A bicycle odometer (which measures distan(0y,tmb o7 yjgce traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you my0t j gby,o7(use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Donwfnj v( ;g6es9(zl s2 x+ rnfxl9:yf2es a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude ofz6(r9 xny92snfv+ f(;2gjfew :x sl ln either component of linear acceleration change?

Could a nonrigid body be described by a single value of the angular veloyn,1s0 dt9cemkvj *plm n+;/dcity $\omega$ Explain.

Can a small force ever exert50 tof+- si/gnp:;umeycv*c)ipvk y: 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 sit-up with your hands behiw+7- zh0tl gwind your head than when your arms are stretched out in front of yol-+ gwi tz0w7hu? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at *a9xk2 6e098r eqwz4a iupb tnthe pedal cranks. In which gear is it harder to pedaazb 4 n 6 pe9utrikeqx*08w92al, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower legs withm;cydr4d i2f 67fspl : (ev4na flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass 647fary pscm e din4vf2:(ld;is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a l6a6 jhr )buxy6,d2u2tvuwzn4uyw ( 9long, narrow beam?

If the net force on a s2hiym//i0 ) hldf oqx1+ph:rkystem is zero, is the net torque also zero? If the net torque on a systeh po1h:/2f)r myx qidi/h+lk 0m is zero, is the net force zero?

Two inclines have the same height but make -cddkto0mgb ysj+nc.i+y k4tj4k0 97 different angles with the horizontal. The same steel ball is rolled down each inclin stgk ot7.-cj0kdb y4c0y+ 9jdk m4in+e. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously8j) dreqmvfz. 91m pf ,e ty7 cx6i*hjana)8q6 start rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the qeh8jxi,* 87q)ftm61 ycn 96. vfjepa mdz)raincline 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 frrluo hq9o205ro*l mpzy+ke35 om rest at the top of an incline. Which reaches the bhlop e0r2 komr95*3y ouqz5l+ ottom 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 angular mah:b/kx7t65b x;b+ jghkfrs9omentum are conserved. Yet most moving or rotating objects ev;b+7 :ats6 fx9rbh khjg5/bkxentually slow down and stop. Explain.

If there were a great migration of people toward the Earth’w36xan5rc (,i aeii;ys equator, how would this affect the length of the daii5ia(ecy,a rxn3;w 6y?

Can the diver of Fig. 8–29 do a somersault without having any initial rotationsl,i m7w :xfgnj23imwy9)jx( when she leaves the bxjji9m:nfx7myl,w2 3 (wis )goard?

The moment of inertia of a rotating solid disk abou l16gh5yo:fgds*vi e5* c;p olt an axis through its center of mass igpey;fl *d6 l1s :5o*v hco5gis $\frac{1}{2}WR^2$ (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 on1mm n-0x n zngn;8s-ee a rotating stool holding a 2-kg mass in each outstretched hand. zns0 emn; -1 n-emxg8nIf you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the suqny-:3kwb7kg (mtjn 26x x7aame mass. However, one is hollow and the other is solid. Describe an experiment to determine wa3x2 n g7mu 6tw ykj:(kn7bq-xhich is which.

The angular velocity of a wheel rotat94s col*5paf2 bbzl-m ing on a horizontal axle points west. In what direction is the linear velocity of a point ocpb2o b9zfm5al 4 s*-ln 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 speed increasing or decreasing?

Suppose you are standing on the edge of a large freely ru66b6nn9sha i:g x6wuotating turntable. What happens if you walk toward the cents 9inu :6uga6b6n6hx wer?

A shortstop may leap into the air to catch a ball and throw it quickly 41w, clfwom:j3n+q1k cy2vkk. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his h k y3:wc 4n km+w2fcj,1klqo1vips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discuss wh+.k7wv3qnqd vy a helicopter must kq7v3 +vnq.w dhave more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in r :m*zr5t*3sqcd8 od hoadians: (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. Calcuh3r i :w: 5blgxl1mce()/r fbelate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and t 5cef:e mh wgx1ribl b)l(/:r3he Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moonmll ysa2.buc u-7ht;p 7e97 ik, 380,000 km from Earth. The beam diverges at sp kl2 u-7ec7i9h mat lu.b7y;an angle $\theta$ (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 moto*r94jx b, xvhor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow drxxjbv 94 ho*,own?    $rad/s^2$

  A child rolls a ball on a level floomc/su k1 obr i48(av0jr 3.5 m to another child. If the ball makes 15.0 revolutions, r80ciabj 1 vsomu4k(/what is its diameter?    m

  A bicycle with tires 6829upj.(: shskga) oe0ik s1nw cm in diameter travels 8.0 km. How many revolutions do the wsanipo:kw2j)u e sk1( .hgs 90heels make?    $rev$

  (a) A grinding wheel 0.35 m in diameterpg xy,,9 ki/em rotates at 2500 rpm. Calculate its angular velocit xg/kpy, me9,iy 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 in 4.0 s (Fig. 8–38). (a s lym)jzp hd1e8iuvu(3m9c41(e ar2d) What is the linear speed of a child seated 1.2 m from mr3za4u(dvj(yu dh91 epcism 28)1 e lthe 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 Ea 4t*(0w-og pte8hberprth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of aav:, .mxk (lj2 livdd. 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) mu1( ;j ttam g,sofaas) wi.+cgh/7hn7hst a centrifuge rotate if a particle 7.0 cm from the axis oftc)g+a1h 7wh7his m.a/s; ano t, fg(j rotation is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its 1fo,po-/ a363bw2 5uiw6pqvz m urhaa center from 130 rpm to 280 rpm in io 2wa,/3 pzhruw-66u vo am15qa3bpf4.0 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 radiuto,zzrcz7 +f 5s $R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboard the Apollo spacecraft put t,i(9uh: hn0d)a xtt 9j7zynuhhemselves 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 o9(0,txzj h9nhu :yh)tuiad7 nf 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 fromwnm l kgyi5(4+vkg6v l)3txw+w9vnn) rest to 15,000 rpm in 220 s. Through how many revolutions di twmwvn)46 9+l3v kv)gnikgl5wx+ (n yd it turn in this time?    $rev$

  An automobile engine slows down from 4500 r.l 2;g,ijbrrsf4kz f+ pm 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 jeb jgjbs r; k70p292zmnts in a whirling “hur0p s 22mkb 9;7znbgjjman centrifuge,” which takes 1.0 min to turn 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 2wy/5+ j tulzq1a 2w0pvjmh-g/40 rpm to 360 rpm in 6.5 s. Hpgj w/u t+m-h0vq2zw1jy/a5 l ow far will a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 8 yd7m*eri w8 b)or3k)i50rev/min It turns 1500 revol8m ) 7 yrk3ibior*)wdeutions before 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 s t*kws if c:im;25s-lwpeed uniformly from 95km/h to 45km/h The tires ha5 c*t f2kwiissl; mw:-ve 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 revoz lj76wyjxvg a *ktg6*qj.90rlutions as the car reduces its speed uniformly from 95k7.vwqkxa6t yg*r69 *jjjg z0 lm/h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike pe/diup w .f yiltg86:i9v.sn/uts all her weight on each pedal when climbing a hill. The pedals rotate in a circle of .nte:lfiivg 9i .pd/wy 86us/radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cm wide. W e* ge,nx+fisl7eg/-mhat is the magnitude of the torque if the force is 7mfglxe-ie ngs,/+e* exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fig. 8–39. g) ag /mtl9*c:tu4erbAssume that a friction torque of 0):m cagru *gt/be 9tl4.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a massless rod w- 2raa2h8dooih a-zg9zrxju5/ * sjrlf 43y;zhich pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then releaserryu*2a - z8d 2ahi4g;jzfo 9o zaxl3hj-r/s5d. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an e00j8vfx uqss ,9:mi:n9ub(aq u;rd jwngine require tightening to a torque of 38(j80nwrxj;dqsbs 0iuf9umva :u:q9 , $m \cdot N$ 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 when theinz82 px p;kk55/l1*jjkd3hzq;hvl w axis of rotation is thrjvh;hz;*i2k/wp 5 5knxqzld k p183ljough its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bika1rn g ( 5nh1eoio/ wqp6685mqxci1icycle wheel 66.7 cm in diameter. The rim and tire have a combiqk h5/m 5iex8qgrcw1o1ia( n 1p 66noined mass 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 8hch ak*n 2() j.aotrdthin, light rod is rotated in a horizontal circle of radius 1.2 m. Cj dr(khotn *ha.)a28calculate
(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 constanmy0 65jyg/pno t angular speed (Fig. noj 65gypm/y 08–42). 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 objects shown in F s0s,vft-*p2nhpx3/v bzqwbi30* hsdig. 8–43 abo 03hqvw2tsfsnb-p*sx3,v0 di pz/ h*but
(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 ato 2ldpi6oeo68 oms 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 spinning at the correct rate, enakydi;0qrm*6t*1 u f hgineers 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 satelliq1r a yt*dm6hu0k;i*f te 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 radiuse rm*w yc(q9-me5v,um of 8.50 cm and a mass of 0.580 kg.uyq -, e*e5vmm rw9c(m 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 be1h8 kj uhh m6k2v7wsocd78r1at, 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 hand-driven merry-go-round8)mkjfp.yy1)-rjk y1 plw ne0t1 lk+h 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 diwk) yplk8+0-1 kfrep)1 yhtny1jlj.msk 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 rotating at 10,300 rpm is shut off and is ekzc l//ar6. syventually brought uniformly to rest by a frictional torque of 6//czsya.lr k 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 3.6-kg babxo*bhvso/ 415ml*a sll 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 action of the forearm, whiho3si drdj-c1d./ gapj *d:);y.wl ejch rotates about the elbow joint under the action of the tricepjyah g1dp)3wsd* j ;:o /jd.-dirle. cs 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 bee zns +z/.ab 28 t6sb;;svpvpa considered a long thin rod, as shown in Fig. 8–46a6ve/pt2;vsb8;z+.za bsnp s.
(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 mass;xdh; nw3grc /es, $m_1$ 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 dj r ao4w/g;(8ao xebl/fj,x; from rest within four full turns (revolutions) and ob;g ro4lf/ awjx/8 ,aj(exd; 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 hashspd . n2 vi+:jppc,+m 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 deve)5sceq0 sh2yz lops 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 majzd00frw*f9 rdm w2phx:u 3: oss 7.3 kg and radius 9.0 cm rolls without slipping down a lane at 3*rr3 :dow0fdp:2h 0z9 xuwfm j.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sdgdy6w8zk6roc95+f v un as the sum of two terms,5zcfgd+r66yk89vd wo
(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 atf, c0rze 0;eqm- bd+work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinderqc 0ezfdr -+ me;b0a,t.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls4fx5:b7ncqyx e )6/ 4f xbjkpf without slipping down a 30:f 4 7cky5xn) jfx6e/qbxbf4p.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 ver1x-s yop76p7fydrio *tically on its tip. It starts to fall and its lower end does not slip. Whatr6yd71*i pof 7 yx-ops will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotaa,d xnq61 y9( okk*mg.f8migating on the end of a thin string in a circle of fm(d kagani8g1. x*y9o ,mkq6 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-kg un py6pk*4 9ctece b*vbh;:qyct93mp t-iform cylindrical grinding wheel of radius 1 3eyc *c 49t6pvp bmhtq:ykpt -;cb*e98 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 i 0216:;ixov h+x6jpzjg qkgga platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speqo ijh2:; ixg1z6p+ 6 x0jkvgged 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 onqszg; m -t)se2e shown in Fig. 8–29) can reduce her moment of inertia by a factosgqms t2;-)zer of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial rate i(p v-bkvj- zitxn d*j a3x+s+rv 876zof 1.0 rev evead+ -8tivbj (+v xkxz*prjv7s6-nzi3 ry 2.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis through its center at +a jmsbha :qj0q +)l)sa frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is tha + )s0jb:) aljhsqmq+e frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momep42 ec(9ne+2wsfz d8v x yzc5qntum 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 of9f-yq q.agg3)7t p qglxr63lq the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is dropped onto an idex 8a8car.r8t 9xbljc* ntical disl8xar 9ta8 j8b xc.c*rk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns i9ds b3* nsq82iv(ywgat 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 rotatk.elg hzc)l*2q rnl q9kx6c adx8, :b:ing merry-go-round platform of radius 3.0 m and moment o *blz:.l2qe6dagxc )8,nc9h rlkkx:qf 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-rounofy4*kki1hd 5d is rotating freely with an angular velocity of 0.f4odikkh *1y5 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, los /s chdfm6sj;k-m er-)ing 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 n6;r - /sk-shfj mc)mdeo 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+ 7sipp72krl a57x jss in excess of 120 $km/h$ at the outer edge. Make a crude estimate of
(a) the energy,    $ \times10^{16 }$ J
(b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 $kg \cdot m^2$) of radius 100 km and height 4.0 km.    $ \times10^{20 }$ $kg \cdot m^2$

  An asteroid of mass 6r;4mrp p t-yr$ 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, init.vy x8q1ba 7h.tbkm;tially at rest, that can rotate freely without friction. Thextq .abtmvhy1bk 78.; moment of inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diameter merry-go-rou, ++3 b14jeauxib gmudnd turntable that is mounted on frictionless bearings and has a moment i+e,m4d3x1 gaubb +j uof 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 euiz07 7e 4ftri7pl:zy nd 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 rolls bu7iz y:r 07zei4tlpf 7ehind 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 thatl;ua85lq4 cmwpt3 2 yt the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) t8a ly;p tw345q tucml2o its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerate an+ y v fmbh14iej,yyu15(a,hs from rest at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shus5vu;s k. oy+97qhmn ape of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a vsu5y.uo 79 ;ksmhqn+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 solil-azwng3wb o t;fc6o h gnj358 c.wi22.w6 nhid 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 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 5;wb 6wo22w 8h .wn at.6n3-icng3ijogfczhl.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angulajr4g9 8v7j3lbl3grd.gn ;er 0k s 6pagr 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 ma qh rh v8aks)s zs,m.;rtx/3a-ss 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what woux.hsk, -3az)mr h8svrq/ a;ts ld 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 automobile is for it to use energy stored in*i6r7ucdx, .yagti9p a heavu pracxy.7t96 *g, diiy 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 illustrates anrqm8w2vkl m.po.w+v6 $H_2O$ 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 a. c+7hdqdkf r1uwb, ;tt 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.vcn g w9+u93qxvnn5 k0, we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear ak5+ g09nw3cn xuqvnv9 cceleration 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 radiusle pb x;.fn77aoop2k*. a dxb, R. It is standing vertically on the floor, and we wan xxbef ;7no *a.p27l pk,.badot to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a turn with a2u mujrs*2o -a radius The forces acting on the cyclist and cycle are the normal force auu* m 2-jr2os$\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.5 lodxpw2 cw 1- z+j5pweji+7t r/y (er.m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a pj-zit/7w+w1ew+. o e2yd p5(cxjrl rrate of 120 rpm after 5.0 s. What is 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 cylinderh8bpd zc 2p1ya7k6an/j*uyc 0 and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angularzy6c u1p*k70 hpjd aacnb2/8y 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 plahf,b oxjd /+3vtes, of mass 3vh+jxfod,/ b $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 q/fs4 dtlc(r01kq wh.the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the(qr .4c/s0h fw tq1lkd 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 li m05,ukinkxufai*kmj. 8 bee()g - )not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uniformegg f 8mhn3:tnun7-u (ly from 90kmn(u 7fgn uh:nm8e3t-g /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