A bicycle odometer (which measures llez ou74-bsti /*3gv distance traveled) is attached near the wheel hub and is designed fs *7vl-e3oz4/li tgub or 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point ono* no-a3s/;( uxtxynq the rim have radial and/or tangential acceleration? If the disk’s angxxo -3ysnuo;( *t nq/aular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single valuc+5qwj6 )ezd62l ibove of the angular velocity $\omega$ Explain.

Can a small force ever exdjlgk50j+ + ubrz-e*ndbn -5xert 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 wi -n0zv5*b9hoq na9cikth your hands behind your head than when your arms are stretqvh* i nbkn0zao59c- 9ched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has 3 , 4thysihk -fof gfh/;s:g7hseven sprockets at the rear wheel and three at the pedal cranks. In which gear is it harhf4s3-hf t ;7g/hsh k:igo,yfder 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?

Mammals that depend on being able 20z,5 qi k ctqwim x-(/ig,nmbto run fast have slender lower legs with flesh and muscle concentrated high, ct(0 b, x,-iig/mmznwq 5qik2close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, naci 9kh1)n +2 evk-vgy94h dyjfrrow beam?

If the net force on a syskdireax +s19b2blgx d1 ,vsk(n(4 sd/tem is zero, is the net torque also zero? If the net torque on a system is zero, is the net force zero? e4vrl akd1xnd2s/b1+ 9 bd( six(sgk,

Two inclines have the same height but make different a-8v msn28x6fp.v ye/nn7 gooej a-z r9ngles with the horizontal. The same steel ball is rolled down each inyp.a2n- o f r96e7-g8ve mxv/onsnz8jcline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) dolq-zc * )lb;jywn an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which*z )cby-llqj ; has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylindq59r+if0uc 4kcgh8 bd er have the same radius and the same mass. They start from rest at the top of an incline. Which reaches theq bdfik 9gc5+840hc ur bottom 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 momentum are conserved. Yet most yq e:pix()j.olm. ued, qr, )dmoving or rotating objects eventu..,yimx ))ru:e dqpedj,( q loally slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equator8gax4 +vp 8gf95an jk8)oxhxgjovd24 , how would this affect the length nv g hd9 j+pxagk42o8jfxax8g5)8 vo4 of the day?

Can the diver of Fig. 8–oy v 7yxl9x25h29 do a somersault without having any initial rotation whexoy9 hvx5ly 27n she leaves the board?

The moment of inertia of a rotating solid disk about an axis throw6;nd7 teuui f6m/x:cugh its center of mx6 n6 eiuu:/d;7ftc wmass 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 nsuv z2 /-xpg.a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will yous.gn - 2u/vxpzr angular velocity increase, decrease, or stay the same? Explain.

Two spheres look idenhz r:y c*6wx.0pz1rdu.h o7g0pp., brnk4lkz tical and have the same mass. However, one is hollow and the other is solid. Describe an exper0,w.4h:b l*zkrxp zp 61.70.nuychod pr kzrgiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle -z+cv*mst 6ev points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decrzsv6e*t - vcm+easing?

Suppose you are standing on the edge of a l,8ca2gj,wg)hz 9 qbomd s)ci2arge freely rotating turntable. What happens if you walk toward the centea2zw c,)hisg82mog,dc9j b)q r?

A shortstop may leap into the air to catch a bal 8;eb nw80q 9zusob/avjx2x-ex,t2iml and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips andquxeiw82,j xbob9 0tevz8n2/;xsa m- legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discuss n 3n:bjayyo35 why a helicopter must have more than one rotor (or propeller). Discuss one or more ways y 3an: oynb3j5the second propeller can operate to keep the helicopter stable.

  Express the following angles in radian8uwu6;ddee / vw-z8qwx i5 6aps: (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. Calculathg1s t4v;0ceo wfo3w0 e, using the information inside the Front Cover, the angular diameters (in rvgh0 13to;se4 fow cw0adians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed p5;o6qxdy-+ h/sv dz wat the Moon, 380,000 km from Earth. The beam diverges at an an5dxy pwzh+ -d /ovsq6;gle $\theta$ (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 ratetpv4pioookg z;pzi2p/z k );;;phy + 1 of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular accelerat/zvi12;poz p pt+go;p o)hy i;;pzk4kion as the blades slow down?    $rad/s^2$

  A child rolls a ball at1s//dqhx: d on a level floor 3.5 m to another child. If the ball makes 15.0 revolutions, what is its diameter?sat /d:h x1qd/    m

  A bicycle with tires 68 cm in diameter travels 8.0 kmv-;y qm ;; htrm ghob.9xavh)(. How many revolutions do to;m.vh;)9 ;rhbtm xqy-vha(g he wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Ctrygu6m 8)i6)f9yoyjll-i ;malculate its angular veloi )jium6r8yy fm)6;lygt -ol9city 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 revol hflc6.u(hus (ution in 4.0 s (Fig. 8–38). (a) What 6uc(fh uhs. l(is the linear speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angula+ z6 w+rnfil;+ qzhd b1hzw47*yyvf2er velocity of the Earth (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 a poinwhq* 9 wgq :7d;4nptyit
(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.s7b6e3 chsj2 o2zv y2s0 cm from the axis of rotatio7shojeys23 b2vz 62csn is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about ih zlij3,zdi,/ ts center from 130 rpm to 280 j3iz,,/ hi dzlrpm 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 radius77 5)2hc3xmh mouefft $R_1$ 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 or0llq5 z*-u84bwg9,z mufqzput 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 1gfw* q o0 8mrub 5zz4-9ql,luz2-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpm in 220 s. Through howvmc;372if wq 6hl+hbld-q3 u r many revolutions did it tu+u2qc fq;ldv3rbhi-6l w 7h3 mrn in this time?    $rev$

  An automobile engine slows do1crgd3)b z; )od0 dtixdc7s ,ewn 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 g-vp. ofl5,0xhkx am,of flying highspeed jets in a whirling “human centrifu,x0o,pf .mv- xg5lhkage,” 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 accelerax go7ajss 4-vl*)/namlk2b p 2tes uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled in thio 7n2vsp gk j*/-aa x4ll)2mbss time?    m

  A cooling fan is turned off when it is runnin oygm26 atctb):4 2difg at 850rev/min It turns 1500 revolut)tt4:2g2a cmyodbf6i ions 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 speed uniformly vlnv-1n) rcg w yzq*3mbzn,s8aku8;- from 95km/h to 45km bnkmzlsnz8-;aw r v)8qg y,1 n-3ucv*/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 2va p ;0,czbjlthe car reduces its speed uniformly from 95km/h to 45km/h Tbzjp l;a,2 cv0he 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 bikejy44i*jb- wpg puts all her weight on each pedal when climbing a hill. The pedals y-g j* bw4ipj4rotate 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 a door 74 cm wid8n7bl z9x ;ycoty)gw8r:a9mze. What is the magnitude of the z78o;l: b y9ct9xngm rz8y aw)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 shown5xf, kx8hey1sn 7z zae7n7z )jc;mw .p in Fig. 8–39. Assume that a friction torque of 0.4 e7s7fjpx;5) ,cmzzk. 87 hxnw1eazyn $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of p u,z4-t yfk+edzi1 6jmass m, are attached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rodjfy 4d- 6uk z,pz1+ite is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening to k u42 jjl-ia7ya torque of 38 -j7uya4 2i jlk$m \cdot N$ 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 1lce -,n a:k 6wc6o.3qso9 spgh0.8-kg sphere of radius 0.648 m when the axis o 3cegqhca: ,wps.k6sl ono 69-f rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 6iye4 i(fh(z 1vmk 2 1synab65l6.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The massah21 (4kv6il bf5me( yn y1szi of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin,,n0h1mf7q jrd light rod is rotated in a horizontal circledh n7qf,01rjm of radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowi/nwtq6lsg)gf ( bwb*r11 ,lol on a potter’s wheel rotating at constant angular speed (Fig. 8w1on)bwt*g(q l1rbl,i g 6s /f–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 o2gi-(jmfn0cq; g yab kk, )ux,f inertia of the array of point objects shown in Fig. 8–43 acu ig, jb;g ,-kaq) 0kmfxy2n(bout
(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 t(q k6xcbuh5v3cbmdd7b(g*t9, h k3ctwo 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 the8uwxgg v*8b;b 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 t*b gu8v; bw8xghe satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform culmxa7k)l 8uzs ;jk x9; p*ts7ylinder with a radius of 8.50 cm and a mass of 0.5807)lxs u;;lpk97*8a szmjtxku 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, accelerating it from restx0dr(viiq(r45oy9 3 pjar 7sx 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 tangx7jb- djnbgbo(g4-6 qentially 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 o(bg4x7jbdnq6- g- jba 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,u / w9lxy0an9i;p1zsz300 rpm is shut off and is eventually brought unifor1 yuws9azlz/ pn;90ixmly 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 ball3i3rqmto *q w7 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 *o4m:h xngl+kball is thrown solely by the action of the forearm, which rotates about the elbow jogh+*4x lko:nm int under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade canit,h)8 judpkjm ut8ab:0( ,b o be considered a long thin rod, as shown in Fig. 8–4tmb88:jjui,)0 ptdh bo(u a, k6.
(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 m.jb(0ph6 lw dg.oet.e asses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest within f+ mh1gjy9 st zn3g8pds dv)o+-our full turns (revolutions) and r+vdsh +m9to8z 1)d3- pj gngsyeleases 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 inertia + .o6nr jsy2gp w5v)qaof $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine : jkmk 4b*s7m1zl3tx 4vq2 rlufs.g2odevelops 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 mass 7.3 kg and radius 9.0 cm rpi7 z1ktn,*b/ w9 oakk /qwhz1olls without slipping down a lane az 1//wba,iwtqzo7kp n9k *h1kt 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth w9a*gws3,zz f jkij+js.ze2o0 ith respect to the Sun as the sum of two ter k30.si*9, zjgaw2 z jj+esofzms,
(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 hl ;y0a4azu5(srf t 8ga radius of 7.50 m. How much net work is required gy;0a5 8az(4t rfh uslto accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest atdil4 iu()y76bn l5 pvnd rolls without slipping down a 35b i(7yp 6)unlt 4vdli0.0 $^{\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. It stan ey1r7 t3hkc.qptlkh5/c508n4 iw lfc0lr8n rts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before itr5nhr0 48lcfinlk5 ncq3/1c.p7 hl0et tk8 wy hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on .78o epp5.xwrz, ;b/xr zser ithe end of a thin string in a circle of radoizw.br7, 58r/zse prep; .xxius 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 cylindri6ob a,,x5ip( iqbk fncu;: a-ecal grinding wheel of radius 18k,p; baf:uqbeoi(i6n- 5 xa, c cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotating at a j*vw.x ro/tz;bv1uh8 rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the /; *.torvwxz1 jbvu8hspeed 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 soarxl* (gwrcs a 4)4-shown in Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the strai4 )r-goswa*xrc s a4(lght 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 6(x(es6 vpf; zcs c6gnincrease her spin rotation rate from an initial rate of 1.0 rev every 2.0 s to a final rat v(nc (z es6xcs;g6f6pe 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 at1nt/vz 8 ,wd.ap2aww vertical axis 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 rotating wheel. What is the frequency of the wheel after the clay st na/,ptzwvw2 8d .at1wicks to it?    $rev/s$

  (a) What is the angular momentum of a figure ;lxtw/dr(sr 7;irkq3 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 on e,k 7r-rpvb y+7-q y)a8vgpqf 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 ide8i5/j6a2yc xt;1 fmal n h1iorntical disk ro m;n i6l5aojatr8ih yx112f/ctating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at k o)1uf 2 tm,z-w.i vj4y5dhxu2.4 $rev/s$ 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 stt --ocy1xa-g. 6bwf qxands at the center of a rotating merry-go-round platform of radius 3.0 m and moment of inertia 9.-xtxo6 1c --gwbaq yf20 $kg \cdot m^2$ 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-rounnc2 kh2exkc *mc1:n3 vd is rotating freely with an angular velocity of 0knnkcec hv3 m22 cx:1*.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 collt 7t*:ko suqe*apses into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new r utqo: sk7t**eotation 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 winm3us 5p1 x:tewds 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 p/) k/esw r+i x9mlvd )z6xi7v$ 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, initikt opjal.j 8,d71s mf,ally at rest, that can rotate freely without friction. The moment of inertia of the person plus the platfojfpa,. stdj 1o7,km l8rm 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 th qm u52rgg)ic+e edge of a 6.5-m diameter merry-go-round turntable 2m)c+ggq u i5rthat is mounted on frictionless bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground wit+jeruykdtj o(i ))g i ui48;kyq7.6gyh the end of the rope lying on the top edge of;oj)7tq gid j i(g +uekyk u)i64y8yr. the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person 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 ,j8/uhhl:y7n *rpn rdn eu) a)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, treat the Moon as a particle orbiting the En7d)j ,yu* pen/)al8 hnhr ru:arth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a ratw46gm hpyd5: t slk8tqb/2 :6ij9pjzae 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(esa/ a5xs1ua;( tjac of a uniform cylinder of radius 0.20 m acquires a rotational rate of ausa (5sja(ae x ;t1/c from rest over a 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,bq) y+ytd3 vy 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 rea,+)3yt yby dqvches 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 is the angular speed of the r 4:7fwgupy sdzq -bj:+ear 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 withg+ p ih*hwuvhnpx-;(5 mass 8.0 times as great, were rotating at a speed of 1.0 revx+h pign ;h-whv(u*p5olution 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 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 usnv1j zrf meym+f8 :g72e energy stored in a heavy rotating flywheel. Suppose such a car has a total mass m vrze7gf1:jn8fy m2+ 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 an ;hayw :(5)ep3;rqshztcr aj++o+xrw$H_2O$ 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 rzij )vu,ip1/xn s+ ho:olling 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 masm:j:-g(4 b0a p1 sz+atkscaads M and length 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 ajc z 4ta1+ b(0mpsgs::-adkacenter of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically os c3pzmb64dn +qlznppegd7, 3gc z6,dl;(c a1n the floor, and we want to exert a horizontal force F at its axle so that it will climne 3lq,p,bd+ pz6cc g3zm6d(7;s p4agl 1dcznb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist travelingiudq;5 h u 6qvei5vp40 with speed v=4.2m/s on a flat road is making a turn with a radiuipu;v0hudv5 q 5iq e46s The forces acting on the cyclist and 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 whi)ub 26ry;to(wxi3d8 hled3xx rling 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 theyhd6x (di33;r l8w)eou bx2x t torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid6ywrcq.s 3.es3e,anb cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spisbyce3q, .ase36rn w .nning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consistpsi3d- qn 40.r1aayym s of two cylindrical plates, ofs1ni3ryy pd0 4qam a-. mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r f;ud(ghi b ;a*rolls along the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the * baf(d;;uhgi 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 assume9f(:ydh3dm-jnu yk k+ $r\ll R$ h =    (R-r)

  The tires of a car makly+1 sn9kyie 2pn- :gae 85 revolutions as the car reduces its speed uniformly from 90km/h to 60km/h The9-+s:2 1nayigpykn l e 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