A bicycle odometer (which measures distance traveled) is attached near the+x o-.fq8* jbqb)jdyp wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicpybj* j) q8o dfb-+xq.ycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a pom-y56ash c;fr ; )pnnaint 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 componehy5n)nm;fa - r6;scapnt of linear acceleration change?

Could a nonrigid body be described by a single value of the angular velolscar w,sj:iwb qng/ o94y3)8city $\omega$ Explain.

Can a small force ever exert a greater torque than a/x 6au abz0aq.0 5psun larger force? Explain.

If a force $\vec{F}$ acts on an object such that its lever arm is zero, does it have any effect on the object’s motion? Explain.

Why is it more difficult to do a sit-up with your hands ba8xucpy:+k :c:bg nh3 ehind your head than when your arms are stretched out in front of you? A diagyc+uhp k3n c:x8g::baram may help you to answer this.

A 21-speed bicycle has seven sprock7b3xzs a;u n* wxjs((nets at the rear wheel 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 bx;w(x7s n(nz* a3sju front sprocket? Why?

Mammals that depend on being able to run fast have slender lower legs with f1o91jpxah vh nu +w dvj3t 68/5pi+zxylesh and muscle concentrated high, close t +j5dv1n 9xxptj io8ph1w3y6hauvz/ +o 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 4 yxq9w(4mt2 mgi;ctba long, narrow beam?

If the net force on a system is zero, is the net torque +yo w)/ cur9c +jsx156s +l9frzzbwklalso zero? If the net torque on a system is zesu+fsy1/w+blz +k6cxcwr5jr )o l9z9ro, is the net force zero?

Two inclines have the same height but make different angles with the hbw80- 1zs j aq4uhxb (7i:rcrgorizontal. The szrx4sjh :i-bubq0(wg7 car 1 8ame steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres sib7hqn 8-umtk8 multaneously start rolling (from rest) down an incline. One sphere has twice the radius and twickqh b8m8tn-u7 e the 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 sq1, 9j bzxpv*nxd*h2a ame 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 at the bottom? Which has the greater total kinetic energy at thehbq2n,jax*9 1xdzp *v bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are co:i 6a8u/p p:iv4ge wejnserved. Yet most moving or rotating objects eventuu6 /:e:ie ipwjv8 4pgaally slow down and stop. Explain.

If there were a great migration of people toward tzrwbp:22sg9 h he Earth’s equator, how would this affect the length o2pbg swrz9: 2hf the day?

Can the diver of Fig. 8–29 do a somersault without k,uv i:6akaa:h-jrs 7having any initial rotation when she leaves thej,hkvak:- 7a a: urs6i board?

The moment of inertia of a rotating -r n.m6duds a:mvd;+9zs s ma,solid disk about an axis through its center of msmrmd9m,n6d.- as:v ;+ asd zuass 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 6o ynt+7c6ho;g)(c vyq(lv o wrotating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the mass nhyv o; )o67((o6yw qt+lcvcges, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identicea +1y(+di8fs vz1tq +wi uem5al and have the same mass. However, one is hollow and the otheriy+z11 qa t5ue8fw+ e(vdmi s+ is solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points west.0f9,yb 2 tvfrv7of ih+ In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, ryffv7 fi,v o029+bthdescribe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing h+tc 6 e7tci;tcyx1,v/zwtv0 on the edge of a large freely rotating turntable. What happensvchc+wt10vt y zt/t7; e6,xc i if you walk toward the center?

A shortstop may leap into the air to catch a ball and throw it quicklzm, v6c0syrp7y. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotat76v,rs ypc0zme in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, dh czxt++sg;; eiscuss why a helicopter must have m;+t+hze g ;csxore than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (a) uq-z,a p :c k/i.gyzh cmg;gwj u7;)-l30 $^{\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 Ef mv2t7*:kjr barth because of an amazing coincidence. Calculate, using the information inside the Front Cover, the bk :7vj m2tf*rangular 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 wm,fu5d7;) ccbo+sx p Earth. The beam diverges at an ao;bsd wpcf5m)+c u7,xngle $\theta$ (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 650j9mhg i9r2p/d1eq* cr0 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is d iqr/19mrp*hejc92g the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor0totvpaoua8-zt-lsvu;*. vs 8d4.yq itc c 9 , 3.5 m to another child. If the ball makes 15.0 revolutions, what iauaz*-st89do , puo8svcltc;.q-0vvyit 4t. s its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revp:1/v; yimsdolx( x9tolutions do the wheels mak oxmsx1t;d / 9li:(yvpe?    $rev$

  (a) A grinding wheel 0.35 mlvaf 0yke,n nd ,-tq7j61g0jog .d.ng in diameter rotates at 2500 rpm. Calculate its angular v j, g0 tdye7vqgn6.d. 1o-,ka njlngf0elocity 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 4hi ik 3,oh-ax8 g0kehcyxg9.*.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 39h-ggoic.8 k, x*i hahke0yx 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 /a tqtar5nni ,om1v/-b0yhv/ sn-2 w, nd*c kdits orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of vruk;ow61e o )l* oj6ja 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 rotate if a particle 7.0 cm from the axq1hzm : (v7nrpis of rotation is to experience an acceleration of 100,000q:n vh7m1pr (z $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its cb,3y e, ,d u)pm(trebventer from 130 rpm to 280 rpm in 4.0 s. De3e ubedp( v)y,,r,mbttermine
(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 radiuspelsoe( oe)+2 ;)rfezb;l k1z $R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboard the Apollo spacecraft put the6tvvvaa7r w 4.mselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they4 6aav7.r vwtv 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
(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.yrh37*h2juncr nc)6 a Through how many revolutic) 7rnr6 hy2n*j3uhacons did it turn in this time?    $rev$

  An automobile engine sloa*z:t6h , /0yomjnv 1p4ejtwfws down from 4500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeed jets in a whirling v+e cwo4oe9nyc7,st k94+; mr vu2 rvi“human centrifuge,” which takes 1.0 min to turn through 20 complete revoluto 9 7+ec uv,rnvk vwcito4+; mrs4e92yions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 2 b ;/5euy7 mpavupjv3940 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel havu 5uja9/y pbvv ;7e3pme traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 15nk*7vd (s9hoaat,k 5 u00 revolutions before it comes to a7k9adhnv ao(*kst , 5u 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 un+. h/v ps o5ixv rutkaz1n4.o0iformly from 95km/h to 45km/h The tires have a d r5k1o/ san.z 0t.pv4i hvx+uoiameter 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 speed 0otyzd) k60elaf2u /uuniformly from 95km/h to 45km/h The tires have a d)6e yak 2fzlo/uut00 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 when climbin9 y9:,wf1epviyj ;prig a hill. The pedals rotate in a circle of raf9ei v p,jwyriy1p: 9;dius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a forcc tg y7/eauux7pfy, ,)ofe,pl*/ gf( ve of 55 N on the end of a door 74 cm wide. What is the magnitude of the torque if the force is eufo/ yy/tvag,pg7)c *ux, e,f7fe(pl xerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fig.mnmlc2,++ df 2gfes h5 8–39. Assume that a frictgms hc2e+ fn5dl,+2f mion torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, ar-lvxh*.6o fgre attached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal pos-vrh6 lxf*o g.ition 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 tightd4mh5.j 3yjbsb l,n8dening to a torque of 38byh85j 3l 4 db.nsdmj, $m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of radium2 7b lch1,vn , flk.tey44(xr etxj(ls 0.648 m when the axis of rotation is through its 7 ,41j.xb 2t l(e elyxm,flvrc (nkth4center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel hdik+ 5f+x*qvygybi k+e564ux h*6yj66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the fx i byq 6 +4k65v+*y*ed5khixuh+jyg hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated ij+c6s o.scpq 1zy69 vpn a horizontal circle of radius 1.2 m. Calcu ps9c.o6zysjc1 qv6+p late
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a;pk++gtkd. m5dq ;u4vxvn;ze potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her h+zn5qgd ;u4;x.v v; pek kdmt+ands 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 ine q//mw(rxez:t yai c(/rtia of the array of point objects shown in Fig. 8–43 aboui:qw(a/my e/(/cxrzt t
(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 1/4 ggmgw5qr l/empd4atoms 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 spinn/t45 xhbn 6piying 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, whpn6bt4 /xyh i5at is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.5z4b podu- awb*5/e4dh0 cm and a mass of 0.580 kgdh*/ u 45pe-abzw 4obd. 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 swin,vr2yz .xmt, 2 ;o zojzm/tj1jgs a bat, accelerating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tanbnq u-gb(nm1 :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-uq :bn1m(ngb has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotatingjl ;:8qudpf3z at 10,300 rpm is shut off and is eventually brought uniformly to reqfu 8z;ld3 p:jst 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. 8–4fi t8u sp8s,m*s1rd+x 5 accelerates a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the action of thihb9d 8cc vq36wag6f w:8ypcqzle13aae 00w(e forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8ye6i 3v p: w0dq3w 9ac0(gecl f8c 8az6h1wbaq–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be consmuj5g;o1 u t*c0-dlvtidered a long thin rod, as shown in Fig. 8–4m jgv*1uod50lut t ;c-6.
(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)wf (e5uycxufp-/ d4i 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 accelerates the hammer from rest within four full zq7-if.3f3gwzd(e s*+cjh kz8 tz ,ngvf 69bfturns (revolutions) and re3+8nt*kb,fzff zigdj 6g.vzw 3-7(z scqeh9fleases 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 ha8a2ob bmy51i8 u qjt,ns a moment of inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine devguxt(no73* kbelops 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 rolls with:sygtdm;a (( tout slipping down a lane at 3.3m s(y(dt t:ga; $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the v5ps 4fqq-vud,i4-d3af zu- hSun as the sum of two termf 3ph,faq -uzd4vv-45 diu-q ss,
(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 s:00p,+f zifhiu,xgk net work is required to accelerate it from rest to a rotation rate of 1.00 revolupf sfkhi0i,+u x0zg ,:tion per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0ugf:z *blp3 )8axi+rw cm and mass 1.80 kg starts from rest and rolls without slipping down a fz:bwx+*ur) 3gilp 8a30.0 $^{\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 tip0oj)k x2)hapg sej+ i*. It starts to fall and its lower end does not slip 2)jh xpgike) +s0*ajo. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum 1on30ne4d oqd2 /0u fogsnsn/of a 0.210-kg ball rotating on the end of a thin string in a cinseo n 3s 1g4o/fo0dqund/02nrcle 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 cylindricalj*jb/,ildz2c(ez(s v grinding wheel of radius 18 cm when r2sjbile zv(/ dz *cj(,otating 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 .zzvc- ++nxl2nw.i 48mq f;dltv lhj+that is rotating at a rate of 1.3rev/s If he raises his arms to a horiz .l n. ft+8-zmvl+z qi+wj4;2xlv hcndontal 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 inertid2g(zio -.g coa by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular .do(i-co z2ggspeed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotrcjnqah e2k18 gh87g5 ation rate from an initial rate of 1.0 rev every 2.0 s to agc er k8aqh15jgn82h7 final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis thro:nglp 1u-d ig0ugh its center at a frequency of 1.5rev/s The wheel can be consideredu:0- gligd1np 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 spinning at 3.5;uy r8.ff6(r+ g bvmlf $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum of the Eartw3dr o2o3e6m rh
(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 +8 t gk-wj;vryof inertia I is dropped onto an identical disk rotating at angular speedv8k-y+tr; w jg $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4a1km4u51o,x5fu3j c a0s qqty $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotating merry-go-round platfornd,5sl aqqe yd5w8j* 3m of radius 3.0qw 5j q,d ye58lns3*ad 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 rotad074rro31s f5/ zppitn0n2 -wduoumyting freely with an angular velocity of 0.8f70 /mzyritn dp05o2 pd1wus34r- uon $rad/s$ 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 collapsesu;cy12m7qga )+vxwfo/ bx,w tjg 3)::dsejv g 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 rotation rate be?(round to the;g ,sv2xco mjtx 3vq wb1uayf) / 7j+:d:g)ewg 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 errp ,1uf;eth+ xcess 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 yd01 mo d4i u01vwkl5x$ 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,1.trf o5sxn7b: p uc5i that can rotate freely without friction. The moment of inertia of the person plus the platfo.fs i ux:5n b7o5rt1cprm is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diameter merry-go-rw v2fq, 2wa(wround turntable that is mou ra2f2w,vwq( wnted on frictionless bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of the rope lying on the m 5lqo6y nnn2q8;0.nf3 z(fzj zi7vdttop edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls bylfqn.jd tnn83 (z720oz qiz65 fm;vnehind 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. De9ks9/9eq ty 5n,gbe zmtermine 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 particlbnt9qks9m,/z9eg e5 ye orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate (8xng lc s1j1xof 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 o4ee5je 1ejux- f a uniform cylinder of radius 0.20 m acquires a r 4e1eje-u jxe5otational rate of 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 solid cyliqt gni -o9a*sh7,tkru9h: 0qc ndrical 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 cn*q-i7 ,s trah9g0k u:htoq9 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 is the anguh*uo.dtfe ;wf lrb5r58 s,3hrlar speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times as great,hw1o bn( q(h-vu;0 nnl3)nz on 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 all times, and that the lost m ((1 o- unvhlnnnn)q;zo wh3b0ass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution autompdpxyhxx3; ,4 obile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, anpd xx ;h3yx,4pd 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,umw uppk 6-4i2xm va* an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal surface at speev f:4re)(jq 4tjl5fy hd 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 7/ll,;ads 9h cl bx- /b(3wax:l yljs7ti7 fujand 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, determinexblllji at s:a yxj-h7db7s3c/(l,7; f9u/lw (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.2hfuqg/xr)vyf(jalc (8- x 4 w It is standing vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a/-wlu2rhq)fvx(8 jfgy cax4( step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4sez4 3oqd; y-t.2m/s on a flat road is making a turn with a radius The foroz d4q-3 t;eysces 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 lenmuvqm)v1x z,9 ;auh0vgth 1.5 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 the0,) zvmum vh1; vax9qu torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder 6lk8 dy)eb eh(n5a5dyand her arms as thin rods, making reasonable estimates for the dimensio58l6ydd ehae kn5( b)yns. 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 x+)u9,+sv k7lwy xchih8rxs 7bfi0p) cylindrical plates, of ms xxk9 7rvbx7ihyf)) lc+s0pu ,wi8h+ass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 Fi f4osyyomtv c689( 6)w xjid(wg. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point of the loopi(6td4yv6ojfw )yc 89(x mswo without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do noteqga7 w* 1 .wldxp8ke8 assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revor;fcf4 +cydusyk 17x3 lutions as the car reduces its speed uniformly from 90km/h to 60km/h Thr u yck37+4syfd f1x;ce 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