A bicycle odometer (which measures distance wj73up5 aro4ql07+x sp :wgtc 7bfk)btraveled) is attached near the wheel hub and is designed for 27-inch xsjk 7wtlwb4rgcpp7 ouab5q)3+ :7f0wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular veloc 4itu-s 4cfivj5r9ze9k hzy w(wi+85fity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have raczs hiw5wfvi8ur-y 9 +ji(ze5 t94k4fdial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a singlerj9tl1 7x4ayev (j1 by value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? E hd) 89gmzjo atd.7a5xxplain.

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-u/)9d . +f/0bubc/u ax bemx*jbal x8ugp with your hands behind your head than when your arms are stretched out in front of you? A diagram f/gebcmb89/u0d)a x +.ubl*b/a uxjxmay help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the pei 1jib* xxb7)8sdhj+qd / ch zs2,n9b irk7*kidal cranks. In which gear is it harder to s r ii z98i*b d)77xbb c skq2nj*i,kx/dj+h1hpedal, 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-b4th/28w yu:u r6bu ivey+ma run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution b u6a yr8it2u4hb/ :e+vyumw-of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow beam?)dhju 7ql63p y

If the net force on a system is zero, is the net torquecu3fb66c( soe also zero? If the net torque on a system is zero, is theb u(6sf3c6 ceo net force zero?

Two inclines have the same he5;nuqan uh p)vx , d,y7lb*gjo xm+/4fight but make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline wiquv;,,*j+ul/)4n 5n7 dh bxxaogf yp mll the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an iq3bfmbt (hud)i 4 gx94ncline. One sphere has twice the radius and twice thhmb3xi) gb(9utq fd44e mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the 4*r,gzj fg1u7r jwh s1s: 3nrrsame radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed a*: 3sw,z1n ughrgr1r s7 rf4jjt the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum( as6/(bk- hytkgoe l.k t*3cw are conserved. Yet most moving or rotatingc(wk(be-lt.yt sh*k kga o/36 objects eventually slow down and stop. Explain.

If there were a great migration of pf*ntg7 ch10zegvh(q-iwj ) m0 eople toward the Earth’s equator, how would th* zhtqm-h1(gvgje w0 7fnic0)is affect the length of the day?

Can the diver of Fig. 8–29 do a somersault without having 3er,twak c 50s i;ok:i zh-hf 28sep+iany initial rotation when she leaves ke5c3iht8pis2 k0,-sf iaw r:z;+o hethe board?

The moment of inertia of a rotating solidov2l am34meqf *x pm:( disk about an axis through its center of mass is4x:*vo32 m(q apmlem f $\frac{1}{2}WR^2$ (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 rotvkas,nzh*tug92r o -6hq.nw +ating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your ughr kq.z 69hwo,* v+s-ta nn2angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the s/66wx 5p*p1h 9 pzhwedgis xl9ame mass. However, one is hollow and the other is solid. Describe *6 9 px1psxdhpz5ge69 w/ihlw an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points 5oc 8d*ppol 5iwest. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east,d *i5oo cp85pl 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 own/xf :e;kjg )f a large freely rotating turntable. What happens if you walk tojke):;fwnx/ g ward the center?

A shortstop may leap into the air to catch a ball and 2b 4fn.l0pgicy:; 3bxczq ls- throw it quickly. As he throws the ball, the upper part of his body rotates. If you0 ;n c2:b4lxzg.i3pyfc b-qls 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 of angular mo2/k,sg xkwq 2amentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or,q2ks w/ 2kagx more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (4yc4kv6j3z qfs3 1+4ugh ekq na) 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. Calculate, using 8,zeuqzb8tvu s2rme 60/ol6sthe information inside the Front Co,60b/s zvleutu86 m e82 qszrover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beam d:bz4tu +h k;wh ypwi16iverges at an angpty +wu1h4; z6bkh i:wle $\theta$ (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. W )zrs.abnw )2i(7cj8b-(mbdkn r uzi5 hen the motor is turned off during operation, the blades slow to rest in 3.0 s. What is th)b2bzr58 -7r( nnzwkj(ib a .c)sdmuie angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another:rxpoxri ,3tzvj)2 jtz:ha 4t,.x7 nm child. If the ball makes 15.0 revolutions, w 4,txj r xozai3znmp27v:xr.jt,: th )hat is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutionsm+z* g::lyxjrq 17wkweg,ws * 9+trjx do the wheextsgxy**r: ,meg9w+ q:lj+w wj z7 kr1ls make?    $rev$

  (a) A grinding wheelp: if-0jtxx- p* hr 3ve2y;tni 0.35 m in diameter rotates at 2500 rpm. Calculate its angular velocityhx;0n ev:- yip2ixjtp3*tr-f 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 revolutv:- m/p rep,jpm dabq)- ;hv:jion in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2;d/v:pjmr ) jv:aq- mepbp-,h 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 orbimt6q; whl1:oh dv )h3gt around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speedg)5of) bm t*xz 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) must a centrifuge rotatjgg b0d6 )-pn)oo;szb2 qc/wue if a particle 7.0 cm from the axis of rotation is to experience an acceleration of 10;jnz)-p/gb c0oguqb6 )ds o2w0,000 $g’s$?    $rpm$

  A 70-cm-diameter whee;oq-g-5(sgnlt 3u ggcl accelerates uniformly about its center from 130 rpm to 280 rpm in 4.0 s. Determigu3 ognc -qt( gs-gl;5ne
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radius * +g re5.hienw$R_1$ 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 Apolhz.i z xm0z1blc jk,1ba756qolo 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 ofh,a z z 17m1ql.x6b5iz0jbcok 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 uniformlyq /j i n/oz6m.b7pm6efr/0 gmg from rest to 15,000 rpm in 220 s. Through how many revolutions did it turn in this tigjmpq o/i06/ gmbm.rf z/e76nme?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. n;u6w/xdwy r y i).a-:szfgecmo l23cmz/+c- 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)2r rb5li s3mn in a whirling “human centrifuge,” which takes 1.0 min to turn thromli nr )532bsrugh 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 diametioza jl 6o)576pp7m gker accelerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have travelejog6 om)z75kppl6 ia7d in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turadfp0 w)y8e kj/ :w4onns 1500 revolutions before it comes to0ky 4:oe8na w/p j)dfw 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 revoo 01i,ctl:i 5uzldqp 6lutions as the car reduces its speed uniformly from 9 1 ol5q60itl,cdp:izu5km/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

  The tires of a car make 65 revolutions as tdl gc-bj8ntg*v6e6x9 he car reduces its speed uniformly from 95km/h to 686 9gjvbl-egtndx c* 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 puts all her weight on each pedal when3rcy hb1 r.l(-o s pbq ,cr)nu6.uklv+ climbing a hill. The pedals rot-l b 1lp3cvbr )+u,shck r6.yquno r.(ate 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 55 N on the end of p-r slq k0qh2:a door 74 cm wide. What is the magnitude of the torque if the forh:ql 2srk-q0pce 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 twne mq:w8b (9the axle of the wheel shown in Fig. 8–39. Assume that a fnw tw9(8qb :emriction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a massless rod which pivmllz;( 3s8-az;u g,ivfx2fxqots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then re8imzl ( vg;sluxa- ;qz,f fx32leased. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cyli7ns j q1bgzt/1d;h js4)ql)swnder head of an engine require tightening to a torque of 38bzq qg;nh s/j l4)1stj )17wds $m \cdot N$ 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-k 50 faok;9;ntsuxhl7d g sphere of radius 0.648 m when the axis of rotation is;5 h79lk0n;stdxa u of through its center.    $kg \cdot m^2$

  Calculate the moment ofc31. qiey h3h):ann*jr vwa t9 inertia 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 can be ignored (why?).nt1 ah3envwi3 yah cjr): *q.9    $kg \cdot m^2$

  A small 650-gram ball on the l( jnll52bn p 2l.rcfja2+c y*end of a thin, light rod is rotated in a horizontal circle of rarj n2blf5nlll+2 cpj(a* .yc2dius 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 ytp:0vd on 1 i5kviux+z12*j d+bvgko u:/b*nbowl on a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and: bu1voyo1u2j td5k*:*x+ pnvgi+0v ik/ dbnz 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 objectpyb.-te xe 5,t en/gs*s shown in Fig. 8–43e * nx yt,-tebg.ps5e/ about
(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 consis x/)febc)1 wjs9i6enyts of two oxygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the 4f7 )b -eocpfasf/.l c82 zxkgcorrect 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 satell2 4- fx cfgoe/.z7a)c f8slbkpite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform wtnu9j. ;a4qwis :p:q cylinder with a radius of 8.50 cm and a mass of 0.580n t;qpa:i4wj9: qsu.w kg. 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 bat, a-it4 9mqhzq;n313*dt6c ad hd )qs sjsccelerating 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-dr6czzv(qtqf6e,n7+f q3 r h, jyiven 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 ofy6zq ,qtf ( qer73f+6njch v,z 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 b r0:xbqm/+t zis shut off and is eventually brought uniformly 0b bq:r mtx/z+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 3.6-kg bam.ojjr y ++z0pll 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 tkb i0v2mqt jo :t.f.jt bv8am39pk99yhe action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from f23q.9 tja.ib8tm: p0kv j99tb ymokvrest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod0q5u1fi kn4kazo8br9 , as shown in Fig. 8–4q 8k1uzoi5nb 0kra f496.
(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 cos 8f)( xq1jey(x wm:zynsists of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accf d6cyalw+.g.gs78x helerates the hammer from rest within four full turns (revolutions) and rele6ad +s8hx7 gfy.gwc l.ases 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 moment of *4lwog0nmn). :(d6ks 0 q cqckh-cvguinertia 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) 2uxtr/bw j/hrd)q7q torque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mas,eiy)b*1l a qls 7.3 kg and radius 9.0 cm rolls without slipping down a lane al*be q)i1la y,t 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as tuzw82mvny d u)ek9x +3he sum of tw)u +zev2nm9x uw8kd 3yo 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 of 7.50 m. How much net wo vo/n q,et;ja.rk is required to accelerate it from rest to a rotation ratev; jqt.,aon /e of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rezx *z;.p0-1ysa7p j,draocp jst and rolls without slipping d0s, j-pazoxpa1prz* jdy 7.;cown 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 vertically on its tip. Itq rrgh arytj 4q-3 6l:; 3wc-iq8x5zzh starts to fall and its lower end does not slip. What will be the speed of the upper end of the lyci-a; xr w jq8t3zr6q3 hh4r5g-:zqpole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular moml(js/(n aic-c entum of a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an angular spe(jaic/n sl-(ced of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momgb 0/ o(m8q 3pa(gr0r kjma)ozentum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotating at 1500 r)g(qozomj8g/m arakp(b 3r 00pm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that ik;6p88dl7pc qz4 gohrs rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed ozqrp88ko7;l dc64hg pf 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 momee * xye fqak,ia97q-suq:l):ent of inertia by a factor of about 3.5 when changing from the straight position to the tuck qekx) alq:q :s7efe ,u9-ia*y 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 rat6d7 0no,olt)r-vce ggyi 8; dme of 1.0 rev 8cirg6d;o, y7vg n-)d et0lmo every 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 ar0+j z i(f-t,ly/lqdgn (5peqiound a vertical axis through its center at a frequency of ji-gz 5l(n0t ipdfey(+q,l/q1.5rev/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 the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spin.61uk grlj36xb,5gskw9 dvksnsuv:0e um3i)ning 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 tkvdu)e+bkavl0gl*:l-r , d: of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is dropped onto an i)ebvhcg9m /;bqn4* 6ey ez;a fdentical disk roe ca nfqb4;e)e6vz9 *h b;/ygmtating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns azmdi8 ,pq w0+nt 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 at3 - octh1bnw rw5lb3/2oaqmh* the center of a rotating merry-go-round platform of radius 3.0 m and moment of inertia 921b3/*ba2onho5 c whw-q lt3mr0 $kg \cdot m^2$ 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(k2rqv q kn5qyo,c97.s9d fn t-round is rotating freely with an angular velocity of 0.9q7 kr.9,2vnot y5fqqc sdk(n 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 *-k 6wz5kmxfna6 thx6dwarf, losing about half its mass in the process, and winding x6xtkn6*mah56zwk f-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?(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 s)w g6lu2 cbn3kv 5t+yof 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 abzrt5,4dixl7j *43o aonu( i$ 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 freely6att)* kko28/n 9ign)v tl wxx without friction. The moment of inertia of k i)*n/)6tk tgt xla9x2vn8ow 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- 3g30obor0ci7bofm j9m diameter merry-go-round turntable that is mounted bo rc7g03b 9oi3m0jfoon 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 groc rr2veu3ir,r os8wb,1:mr 42*n oxrf und with the end of the rope lying on the top edge of the spool. A px,,r2rrbursi1m*fo8vo cre42 3: nr werson 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 faces the Earth. Deth5vhg( ygn7 n;gz* srljr4e 4*ermine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting;4*gs (lrnzghy *ng5 jr vhe47 the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at l:mv2 9l7hr p:*+kes(yohn xta 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 shape of a uniform cg-q znmbt ;/b0ylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constg;n bzbqmt 0/-ant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 13n m6xrx.ntz8-7bx2 ; 2 )yooam cr9fsyuvml0.050 kg and diameter 0.075 m, joined by aoa- ym8rv92cxx obnsmlrutz;)f1 7x3m6 y2n. (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.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how i1.eo2 k j sg)e)n aml75ydv6rss 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, bux4eikj* *.re at with mass 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 would its rotation speed be? Assume that the star is a uniform sphere at alke ix*jrea. 4*l 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 en2sep0qo 8v0v eergy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diametevp2v 0q8eeo0s r 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustra *gb oy)+0wj yxy,zdaww+) my/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 hor tdg *11rsci5 5laf)b7:fcvis3b; rxq izontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.*(kj9adb hiy /.5nx spe., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally annibxd j9*s5(ya p./ khd 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 standing vertically on the floor, and we o3e5 6 deqokt-want to exert a horizontal force F at its axle so that it will climb a- e65oo3dq kte step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.5,ccb-npt*ud;spm.m 2m/s on a flat road is making a turn with a radius The forces acting on the cyclist and cycle are b,-;.tc cdpmun p*ms5the 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.w ixu9bk;0/dx: pn:ve y9.wfa 5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and wxide: n 9v9 upw/fy:.abkx; w0here does the torque come from?    $m \cdot N$

  Model a figure skater’s body as+hz8pqx;hui9lnj wn a2 xg 4;4 a solid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against4liw8n u;x a92;xqn4p zhjh+g her body.   

  You are designing a clutch assembly which consists of twb 6ax3v hof.l1o cylindrical plates, of masb6. o vlha1xf3s $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the looped rough track of Fibfkl3v, 5ew s9g. 8–58. What is the minimum va ,5eflvs9bk3 wlue of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assc 9grh;ds (cbylk5es6zb i283.,fa mnume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduceyd)l +) mrsaod)k/7bs s its speed uniformly from 90km/h to 60km/h The tires have a diameter o sm)l7/sdo +rb)d)akyf 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