A bicycle odometer (which measures didhkxw fl*q 7,z9g;+n lcc;n,mstance traveled) is attached near the wheel hub and is designed for 27-inch whelncwg* mfxz,9 h7+ d;q ;cnlk,els. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on 3 ;*j1nyl-iuunkoclj1 ot;/ uthe rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration changno/3-1y;o1ku uj jl*cuni; l te?

Could a nonrigid body be described by a single value og8t ,hflx4zv mmx*jwc-nct1:w: i( o9f the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque s 82fkfkt. kc 8x ni73)c)lkgjthan a larger force? Explain.

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

Why is it more difficult to do a sit-up with your hands behind your head than wg hs29f:+ j;zc,nq ibqhen your arms are stretched out in frontz :c ;g9ijb qf+q,shn2 of you? A diagram may help you to answer this.

A 21-speed bicycle h7satiun- 7/orot 08ts as seven sprockets 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 haroot 0t8 -/asit77nusrder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run f* z*apq3qlq c +/glyskiw l*.2ast have slender lower legs with flesh and muscle concentrated h.2+ 3*zal*g iqlc lwqpk* sqy/igh, close 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, nwqmst 9 aie.n-5ok8oyl ,h,m2 arrow beam?

If the net force on a system is zero, is the net torque also zero? I:sugw+0amnp *m jnfhm7 iz 91(f the net torque on a system is zero, isfnm1w uniza(g9+:s7 0 hj *mmp the net force zero?

Two inclines have the same height but make different angles with the horizontvuazyq61zrbe673 k 0w al. The same steel ball is rolled down each incline. On which incliqewrzy 3a1vz0ubk 67 6ne will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (fromc7we r3h j)9nz rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater e9r )czw3h7njspeed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and i m6et6 0f-1zz dxll7nthe same mass. They start from rest at the top of an inclinlzf-1t6 0nd ezx6m7ile. Which reaches the 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 moving 1h zmnfa f.q5)z+bl/lor rotating objects eventually slfl/bm qaz)hf1 5znl.+ow down and stop. Explain.

If there were a great migration of phb:cup 68s (kueople toward the Earth’s equator, how would tcb(uhu sk6:8 phis affect the length of the day?

Can the diver of Fig. 8–29 do a somersault without having any initipw g - lon(*lkheg09k2al rotation when sg ( e2k- 0kghnwo*p9llhe leaves the board?

The moment of inertia of a rotating solid dis an9gyjeixgc:0 e(5u5 a3bmi3 k about an axis through its center of massign ey(ecix35u0a 3jgm9 ba5 : 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 holdinivun y qqhakh) t) 21jx6m0(8bg a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular vmi 8b1ku)vhqjy0 at (2xhq)n6elocity increase, decrease, or stay the same? Explain.

Two spheres look identical anj-wa vs6m ur)b-6;4sep9; iwp rl fjb*xv :;kkd have the same mass. However, one is hollow and the other is solid. Describe an experiment to determine whis w:b9rx j;bs-4a- ; w ;u)pkl6m vepkvr6fij*ch is which.

The angular velocity of a wheel rotating on az/ 5.qm3qqwnxg ;rc w3 horizontal axle points west. In what direction is the linear velocity .n; q3z3/q5 wmrwg cxqof 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 decreasing?

Suppose you are standing on the edge of a large freely rotating turntab7tg xi.q93nyzle. What happens if you walk gt.yq3zn9xi 7 toward the center?

A shortstop may leap into th 70wmv. c77ag /dv hpsi6aku9we air to catch a ball 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 and legs rotate in the opposite direction (Fig. 8–36). Explavkw77 hmw v7 iu6pg c9dsa0./ain.

On the basis of the law of conservation of angular momentum, discuss why a helirjgr+w r 03rou z7rj8xp23z( ycopter must have more than one rotor (or prop+orxrw(gyp2j0rjr3 r 3z8u7 zeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in h /uku, ctf(;aradians: (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 Earthowd. bk) ;jb,t because of an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as w;d .b,ojt b)kseen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 3z z7eaqixk m;wbx +tdq*8 lowd(9//9c80,000 km from Earth. The beam diverges at(dtw we b7;*/+qx / cxlzadqm o9i8z9k an angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. When the :3knbtkfy 0s38xr 9 , z:iogyfmotor is turned off during operation, the blades slow to rest in 3.0 s. What x3 gfo9y8k z0,:nkt:fsby i3 ris the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If fnx t;n/n9 8+) e/ootc8nfeirwjb)9p the ball makes 15.0 revoluo inf/xtbp+t;8f)8e/ ew cro9 nn)n j9tions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travel mdlf /ygmc9bnb;8 c408 liiv)s 8.0 km. How many revolutions do the wheelc4vcidly8lmgi ;0mb9/) b nf 8s make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at fkb358y)/r c m b89bonfnmwi- 2500 rpm. Calculate its angular veloci9n m b)ofyb35-ifbkcm 8r 8n/wty in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.xm shlrti2dmxf.v f218(i .e2 0id2m( know0d0 s (Fig. 8–38). (a) What is the lis(ilwmx2 hixf (e2dimdt08k.ofmn1vr 02 .d 2near 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 angular velocity of 7m3 . fit 4vfwlg7ggn5the Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear spe*hu 6 aiblmrtv ,:8(wh+b ati2ed 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 ceny 5; s(nbsw)v3 iug9npq/,tgptrifuge rotate if a particle 7.0 cm from the axis of rotation is to experience an accey(3pb vn sut, ni);g9q/sw5 pgleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniforml x(wf mbjeam1u6mb4 5;y about its center from 130 rpm to 280 rpm in 4.0 s.x ;5fa6(embu1m 4wjmb 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 radius kijkl r,s6ig+prr. 4 t3,bi ,s$R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astr*vc ( ejuyrw:gom+:tm0wkdq8:nv6o9 onauts aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they acced(wyk m v*gwov:+m q9:ceo6 8n jtr0:ulerated 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 res-l-elrj 3w ov7s3c:d ft to 15,000 rpm in 220 s. Through how many revolu3 -3jrcellfwo7:ds-vtions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in(m 7 0xx-+ 0 :m2waeukneolxcm 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 6 iop3.faw 8xnn2ripj: us 8,yin a whirling “human centrifuge,2fnpu 38p., 8:6rniaiowxjys” which takes 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from *rnppy9 ok1-jd a:k3p240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveledpk pjr1-k9ndo3a p: y* in this time?    m

  A cooling fan is turned off when it is running at 850rev/mtnf7fii8 3vs9 in It turns 1500 revolutions bef93 8vtfsni7 ifore 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 re7gwxjqpl*pz.2v te/9 duces its speed uniformly from 95km/h to 4pv7*qx t2 wg/ 9zelp.j5km/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 the car rewcn 0 )z,k4,*udfmvlel)l5 yrduces its speed uniformly from 95km/h to 45km/h The tires have a u *z 5,)ymcl, vl0f4n)lkrw dediameter 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 bsjd1l ;8 sz)zjike puts all her weight on each pedal when climbing a hill. The pedals rsjdj81l )s;z zotate 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 e 6*;qiq rh(zb74 cm wide. What is the magnitude of the torq e6ihr(zb*;q que 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 shown in Fig. 8y-a i /ngs8w6o–39. Assume that a fr isg -yaw6on/8iction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are atta y.xfy,hlbo3*7fx 8fvched to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the nely, .ff 7 3*h8xyfbvxot torque on this system.

  The bolts on the cylinder head of an engine require tighteniha,u.ja lw.9 o/ ;up(7iz1vtou rcr+tng to a torque of 389 r7,1z; hurvw.o.o t(ua +ijulc pat/ $m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of radius 0.648 m when the g4+djl scb -1 *sdpn.jaxis of rotation is through sbnjds d41j .*-pclg+ its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. 0wi-ev ur1r386amq / 1r,d s(p m4os u.xjrbbmThe rim and tire have a combined mass of 1.25 kg. The mi v-epxmb r m8.1ru0wsuo3rsm4 6 (aq,1jrb/dass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on tdmew+f ywhwbnwz6d a8:j. (48ayb - bk7w6d3i he end of a thin, light rod is rotated in a horizontal 6 awwb7a b4d3enyi kwwjzw:my6 +88.f(dbh -dcircle 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 bowleh5 :jnl1+ nbnc 5agg8p:sx;j on a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force1acg;pb +hns8j ::njl5ne 5xg between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of*731)txvkoto 5eru zj inertia of the array of point objects shown in Fig. 8–43 aot xk3*7z5 trev1ju)o 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 two oxygen atoms whose tota d ;rg*:. iu ta4um20tpyo rvv,4mu7pql 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 0 5+sbqxrrse 9( jt7unat the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what 70n9 qstujrxb5+s(r e 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 8.50 cm and a mass of +ijwp/b.zi6q w /dgw 00.580 kg. Calculatdigwq0jbpw+w/i. 6 z/ e
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, ec2di566ektnbvcmiuu) 88 ;q accelerating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven merry-go- :h*lgo:6r 9-n7l 9a-yuqx9 oodm0wmy v0xycxround and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting friction-:lx lad 9 qymmuo9 07o hrxvxwyc-n9y0o:*g 6al torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating u3bnc 2pb*tq;at 10,300 rpm is shut off and is eventually brought u 3bc; putb*q2nniformly 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. 8–456pqi0 *x5hsj w 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 the forearm,,shkg-obs sa),hdlh :2be ,x+ which rotates about the elbow joint under the action of the triceps muscle, Fi h:2hxslobs,kg,s +hb- ae, )dg. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, as shown in Fig. z uhbr- 0m)/c*65c mrp(akt yj8–46.m -r5tu*zpryccj 0k)(/bma6h
(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 masse. mb msvb,o)fg80u hy4s, $m_1$ 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 tur/) ymfgos:i4 b8daf 5d(rrn.ins (revolutions) an4r/58rd fsy( i ndf .im)ab:ogd releases it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has aesq s)fb.lk sdv1n-s b/5tg9dj, 0p5e 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 develops a torque 61mk;o3- +kzaqi krxcof 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 wivx. odf6h;m6dthout slipping down a lane ax.h6;dm6dvf o t 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect 0l7 okxbv lst(ic17d 6to the Sun as the sum of tw7 xliv61sdk0 7 lc(btoo 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 worf lbdc m 1(yq l1ax*q7v3 b2)icg(;kphk is required to accelerate it from rest to a rotation rate of 1.00 ryqbc11 q;ba7imd (2 c(p3*fkg)xll vhevolution 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 r. ;hzg mwpmx.g928 gwlest and rolls without slipping.wxg9z wg.m 8;l2g hpm down 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 iwfk /:p *baaoy+b dt,sk35k;t s balanced vertically on its tip. It starts to fall and its lower end does not t;+bf:pd/oswt5 *ka ybk3k,aslip. 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 of a 0.210-kg b :hh4-ny 1xk-rc cng.-h,gawiall rotating on the end of a thin string in a circle of i,hcwgh -xa:--ykr 1g.n4nh cradius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular m)s8bme njg:5f7c9ijlv* m tr7omentum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotating at 1500 rp m fl5*7jcr)s t8jige:bnv7m 9m?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands au1 tj7 ktl9vkora--0gt his side, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal pu7 k v0g t9ojk1t--lraosition, 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) ch awlv +c lr xf92bd2jnm;+6l2an reduce her moment of inertia 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 positionv2 lldbw+jxhfmc6 r9n22+; al , 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 r 6 s7(xpyarmg-302svv xwlwg, ate of 1.0 vlwggm(say v3p0rsx,xw 72-6rev 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 rot j)g) kv /*x+r7xlxn(ryja v4oating around a 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 clakylg)+o )xv/ *rvj xaxjn4(r7y, 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 angu m)imd5y/v*rgs0 s(i llar momentum of a figure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momen g 4. 6ez;*qx:u ejqiq9e x1ezq7lnpc5q rt4n0tum 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 1 ur88 ql(xduqmoment of inertia I is dropped onto an identical disk rotating at angular speed lqx1u 88dr (qu$\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4fk;::e680hzveru en:yh.g : dtrh-hk $rev/s$ 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 r6 zqrmkx(6 2o3xs0 mahotating merry-go-round platform of radius 3.0 m and momsmo02krz(6 xq3 6hax ment 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 merr2 (g:y ;zfcno6cg3i7vukkm7vy-go-round is rotating freely with an angular velocity of 0.8(yvokg cgmf c;v :zn2ki77 36u $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, losing(x11,/gkb wvbfzp+ cv;l1(ho ( du jzg about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assum1g(d 1(ojlb+zb v/,z(hpgfx uwc 1k;ving 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 ivh + rx;qth*y li/74ws+h hq v1.+akznn 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 nu,1knf c 9i1g$ 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 platntdfa wp,)vt ,, a;/hui;d/ nvform, initially at rest, that can rotate freely without friction.dp/;dtihf v,,tnwa, v a;u /)n The moment of inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diameter merry-go-:v 8n4niiutdrf4n5 d ;round turntable that is mounted on frictionless bearirntui dn8v d;i: 4nf45ngs 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 3. wxxb4fzooo7 ( a:ihwith the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. Thez(.xo boa ifo h4x3:7w 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 alwam*ocr7edf8 *hys faces the Earth. Determine the ratio of the Moon’s spin angular momentum (ao** hd87rfmecbout its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest a k,*y pikj*-avt a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of radius 0.20 m :-tx.zb3czndgad1 )t5dllk 3acquires a rotational ra5l td:kglzb)- ad.dzt1 x33nc te 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 cylindrical disks, each of mass 0.050 -po:1o . wfy9xcrmlcd- 4bd v.kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg anvcpd -fm.odb. c-1r49o: x lwyd 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 is the angular sihrz3b5dn +u,od mlkc 8(05kvpeed 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+zax /2h7 qk0ngfgwx: y z-fw, our Sun, but with mass 8.0 times as great, were rotating at a speed of 1.0 revoluthg,z7 -0/xqz fyx:2n gafwk+wion 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 use energydd m 9)n* n3-hlex cb6hm9x.ri 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, and should be able to travel 350 km wi*n9 xd rbm)-3ndei6ch.hmx9l thout 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 7fi,o35gn0u zm nyct*$H_2O$ 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 hp6)l-3v c1 bsai8 sjpm3m0x ssurface 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.e., we ignore frictjlxou8(e 3jvwqh f*6 *ion) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At e *(l63ux whv8fq*ojjthe 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 radius0w o+9tiuw3,ipru z, b R. 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 step against which ir0+w3 btizp, 9owiuu,t rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a turt6nq/,w7s zp rn with a radius The fo6 q7rn/ws,zptrces 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 lengthbl,nc sjk1- (b2dhe e- 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it esh2-1 nlc( d,keb- jbfrom rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s bo) hv)nx1gz1 0fqx wfzapx*f +hxhd(2 .dy as a solid cylinder and her arms as thin rods, making reasonable estimates for the (x 0zphhxz aff1d+n2w. qf)*)xg 1vhxdimensions. 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 whickm*me h ea-bkfyh4 3w z4n53)ah consists of two cylindrical plates, of maemywb n 3mkh f4zhka4)*a- 3e5ss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 ra o-2jl/,zei1o9dcze nb3s kc )dius r rolls along the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must droz2)s-l k39 id cco,b ez1jn/oep 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 assume ynjz) a/i,z r4bt tl8l1,bxh0 9rqe5u$r\ll R$ h =    (R-r)

  The tires of a car make 85 re+z7)p; wqa jlra;(t gybp;sf0z/7 d m.cmrvv*volutions as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of ear )sv/c;j*7 mdz;myvfgbal 7+ .(0 a;twz pqprch 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