A bicycle odometer (which measures distance traveled) is attached near the whee(2g9 m oguo/vol hub and is designed for 27-inch wheels. What happens if you use it on agm /2(uov9og o bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocitsc9u q)1w(npgy. Does a point on the rim have radial and/or tqs )n 1puc9(wgangential 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 change?

Could a nonrigid body be described by a single value of the angula1bv6 8kr*1xas:3nsrr x, wfym9q +dztr velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Expla-e/m9tzxqcdb1.re i- in.

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 x7 jp4vzph(z tlx x )q ,*v(x1i/argz2head than when your arms are stretched out in front of you? A diagram may jx) l (,tzxah/ pxv zzi7 4pvr*qx12g(help you to answer this.

A 21-speed bicycle has seven sprockets at the reat3n)3ef-q d zkt.qlxw wzewgf 079t30r wheel and three at the pedal cranks. l3fz3)twf exw7 .wq3 0k9ezn-tqg0td In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend - azg 6aevsu,,q k+; u1bd;ual ;vl7ovon being able to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rot,qv;dbe;v+a kau zgua-l7lv ;us6o 1,ational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow bicz 510ec gg ;j-hrjs +v-uk0keam?

If the net force on a s (zm85ur9qumc ystem is zero, is the net torque also zero? If the net torque on a system isu(8mm5z ucq9 r zero, is the net force zero?

Two inclines have the same hylr)bd9og :1qinysh ,545mg z(pe1l op ha9b0eight but make different angles with the horizontal. The same steel ball is rolled) asobl1d mho9pe y5r9in z4y0 5lgh(qg:1bp, down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) ,wz 7dvtmmwe*c-;hh/ipsg l.rsl7if7 .1p k5 down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has mzpi 7s,he./ f*-dh.;gi5wp rv77k sl ltwc1mthe greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. They start fromn; ,, c;ki;bd(zbfue)s ba ;hz rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the zdbcba;) z;;( i,fh, esb; unkbottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and anr bmm( ywvh/36:fo/yogular momentum are conserved. Yet most moving or rotating objects r3m6b v (/o/fhyymo:weventually slow down and stop. Explain.

If there were a great migration of pxyjzz.)z+ 0 sveople toward the Earth’s equator, how would this affect the length of the dayjs.yzz)+ zx 0v?

Can the diver of Fig. 8–29 d;+j/ll7 - xnwgi4mkc do a somersault without having any initial rotation when she leaves the board?wml;+ni dxj /kc4g7 l-

The moment of inertia of a rotating solid disk about an axis through idk ; f3b1l8wmqts center of mas3m; q18wklbfds is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2-kg mass in each outstre 5wuvqc 8nzd 6 ss.1 i+9tv01kzh4w a(mt6fgnftched hand. If you suddenly drop the masses, will your angularzi0tsnw9fv+5 s1mw1 z 486h.c kqtfugvn6da( velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have th q(:6 tgok i+miatm1 ag(5/yxce same mass. However, one is hollow and the other is solid. Describe an experiment to /(mc:atgo 6m5i i1k+q( t gyxadetermine which is which.

The angular velocity of a wheel rotating on a horizontal ; i3.zxk+ w-.mq1xjie1agy c uaxle points west. In what direction is the linear velocity of a point on the top of the wheel? If they c1;i ukea.3xjgwm+zqix1.- 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 turntablepva w3,hqi z84. What hh,vz4 aq ip3w8appens if you walk toward the center?

A shortstop may leap into the air to catch a ball and throw it quic;a 4 2uwlfmccu;h4g1e kly. As he throws the ball, the upper part of his bode1 c4m2;cuahw;fug4l y rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of a, a vc+1vxjrvyq(p0/ql9jz8v ngular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways th/(0qvvrpxv9yvzql + ,a jj81c e second propeller can operate to keep the helicopter stable.

  Express the following angles in chj;7ty p0n v:radians: (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. C ka g-b 8hh5)4rchzu+fd,ecr /49ojmaalculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen our+c4dzc)hk f 8- g5maj49ra,eh/hbon Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km fov ela(.2a8 ;bb(tzbj p2q*ga rom Earth. The beam diverges at an anglepjgo b 8 lv(taq*2zbb.a a;2(e $\theta$ (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 6500m 3qioad0lln ;n2)ty 4 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is4 laymd 03 oqt2;in)nl 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. Ifas fzen3o rav o*2f7+4 the ball makes 15.0 revov2*a74r+n f e faso3zolutions, what is its diameter?    m

  A bicycle with tires 68 cm j.n; rn(ud +:bz w3n9xu 6uscfin diameter travels 8.0 km. How many revolutions do the whc nzs( j w;f3u9nd+urubn :x6.eels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. C.wr5zm5 wlx y2alculate its angular velz52lmr y5wxw. ocity 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;wc+:k 3od/;mrr9q 5j rkdzeysb8- l l6x.y qf.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m je k3xdmyo w lrr8kb9-; /; cl+6:.rz5fsyqdqfrom 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 Eam 3e1 jkb/t8gxhjc24v( qh9mkrth (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 po rwo6:ddv7r;q aix xxjicr *2:l :tq 7dhv8;(tint
(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 particl sq88u/f9z*altcx, mge 7.0 cm from the axis of rotation is to exzg/u lxf taq 9,c8s8*mperience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel acceleratesfx kr9 90*byzvmf-0yy uniformly about its center from 130 rpm to 280 rpm in 4.0 s. Determiner0f f- *0yzb99xyvmky
(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 radiu 3q98 k: s-ruwdkwr-cws $R_1$ 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 spacecrguyggb4yd98 ; * hv2otaft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they acceleo ydb ht9*g4vg8 2gy;urated 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 g6j i5-at1xcs:(tt ucto 15,000 rpm in 220 s. Through how manuix(jag c:t1 56s-c tty revolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200yyqa/9dqzo/iuly* - *m2j1s d 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 /dd hq/sdcr+1trgw0q;l 1,jdstresses of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutionsdqd0h1l/sj ; /cdgt+wq,drr 1 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 diametb b9sifdzp dp j2v5/0lg8 ;5qker accelerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will aqfp8bb2k dp95 z5j igl/dv0;s point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned o y38gq0drs0h6 lgl 1i ukoitwd)3- il4n)gkl 7ff when it is running at 850rev/min It turns 1500 revolutiout6i4ldl0og iqyg i83l3 kr0dkl w))1 7ng hs-ns 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 uniforn;wc o-0 x61f apfdw1emly from 95km/h to 45km/h The tires have a diameter of d1o0 xawwep-f 6n1 ;cf0.80 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 ca(5 k:jlgf6t2+;avynga 5 akelr reduces its speed uniformly from 95km/h to 45km/h Th(a6vk5fagtk5g; 2 +y:elal j ne tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each 6q9r ghd4ye1z0 t;amkv ah/w 9pedal when climbing a hill9thhr4/mawe9y6k1aq ; vgd z0. The pedals rotate 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 o-n3bg(:bh()qucgwz ta9 mj-fu.lr* on the end of a door 74 cm wide. What is the magnitude of the torque if the j (w. hugb -fqzmrta-)93u(cnb*o: glforce 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 ex7pl sxr6k n3vw5 *qas +cu7-b2ihj; the wheel shown in Fig. 8–39. Assume that a frip7cre*q6i3s5xsu;x+h-a kjbnl2w7 v ction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to thw4ah8 a* mt91(xvf;:g1ys tgmkr ef8 te 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 an4 * rtx8mh ;twkf1m8af1ggvs(ae:ty 9d direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tighteninp7:emkdj t6.jg to a torque of 38d.jpem 7t: j6k $m \cdot N$ 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/sx.x7 fkek7j axis of rotation is through ix/sjkf x. 77ekts center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bimx/p-an yphncs l(* ia*r,/i)cycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 nixm )*pc a/hni,rl- (/ay*pskg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on thecs-,l23 w,wi.lrs ooz end of a thin, light rod is rotated in a horizontal circle of radius 1.2 -.,ol2c sz3sl wior,wm. 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 bowl on a potter’s wheel rotating at6rm4 n. vamwpwdne055 constant angular speed (Fig. 8–42). The friction force between her hands and the clay is wa5 mr5 n0nd6wm.epv4 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 bad -ev, sd ;j53jlqicd4.d2jof inertia of the array of point objects shown in Fig. 8–43 i d 4; s ad-ebv5.3lq2jjddcj,about
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen atoms whose)cbd72o+ gbfsi bma+o,ix /x+l-gj 4p 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 spinni8)t .1(g tpcpwz ;txbrng at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rockeg;tw xztp )rp8.c1tb(t 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 hn-1e; 5.cpalai 8 gsdl)h)he radius of 8.50 cm and a mass of 0.580 h)n5.c)eag;1h8e-pdhi s lal 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 it8a ) vk097 bppya 2 7hzoznb613gwnbnm 9m,lgv 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-round nx 9yb7cvnn)b3)cqx 2and 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 acccnb3xvy)7 bn9 2xqn) celeration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at aqp5h(mv ck-z4-dn0 t10,300 rpm is shut off and is eventually brought uniformly to rest by a frictio(k 0d4-h 5zcptnv amq-nal 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 acceler1) c/awwe*dru ates 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 byowod97 64uqyw9 eyo -s the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball y sw w6 do99y7qoe4-ouis 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 shacxp-7b*v(q p own in Fig. 8–4( vpcxbq a*-7p6.
(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 consist )lw*;n.flz+3finit w s of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hame7d,(u pda4,8, ethnwg *oo5oac1a q vmer from rest within four full turns (revolutions) and releases it at a speedv dn(,o5weotqu,,ca g*h78 o 4d1e apa 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 ha5kpc (r*: zgszs 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 develoi +)um/iv;pl i8 mvhb9ps 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 without slipping n2vl3mpilgu da2k3;)down a lane at 3.3g )mlakv2u23pn3 l d;i $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to thrggotksblzrq/1 //5t ke+6w4 ve xf )(e Sun as the sum of two te oteglr1/r/s fqkb6 4/5v)( + wgkexztrms,
(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. Hox9kygu*7qo -b6* l ugqw much net work is boug7lkq9x* - 6gyu *qrequired to 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 l.y1 +,jram/vj 8orpb cm and mass 1.80 kg starts from rest and rolls without slipping down a 3p1yr. l,rmbj/a +8voj 0.0 $^{\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 balaf qj e9ly885tt.jad7zc, lb; rnced vertically on its tip. It starts to fall and its lower end does not slif;l.ytbt9z7 8 j,5ae j rcq8ldp. 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 0z6i(m ;rrx7 zhe7rgpball rotating on the end of a thin string in a circle of radius 1.10 m at an angular speed of 10px hz670 ; rreg(zri7m.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular (*2qe5jtm k3q b4ye tgebw-t, momentum of a 2.8-kg uniform cylindrical grinding wheel of radiumqjq (-32b ttetyg4e,* keb 5ws 18 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform thatkmn41gtp, w g4b4ho9p is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation d,1wp pg 44bo 4ktng9mhecreases 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 redh1z,h c2qgo osh27+6quv: ddduce 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 position, what is her angular sp7qcd 2h,dovh2 h dgz: sq+16uoeed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial rate ofc*r e +uvf. uov3gm4f4k2 mw2c 1.0 rev every mue rv*kf2 +3.c4ocu 4gwfv2 m2.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertic7xv*v dp5(o+kr q b40iesqp9mal 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 th e 7sb 50vp4*rqk9(po +xmqdvien 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 s230 23mcvbg kfx 8qgba 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 momentum of the Em -hibka(/r )c;m ci3jarth
(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 mi0pco, 1x9b c9z5uwsooment of inertia I is dropped onto an identical disk rotating at au1x sb0co,z ipwo5 9c9ngular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4d7ohj0 re mm2q :ky2p+ $rev/s$ 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 centerq dm7 tz-pgy4/*.jrz c of a rotating merry-go-round platform of radius 3.0 m anqycgr4z-mt*7 pzj/ d. d 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 rotating ;rp(r-/yujnw wxk jix7-d7f . freely with an angular velocity of 0.8 7u w -xf7/wjdjk. prx(;y-irn$rad/s$ 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, los4qaji 7ig:nt(uf+v 9qing 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 nearest integer(a 94ni7qjtf+qv :uig)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in excess of 120 rs4 p4b(/gmc ,c6 um2i sue*jj$km/h$ 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 3 /ei/) 0 znbrpx7jpdv$ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, that can rotate freely withou1wjl:2i.*h dzdmz: wc t.oqo 0t friction. The moment of inertia of the person plus 1zlm:c2 hoz o .td0 q.jdiww:*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 l) 3vz6hyu hn3yu2sib 6s*yl*stands at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertivu63ys*ul*hn) lyi s2b6 h3yz a 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 thjz;m :xv/c :mze top edge of the spool. mm:zz/cxj ;:v 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 al awl5r xokb.00ways faces 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 Elxb0 rw0ak.o5 arth.)    $\times10^{ -6}$

  A cyclist accelerates fr11lh :rt vkaq *h3v:lhom rest at 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 cib-m py 1)oca:ix,quflzi 69* ylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s iayii :-pl)9x mcu1bz q 6f*i,onterval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid.g*lkcp p+/o -pl)zz x cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released ozc.g+l pp/-xpk z*l)from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of the rear wheet vl+o1r h2sps)ne(6max of+4l ($\omega_R$) 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, wev z*:+4ct* c6m4g,qg,.x olysfdct stre 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 mass carries off * g.stmtdlx4zt vf4 ,6 g*co,cqyc+s :no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use ene 85pz z*hsukr/r/i 4tgrgy stored in a heavy rotating flywheel. Suppose such a car hasp8ri4gtr /u *z/5zhsk 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 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 pusg4 l*j+ jy9an $H_2O$ 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 spe rgul8g8t5lc)r+k z 0ked 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., wemz w ,/krr;n6r 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 acr6rkw ;r ,nm/zceleration 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 radz7wnldf)x qp o)i538nius 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 it rests (q8 xdo w3 filz57np)n)Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/so.j)7m gmuc(nirs z s1q n,+:d on a flat road is making a turn with a radius The forces acting on the cyclist and+o njm,1u(sq7 zms) i d:rngc. 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 lenga-c,vh 4g ob+gth 1.5 m and begins whirling the rock in a nearc-va4g bh ,g+oly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her armszmt gtd 1+ yw;s+ 9kk4+3iao wlasz;z0 as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio oftz yigsw+sa+;3l+kkzmo1dza90 4 wt; the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindrical plates,uhl *-s1 gz::.8npjjhrq16-wxanh o w of maahqhlu6. rwjp-h*z:s81n j1x w n-g o:ss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 Fig. 8–5kv g;d 6zzda:j2 w))sr8. What is the minimum value of the vertical height h that the marble must2vjsr:z)6dakw ) ;dg z 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, butoemf2hi b iup1/rsk+0mp m 85c+r zd72 do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 c o 94r.aerq/trevolutions as the car reduces its speed uniformly from 90km/h to 60km/h The tiresr /te9ac4.q ro 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