A bicycle odometer (which mes 01udv 1oj(v392hfxnww(d rsasures distance traveled) is attached near the wheel hub and is designed for 27-inc uvdo(s1sjxw9h31wrn( 2vfd0h wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocmry t:s49ljyp a7av6- ity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration cmst6j47ry-ava :l yp9hange?

Could a nonrigid body be rv 3kv5c-pjb0dna 6ji 2k.w)idescribed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torqu(86gfhv:o * cq0 +rlbizr1l mz;vs j2 ixkf.h5e 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-upl4ucshyo012pz,u 3s 6. fprti with your hands behind your head than when your arms are stretched out in front o zuprt.4f3pyics,6h u01os2 lf you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wh. swynx+0qzec 1hzx- )eel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?1zy0- )xz.he qc+wsx n

Mammals that depend on being abwffmwxi5z* s/ov t3ql(vf5.9 le to run fast have slender lower legs with flesh and muscle concentrated high, close to to3q wf5ix/vt* w.(flf5zmsv 9he body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–3szj-ac4, buq8 (e3ag n5) carry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero? Ifx 0twogpmhl x752 (p+;t-wlbq the net torque on a system is zero, is the net 5t0xgptw b2ol -m;hq7pw (lx+force zero?

Two inclines have the same height but make different angles with* 7:tn21 x9ohqalcodv the horizontal. The same steel ball is rolled down each incline. On which inclxdvqc19a* o:7nh 2 otline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rol4xntubwli1k+,m2 3o mpt1 yy (ling (from 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? Which has the greater speed there? Whic1b ylmmwu ,43ni1o xptky+2( th 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 z.zes,)u r4e*h(jb1vw4 s fsn8utz/ k rest at the top of an incline. Which reaches the bottom first? Which h4ezh j)4vu,s1rfsk.z8bt ( swu/*eznas 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 o:fz-, j /u7 ug.n7ay;en yizbangular momentum are conserved. Yet most moving or rotating objectsf7.,auog7nbiye:n-uz jy / z; eventually slow down and stop. Explain.

If there were a great mi,f:9cngp+ s/m o:c8 qe :jcpeo *3uyqogration of people toward the Earth’s equator, how would this affect th8e/yc: 9o+q :cj *q3 u eo,cnspfgm:poe length of the day?

Can the diver of Fig. 8–29 do a somersault withou. g2jwuvga timl0;m8 )t having any initial rotation when she leaves t)2mwg8.ag tl v 0uij;mhe board?

The moment of inertia ofel oo)(t ;vd,ad fo2e+ a rotating solid disk about an axis through its center of ma; e+,vod f do(eo)tla2ss is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2-kg mass in each outqh246- vjrb 5ef bif)(sd:rkx2 i nvf3stretched hand. If you suddenly drop the masses, will your angular vei-) 2r jr6 qvff3(enxd i bvh2ksbf45:locity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is hollow and thra(y2ax ;z/y ke other is solid. Describe an exper2 (zr/ y;xaaykiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points west. In wg1 8 jn)bctdqfx37+nv hat direction is the linear velocity of a point on the top of the wheel? If the angular acceleratt +v7d 1)n8cnbjgxq f3ion 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 cl +jy16c annba8hzvqz-3 vu +-6:gpoedge of a large freely rotating turntable. What happensv6h lcv+y -- n68nau a1o:gbcp+jq3zz if you walk toward the center?

A shortstop may leap 4bor e,;ijz pt ec)7r:xe3 0su:7endfinto the air to catch a ball and throw it quickly. As he throws the ball, the upper su:ed3 ,:j zete7p cri) no0rf;eb74xpart 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 ogrpd/ w n.rg 53kp3*qconservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helico.3 3p nqg rdro5*wgpk/pter stable.

  Express the following angles in radine jy:3m -;-pd-0byigawak :jkv, oj:ans: (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 Eart g)/lu0i wg;i av(s+upl9f( omh because of an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Eartio;g+( )l9a0(m gsf uwupvi /lh.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km f-xq dpm1dd)h9 rom Earth. The beam diverges at and1 )9xpdmdqh - 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 a l0kk ;qp+h1yof 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular accelep10ayqlk kh +;ration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level fl ;dqvyx,w,d0l oor 3.5 m to another child. If the ball makes 15.0 rl;,vyw dq0dx ,evolutions, what is its diameter?    m

  A bicycle with tires 68 cm i ./ woyhm9ua6dn diameter travels 8.0 km. How many revolutions do the w.9y/am hu o6dwheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500bgulp hd4 y s:n3/jf wa:kne-s*0h.3r rpm. Calculate its angulr-ujls bd:e/yfh04gs*: nh.apn3 kw3 ar 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 in 4.0 s (Fig. 8–38). gqo2o;n :spc,0n /en 8e5baav (a) What is the linear sn,/e:8; onecg5 n sp0boa2qavpeed 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 orbito*o ./t3l yql+xns y6c+.v/b hulle,c around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a poin imqfizt io/*cwi,5b: 1f(zn( t
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.0 cm vm xpd t :k*/t1 7xnv6/pl1tibfrom the axis of rotation is to experience an acceleratiodv 6t/i1l/*xpv pk tbx:ntm71n of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniforqh s7r78j .fnymly about its center from 130 rpm to 280 7fn 7hrq s8.yjrpm in 4.0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radiub4dyq q0 e:ys-s $R_1$ 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 f ,f:s:v0r xy-b8jz/cpnr, tsMoon, astronauts 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. Determine pc zfx/8yn:b, -sv0r,tjrsf:
(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 rest60 yh2nxvcp xi(7hgc6 to 15,000 rpm in 220 s. Through how many revolutionhx 0cx cg7i6p(v6y 2hns did it turn in this time?    $rev$

  An automobile engine slows down/lj+x0e c-4b+cnkpmqi fbu ;1 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 w49du;zd2;v qhvrs7jre anb* (hirling “human centrifuge,” which takes 1.0 jr(b s2ea4nrz9v* ; u;hv7qddmin 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 acceleratesp8dl)pdzuk 7 3r;vmcsp6m h :*kxj76r uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the e:ux *)73d6sppj rdk z mvl;r67kcp 8hmdge of the wheel have traveled in this time?    m

  A cooling fan is turned off whui4ka3i, x+p3yov(5 kwz* 8yos5 fnpxen it is running at 850rev/min It turns 1500 revolutions before it come3vpu*azwoykko,s nf3x4x 8(i 5y5 i +ps to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its skhx) ld/rnyk0) k3n 0ypeed uniformly from 95km/h to 45km/h Th0k0y)nryn dk/3lxh) k e tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces its spejari y*1g94is5fq+75 jlfu nled uniformly from 95km/h to 45km/h The tires have a diameter of 0.80 m.+q*l9u gafjy jf7 i154si5rnl
(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 eac (41hq a bvl ;h7ybhu )(jws8xq)omvl6h pedal when climbing a hill. The pedals rotate in a circle of radius ubsqj ; )7l8(by6o x almvhqhv4)(hw117 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 5j1a 1 fs b6e 53a4rhm4vdze5qu5 N on the end of a door 74 cm wide. What is the magnitude oudre3v q451zas4a bh61 jm e5ff 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 axle of the wheel showntl:gy2r.09 knz1r ow a in Fig. 8–39. Assume that a friction torque of 0.4 an y2 : rzktg10owr9.l$m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of4 xpeqhh f9r7 jz0+ cuo95fxe: a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.uh:h9 4e po5jzxrc0x+ fe9f7q

  The bolts on the cylinder head of an engine requirecgp jd78; 8v /phhtd3w tightening to a torque of 388whvdc78/ ;djhp3p g t $m \cdot N$ 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 inet rcn gl4c4/hoe6rd,6 rtia of a 10.8-kg sphere of radius 0.648 m when the axis of rotation is t 4t drocc 6nl,grhe/64hrough its center.    $kg \cdot m^2$

  Calculate the moment of inecse f,83 . xkwehx3sw7rtia of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub k8ch.exfw,s7e3 3s xwcan be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of 3 z0agh1b- fwmegt;o6a thin, light rod is rotated in a horizontal circle oht;3bgwom0f1g6a- e zf 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 bot9 )e-q,invw o glo2fae;/t b8n,,h ql98gv nywl on a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her haney9wnifb,)h,l/otvqag o ,2 8l-gvn ;n8e9tqds 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 arc58 v2 cngpy3mv8)c3u w/alh wray of point objects shown in Fig. 8–43 g5aplm33cc 2 vwv8 w/unyh)c8about
(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 whozk fqxt+m, g wz75+k;p:+c lxpse 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 sak p fyu5y(nw4vg:x-y 6tellite spinning 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 is to reach 32 rpm in 5.0 wnv 5fy:yp xk(4-yg6umin? $\approx$    N(round to the nearest integer)

  A grinding wheel is a ej0u7bj2,d q+ 7ql(ltt0gi n s.qaf;r uniform cylinder with a radius of 8.50 cm and a mass of 0.580 kg. Calculat2gjtlbtn0q ;sl7a0 (q,r j.ifqe7u d+e
(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, accelerati6/:xndde1g5 jo w ljzjy9ia7w30pyq 7ng 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 tan4d rta9yzyc-3o 36ewm gentially on a small hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other oyroc4yzt a963e wm-d3n 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 roeph/kg wt560mr yp6 m/,ckkp0tating at 10,300 rpm is shut off and is eventually brought uniformly to rest by a frictional torque of 1.2 ey pgk0c,/rk0 kw5m6pt m6h p/$m \cdot N$ 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-kbyl2; 5yhxu 0bg 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 action of the forearm, whicwn v )szibkkn p3).9d0h rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball 0s3p)). k wdvnknz9bi 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 ca3 m5 mi:1wkige;:ofm pn be considered a long thin rod, as shown in Fig. 8–46.5 i1: 3m:;kigpwfome m
(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 tw al a euu8/3s9d(v( 7jwc3yiqmo 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 accelk u1wti*xf)56skbf3v( 0d sgherates the hammer from rest within four full turns (revolutions) and releases it at a speed of 28 kgs 6fhvtsd 3x(5 i)ub1 k*w0f$m/s$ 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 mom :fep jw30q.weent 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 tor./n8a0gxttnd que 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*0msxtr pfv) wojxkqtq7-.(8 kg and radius 9.0 cm rolls without slipping down a lane at 3.3 wqkpm0rt(. xoxtqs7- 8f* vj) $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Suna7qd(ku ai-rn f-2) tlrqgje 32)d1vu as the sum of two termv dajq(r3d272knt-l) fu1 aqg-re)i us,
(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. 6 d 7*e.zy:mlumxpb ts/;7tmbHow much net work is required to accelerate it from rest to a rotation rate of 1.00 revolu:tm*7l7/x ;etsu m pbmd. yz6btion per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20gyl.o/h p.f4v .0 cm and mass 1.80 kg starts from rest and rolls without slipping down a 30fol yhp./ v4g..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 vertically7-v4f2 vx- slfod if0t 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 f-td- 7 4i0svvf oxfl2pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end+e fc4,js ya*sqq, ys-xmy.p: of a thin string qj s+f:,yqex,yc*my-s.4 spa in a circle 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-kg uniform cylindrici h)k3axl66h 9dcdn ahnj(+9bal grinding wheel of rdhcdh63kh abjxa n6i9(+ ln )9adius 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 pl2fo qo xjf8mp:bp p7voj s3 4bz--91r f)4rbbqatform that is rotating at a rate of fbb9ppqo2fqzfb-ro-3o b srjx:1 j7448)mvp1.3rev/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 her moment of inefpa cg7:4o/q86ue ftortia by a factor of about 3.5 when changing/fa ctpq u 8e:76gfoo4 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 ratopmc,yfy.k yb1dc v) h) ps/);e from an initial rate of 1.0 rev every 2.0 s to a fi1cfp,.vkm o shy/pc)yyd)b;)nal 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 ax / m-bbq,hgf rzyglzj)fw 6* h/o0.ckue(h;7zis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the-e*wjygbqzu/gfck l,6z)( hmhz .ofhr7 0;/b 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 figure skater spinning at tj j6c/r88r0u;mj wdt3.5 $rev/s$ 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 lan038/vpayr z3 8h 1 usebv0dmomentum 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 mgkax2xgnw :x.9 /td/r ra-q-of inertia I is dropped onto an identical disk rot.tx-dx9qraw/:ag kg/n2 - mxrating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 5gawnqie zc8s7/i*4n 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 cdnvm 7. tsb4o,m+6nqcenter of a rotating merry-go-round platform of radiu6 +m ,bvn.mosndc4q t7s 3.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is r ,.gw *(qydi.7sidf jzotating freely with an angular velocity of 0.8,zdydifws*(gq i.j7. $rad/s$ 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 wh0dk n.ei*jc 1 s76qoxcite 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 rotation rate be?(round6jcqnco e7sd 10 k.x*i 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 e1 n;a( v jrxdsuto 3ner,3(oc)tw1tx7xcess 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 prvj 5zli-v:aq*y)d qh-nzw .c,rc3+$ 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 freel5 f(p5b 4grvyk9o3lbpy without friction. The lb op v4(k5f5b ry9p3gmoment 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-zhzybwxq :jq77gmcn 1y(*4i 8 m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia of 11qznc7x*yyi(4b h:j g7wmq8z700 $kg \cdot m^2$ 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 onir yd3j9 gai0 f9 k+:s0kh:h: aat(9rb oajgk+ the ground with the end of the rope lying on the top edge of the s ygr r0a:ih: a+db:okf9(k j9 s 3a0htajg9ik+pool. 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 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 facwpu(i p);r1pg xzk(6 qes the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, tr1q pw (p()rgupx6;kizeat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 1 m/ :pet6b+gr5pnb m:. zj$s^2$ 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 cylindmwm4 1h;wj x,0sxzqp/q k;3 over of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torq4v wszxop 3hm; qjw; k/xqm10,ue delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two*pnt m (s6 9o wgmxsr*t;ll pjl2:/)jk solid 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-r * *wl/t jst xj)lp92;o:lmnp6ksg( mlong string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of th az ro;,jdv6uevmygo,8(.at, e 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, were rotak7u) 1vpbe +tsal auoctoxj,.3/65ahting at a speed of 1.0 revolution every 12 j)6a ,su b7xtl5. 3k+vaoah/e1cuptodays. 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 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 automobile is f 3 uz-mqw h46epc*y*q:efmg,ior it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindr *q, em-cfe: um3pqi* y4hgz6wical 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 anayrkajll:1il j ( l,*vl wkitf4n1 *9, $H_2O$ 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 ro vcq-c-+s;h *rdlq ac0/ alghth/9/t zlling on a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and lengtue j4m kj /n8iijep fy0ib8r;16d:e (hh L can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. e81jiy hemf/i 0 b6urjkp 84;(:idnej[Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing bt:k6i *mhx7n*rsx2fyksh2 jz)x-4i vertically on the floor, and we want to exert a horizontal force F at : zmni 22 6 byh4rkskj-*)sxx*fxthi7its 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 bg gq ;exwb7w 3q,gk qbu82b*:dri4*hflat road is making a turn with a radius The forces acting on the cyclist an;2qwk7:gi*bb g u db 4rqb,8h3*w qegxd cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.5 m and begins wh.s j28f upxhj-fx3ow47 x8afzirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come frx8auf7 oh xwsf-fp3j8x j .42zom?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thin a688 oekgspx-w9yo yu izy888 rods, making reasonable estzy6 yw os-o8ieu8yp9k88 gx8aimates 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,osw* vxi8m90 a of swxia8v m0*9omass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 ok()wi2kjb gaej7z(,qcai58 rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble musaij(qk), gj7a8 ( 2iezkwbc5o t 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 nei1c:a,sy )n0 sywep(ot assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed unif i gd* y34j;;mldh)2 rsmheoj73;f e.xssg9d yormly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration ofg sj;sj3df; ;*edyh4xryml7o9 ih3 d2sm eg.) 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