
(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:n2)cz244azw ub7kajj37 z9vxjmhq h 65 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameterz72n4ja xcw4ku3bzzv2:) mqjah7 h9 j 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 wheno 1kal1 x zfe3sv.))/ m/r3f)mpgmdu m climbing a hefrmdma3u )xmo) )f1 m.v 1/3lsgpz/ kill. The pedals 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 55gmpw: i9*;qh k N on the end of a door 74 cm wide. What is the magnitude of the torque if the fogp9:hw;q imk* rce 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,nrzn8;h b z9w 5qot/u about the axle of the wheel shown in Fig. 8–39. Assume that a friction torque of 0.4 58zt/b9 ;nwqz,oh nur$m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to t ldck8ii:s)h 1k)b y k0wu;7rfhe ends of a massless rod which pivots as shown in Fig. 8–4k 7l ;i:cy)bh0rksi uf 1)dk8w0. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of uoc.7q9j83 del.ws81gup xpuan engine require tightening to a torque of 38 .eupdx381lscu .g9woqj u7 p 8$m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of radius 0.648 m:wmv :3*asd ;mnbfk(a when the axis of rotation is through sav an* 3(dmmw:k;:fbits center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. T+qz sm d,b97wp-vme*che rim and tire have a combined massm bdc*m9 wq+ e,v7-pzs of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod 3f::w lmhwplrqu x( 0:loj 48nis rotated in a horizontal nlol: 3j0f mw :4xurp 8hl(:wqcircle 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 d)8f7,drb 2u0:r2ti inbc cj6l:wajvspeed (Fig. 8–42,2wjf 0d2u cd: jivnirlbb67t:)ar8 c). The friction 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 inertia of the array of point otf1 kue((l r9yk wu(r6bjects shown in Fig. 8–43 abor futy1lw9 k(e(kr(6u ut
(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 at/x+2 b agyyo83 uz8eecoms 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 cy w5f:ygg8q+vklindrical satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. gywk8+5f vg :q8–44. 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 a radius of 8.50 cm and ) gid:oc ,o,-s-johib a mass of 0.580 kg. Calco,hic )sid-:go job ,-ulate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerayu9nu/ph 0)wlp:.q b fting it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven merry-go-round anr7xd; (pz ayyw1x ws;*d 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 ops7; x;xa ywp wyrz(1*dposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is eventuall avw0k4z:l/5uu -efp-ai* txi y brought uniformlf-aivzupt5ku: 4a-0ie*w xl/ y to rest by a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a 3.6k38wbzk6:t s8xr 8lpkx*r,e k-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 isyigx+g wchq n6/qi)e;*7sep ,d e;5 action of the forearm, which rotai h,/7w ;q)cedxs iny;5g pg6is+e*eqtes about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be con ;ugfj. g)rm-l kzsk)-sidered a long thin rod, as shown in Fig. 8–46.)jl ks.--; fg rukzmg)
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine consists of two masse ry9d)d-j w,7p sovp. mvdly:3s, $m_1$ 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 (re)zw9hlu h -1xkvolutions-91xh) klhu wz) 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 moph.r7azzpkc v a d5me:ub)*44ment of inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a fwj5f.fp nwuf*u t4 48torque 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 dow7q 7crb(3e mdy(wd +wxn a lane at 3.3 yr( 3w+w 7b xqmdc(de7$m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect toqxbof1 ozmwzrz8ik0 v a8b;,/ (+,smi the Sun as the sum of twowqi1si 0z v;+z,( rxz8mk,bobom/fa 8 terms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius o.)z v7yc ou:xcnjsn /2.gug3of 7.50 m. How much net work is required to accelerate it from rest to a rotationu z)cxnj vo :ogyn.s.c/3 u7g2 rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass swezvu38s7+x1.80 kg starts from rest and rolls without slip3vzws +s xe8u7ping down 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 versw *ca z7gs04vl6ge6m tically 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? [Hint: Use conservation of e6wl6msascv* ezg 04 7gnergy.]    $m/s$

  What is the angular momentum of a 0.21 aqm1y7rp9wnppsy 0. 10-kg ball rotating on the end of a thin string in a circle o.0y w s p1rnmqp19pa7yf radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular me4 bqutm+)ex7f 8ewp( vbc)l)c9nq6oomentum of a 2.8-kg uniform cylindrical grinding wheel of rax) 8)4p c6m l)(f7eb9ocvtqwe bq+eundius 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 platfs4xa pq3 tosf*xwd: +w-*ab:*in qb8borm that is rotating at a rate of 1.*8ib* bwb s: +qofxpda n:q 3twxsa4-*3rev/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 hez+i,uvv-::7z jwdw 1 3zsvgmrr moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0:, i vmzzr3-7ujzwvs1+wgv :d 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 ro)aobrbne*74 +ew .ghv tation rate from an initial rate of 1.0 rev every 2.0 s te +awboe4*7).hvngb ro a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis throuhie0+ axu ahu re6;k qdwk-::zr;b hn7u6i/9ngh its center at a frequency of 1.5rev/s The wheel can be 6u-d; rh7qwih zn a9 kku: i:u0e rhx+6/aeb;nconsidered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular m(j7qrq zy49a 6 y5leyiomentum 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 the yju qwt4h) qofye b- 2;jg5bmz 71qc/c+ ldg3,Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is dropped onmj5kmct;swajsr/ m/aev + gs(w74l;a4 rz6n(to an identical disk rotating at angular splm4rk;jrs ze c/mja6awta5;(/(ms gs4v n+w7eed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.wbw ,7lr.nx; o4 $rev/s$ 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 ox 9c:yr r(si9ef a rotating merry-go-round platform of radius 3i xsrr99: yce(.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating cx 8ty+p.nx* ft87slcfreely with an angular velocity of 0.8x npy8 cc8.+s*7tlfxt $rad/s$ 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 intotz au-*jmu2r5 a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new 5a*u-z rm2ju trotation 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 xwcojz k b4e3bm4 r);o:2k5*w+d+-eujm qtkl120 $km/h$ 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 r w* hte/g1u2zz*xf)h$ 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 rotate freely withouttar164gv5z exris 4c+c k2an( friction. The moment of inkr iav6c tra(+s c4ze2gnx 514ertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg pers0 /i.hkdsydn 4on stands at the edge of a 6.5-m diameter merry-go-round turntable that is mouy k.0sh/ind4d nted on frictionless bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of the rope lying on 1x.ln i;jkm 7k2j1h m3 ge4zgjthe top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person witg m 2k31 jjz74 gikxe1;.hmjlnhout 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 sic pt il/c1v g+ a:a3go525yjtide always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital an a:cly goit pv3/j51tgc 2a+5igular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 1 g uo *wt3z*btyb3x5c-m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a;1)fy+4sop 8g d3dqefyfe yo* uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate 1 dpffeg+s4o)yey o8d fq;y*3the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical diskrogcnp7 w o;8w jbl160s, 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 conservation of energy to calculate the linear speed of the yo-yo when it reacwbno opw0jl 1c ;867rghes 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 ig,k5:em,tb psc,vkag9:r 6we s 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, wer9c7wo2 9tmscz*pi km .e rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 1pw7 mc c9.z9oki2tms* 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 l90 l(7+w xp+spe 4qi gop9mlnsow-pollution automobile is for it to use energy stored in a heav p79p4i(gsl++9msxwo 0pneqly rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrax:9g3 cohna4t tes an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal surface at speed v=uc-+dbbah h/+ ls b.d03 eb1to3.3 $m/s$ 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 cfk g*jch 0mc:2an pivot freely (i.e., we ignore friction) about a hinge attached to a :0 2*jgfccmhkwall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standi)o4+m4cm x caqng vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fi4a)x+o cm4qmc g. 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 tu6-mgi+bdvs6c rn with a radius The forces acting on the cyclist and cycle are the normal6i6bv gdc-m+s force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length uy xshw-l-d9bm-pbj8k43,d lk x a 0a11.5 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,bkbasa9mxwu- 1lx-ljy8p-k d03d h4 the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as3q2zmp/es ,du thin rods, making reasonable estimates,uszde /m 3pq2 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 which consists of two cylindrical plates fiy:qj,x4p v( iu4jl+0qi/by, of mass vq bpi0,(yj yq4:4 uji+/xlfi $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 alon 2s bqw8obrwkdi:0n 1qd1jg5+g the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point ofwn k+2b5qdw:j ob 1d1 ig0rq8s the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do nott,e. zgao.q0)ppld1 j z/xm8j assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed unifor* u bos.(acg-2wdt/z7x2ww 4h rpky o+mly from 90km/h to 60km/hbs p(wgzdu*kh2/c274y- rao+xt.w ow 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