A bicycle odometer (which measures distance traveled) is attached near the)x xbx ;epwc 4(vy5)qm wheel hub and is designed for 27-inch wheels. What happens if you us5e )bv(xqmxxc; p)4wye it on a bicycle with 24-inch wheels?

Suppose a disk rotates utc7(nxl 1gb3 at constant angular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or 1x 7cbnut(3lg tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by .o.9jnh1(sb u;rs xhsa single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torkkdu 5/ouc0u6 que than a larger force? Explain.

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

Why is it more difficult to do a sit-up wihinwp xo33s-r+.pb 0y th your hands behind your head than when your arms are stretched out in front of yornps3i0 x -3.b +hywpou? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at them 6;m+ gyd qnwel/6sl3)6 avil pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a la6veqlsm )6y mdn3g+ l/i6lw ;arge rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slend38.1q mv p, dejuqjrz5er lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, expl8 vqz5pjm1d.qju,er 3 ain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carr )6e 1m.m-tdx pfea;pky a long, narrow beam?

If the net force on a system is zero, is the net ;f p ypjn --*jfvvj(3otorque also zero? If the net torque on a (3-p* vfy-njjv f;jposystem is zero, is the net force zero?

Two inclines have the same height but make different 1yyco db(-:t m kz3m+langles with the horizontal. The same steel ball is rolled down each incline. On +z -c(lkm3ombyy: d 1twhich incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneous/w.r*pg bd+qwna/b4.:2 dmrx wr,hkn ly start rolling (from 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 speed there? Which has the greater total kinetic energy at the bonpn/rxd h/r 2ab mgdw:4qk*+b .w,.rwttom?

A sphere and a cylinder 51xrrfje 2s x).f,u6xjgf:m nhave the same 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 tf6fmgnr,rfx)2 5:ej u1.xjsx he bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentu faxcy:; ewup8 l-0k2/8qgsjum and angular momentum are conserved. Yet most moving or rotating objects eventually slolk;p x-82/:a8gjucqyus wf 0ew down and stop. Explain.

If there were a great migro1t,gfhrc7qns.k 9 *w ation of people toward the Earth’s equator, how would thionw7h rg. tq1, skf*9cs affect the length of the day?

Can the diver of Fig. 8–29 do a somersault without having a7yz /f*qish 5sny initial rotation when s iyhf/z5qs *7she leaves the board?

The moment of inertia of a rotating solid disk about an axisww mc7-: wbnkkun:s8( through its center of:ubw cn w-wk:sm k(n78 mass 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-kvhi9w .goz3/ y6llos 5fb:: axg mass in each outstretched hand. If you suddenly droz l5xswylvg:6o9 .3 ofba/hi: p the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the sam;25qmhy y f-85 jdyvnpe mass. However, one is hollow and the other is solid. Describe an experiment to determine which isj-5qfpvy m; 8 y25ndyh which.

The angular velocity of a wheel rotating on a horizontal axle points west. In w,9qt v. vug6 7en3zcachat direction is the line. ,qg3u6acetcv 7znv9ar velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely g2glw d7 x01 )bmvavx)rotating turntable. What happens if you walk t)lbxxgaw vv1m2) gd 70oward the center?

A shortstop may leap into the air to catch a ball and throw idbc3o1*hjit aew5 1nj nx1,*q t quickly. As he throws the ball, the upper part of his bd*ei h 1,t jo 3bj*1xcwnna5q1ody 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 angh: uuegbq8l1 -ular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicoptue-q: ulbh81g er stable.

  Express the following angles in radians: (a) 30a 7)tyj/fms(zk5wi x 8 $^{\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 Easvk-6mjcbt 8l r.je+,* bu8xarth because of an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of tmbsv+r*xjk u6c , a-l.8 8tejbhe Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directql+4l0 (i -xa4 upatdred at the Moon, 380,000 km from Earth. The beam diverges at an ani lq d(apruxlta440- +gle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate ofaf 1n1nw r1b;n z)7gjq 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow d )j 7a 11gnfq;nw1zrnbown?    $rad/s^2$

  A child rolls a ball on a le4ib ak9e ie5e,f o)::8 v(hsboyl-spxvel floor 3.5 m to another child. If the ball makes 15.0 revolutions, what is is , fke-s:voiepio:b5ba lyh94 (x8)ets diameter?    m

  A bicycle with tires 68 cm in diameter travelsn7x /p.z9dos fyabc*;sn ah b8w; lr;4 8.0 km. How many revolutions do the wheels makep*/;b asz ;84bo7;w 9asxh r. ylndfcn?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at l ejgy1jdu* k foc8.6,3yh)o j2500 rpm. Calculate its angular vejl d38u*1 k), yfjo ycjoh6.gelocity 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 revolf 81r5snp; ;n yr7gafsution in 4.0 s (Fig. 8–38). (a) What is the linear speesap f71grrns;58 ;fyn d 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 the Earth (a) in its ooi/uftmcr5phsg;p4)4w 2 4 t zrbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of ltnaax a*h;+4)oik6ta 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 thjlv- ,s bi8)*(qqg6q edw az-we axis of rotation is tov(gsb )--8q l*aq,z weiqw6dj experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about ig:ts 7ikrf3e6ts center from 130 rpm to 280 rpm r76if3 gs et:kin 4.0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radiusyzq6h : v8ve.r $R_1$ 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 by6p buu*c(p h k.)oh+jcj1/cthemselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they acceu ujbc kjp.*)+ poby cc/(1h6hlerated 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 uni16wd;ey o+vwqjk76oq zldp av) .e41dformly from rest to 15,000 rpm in 220 s. Through how many revolutions dvkood p+dq eyza6e1wj;d w7 4lq6. v)1id it turn in this time?    $rev$

  An automobile engine slo1z w5t.i,a uofws 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 whirli+pik* c+ajczd 00bev9 ng “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching ibp0*i ca++0ecdkzv 9j ts 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 fr3,scq8f/ jgixvc u :j-om 240 rpm to 360 rpm in 6.5 s. How 8jc vsc/qgju-,x : 3fifar will a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off whe+rygw01(ii*tk l 3rban it is running at 850rev/min It turns 1500 revolutions beforet0k+gy*a 1l(b3wiirr 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 retf3sy l eend:m9p 04b 7ludr,,duces its speed uniformly from 95k4,ly7umse0febt ln9, r: d3 pdm/h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolu oox8-, f)5mxox1wan-iosv4 ztions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter ooz fxm5owxi41 -vn8o, sx-o)a f 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 her6 8rvz1n , gkp2c;ebuick)0ec weight on each pedal when climbing a hill. The pedals rotate in a circle of radius 17 cm.1cvc0k6bku gnzc, pi )8ere2;
(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 e h). 7n38unbf lydsc,c8 dohb.nd of a door 74 cm wide. What is the magnitud)odl ub.ynfb8 s, c3 n8h7cd.he of the torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle o0kpz t)ar, o/af the wheel shown in Fig. 8–39. Assume that apkz traa,)o0/ friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a massless rod which fwz.0gi8l+(am1savq pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and d+ a( afm0q lsvw1.zi8girection of the net torque on this system.

  The bolts on the cylinder he3 0qs tc5,br gjlrg0*x5upyu vkgq434ad of an engine require tightening to a torque of 38 r04gcbqg54 tk v* rys, jx30pgq luu53$m \cdot N$ 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 ntr . 9z(ob.suf a6l-nkw7 px x3(cy*csphere of radius 0.648 m when the axis of rotation is throupn sf- 9t .nol3.xbky6r( cca7xuzw*(gh its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a t9jg7ssgin+j5o8 a +x0.tz s nbicycle wheel 66.7 cm in diameter. The rim and tire have a cxn+s9t85+ 0jat 7go i.gjs nszombined mass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a wcq fd0o5:my8 thin, light rod is rotated in a horizontal circle of0c 5y8mo:fw dq 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 bowl on eohazzy vg,of/w6++0,a h.dl k m*p0qa potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force betweena6mo k ,qhp.vw+eofy+/ h* a0ldzz,g0 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 inertia of the array of point objects sh ( +ilcg.*i )d*nz)dr j6oiausown in Fig. 8–43 abzu*n)g(lrd 6*ic + joii ).sadout
(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 consij vq):x /lqss( 7+xxkwako9 1g;g8xybsts of two oxygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the correct rate,c hw,d,tpak 1-q1kk bcghc 2:/ engineers fire four tangential rockets as shown ik ,ckgw1hpc: ahk-d21,tq/bcn Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform ccal95 ;,jq:onedxzjv0v fx , .ylinder with a radius of 8.50 cm and a mass of 0.580 kg. Calculatvond5,q. l:0cz9 ;v jjxf, xeae
(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 swingsh9v*()b sbl ou 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 tangentially on a small hand-driven merry-go-roxrelq)g 99 tazs,krz r)g04.jund 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 frictx rzqks)9rl. 4jr 0t9ag,)ezgional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is eventually brought 7,p* k;axg:7omta tm+umbwd 9 uniformly to rest by a frictional torabtmam79u w,:p+oxg7dk t;m*que 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 accelmwge2:jf:l)0y xe 6v .gv 9fvgerates 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, 89bm( bok njrne) )tg-which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ballkm9gn e)j)rntbo8b (- 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 k 9db w 3ibz*(/n0ukh,zokes)b lfo*ga0xd8( be considered a long thin rod, as shown in Fig. 8–46o( k)bl (ihazdoe3ks90g b d nwf0*z8*u/bk,x.
(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 o7 t43,kvhtoot .plib*f 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 hammer from rest0s,wzr4zusk;qocs:n 5.0*x hr /dy9xz c jbw 3 within four full turns (revolutions) and releases it at.:czywzs4xwo /kcd r3nz5 9b x0 qs0*hjr,us; 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 hd* kno sn6 ; ds1tqpopu30q;e8as 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 devei*s q3t1 rtegn03tk23w j8zg wlops 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 wg44ow k0 -(7fwunf6s.rnt aetithout slipping down a lane at 3.3 uf.-ftg 4awto0kwn64n (s er7$m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the;e(4 kqspuf3h3wlae -g;tlwfo s f*2 7 Sun as the sum of two73 tl(ff * esqgo-u a4wkhfwe3;p;2sl 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 ofr)ydc4 kf42kx )*8jd wu.qbxa9d *khm 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 my *d4 qx.r)h9wcb* u4d8kafd kk) xj2s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls:k01 9cdewftaaml 23bf;s5 v fbg/ g.h without slippi.s;b:evcftg1bw3 gafa 2/dmklh0 f9 5ng 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 is balanced vertically on its tip. It starts to fall ankn) 8cr u 9+x*ar6truyd its lower end does not slip. What will be the speed of the upper end of the pole j x *8cy)6rar9tkru+unust before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentu8tl4 m raw.9 c6y4*xrldem7lh jf 1t1bm of a 0.210-kg ball rotating on the end of a thin string in a circle of rad*6tfy.89bx4e7 w llcrl hta1rdm 4m1 jius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angull9)ae2x77g nqe+cooa ar momentum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 ce a+a92e)oq7 x lgc7onm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotating at a ra uz 9rpq)g,df,te of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the spee upgz,d,r)9 qfd 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 q40pm xrmabk*s)vwj xw.*z: 2inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If shem.v 0:k2amx *jr)sx b*4pwzwq makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rateni3df8uqae /- 9g, cym from an initial rate of 1.0 rev every 2.0 s ti/cq 9 3mneug fa,y8-do 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 vertyn3( uq7)q:mf7oisg+ohc2, c; cfyakical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay stick h+7q:fi2s,fac ky7c3o g; m(quc)ynos to it?    $rev/s$

  (a) What is the angulargygzs*9thvhscz.7f 9m 5 5 l(b 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 momentumxv a4cn2d tx ku em1:hwrt3-/6 of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inerti1g. upnxsm. o;a I is dropped onto an identical sux1.o m.ng;p disk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4 7bofzb (p6 r3y$rev/s$ 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 ce wl abufr z/ep,8):t,cnter of a rotating merry-go-round platform of radiu:, tz p)b8,aue lfcwr/s 3.0 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 rotatingl2w1 tw qvh77a freely with an angular velocity of 0v17w7q2 htaw l.8 $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, ln :z-ph a-ck* wza) *m9tj5fpkosing 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 inte9tazmc5:fza kjk-wp*n-)*h p ger)$\approx$    $rad/s$ (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 excetisd3de p; x14ja/cc0ss 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 y( jpi6vmu3 ;h$ 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*qp6pq 4zbms/axs 9o) without friction. *o6ap/q )sm q9pzbs x4The 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 mepboiolsp. g:frn0 ey202*hz - rry-go-round turntable that is mount h 0ysfrbon*.:il-0e2z2 po pged 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 24sm/.,sq+oo -zl wiyx- gjwqthe top 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 behind the person without slipping. What length of rope unwinds from the spool? How far g, /zql.my w-os-2o4 wsiqx+jdoes the spool’s center of mass move?

  The Moon orbits the Earth su m0vdf3y 7ivb5v.5cq0trlyc , ch that the same side always 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, tv5cr. q7 fvi3y ,yc lm5bd00tvreat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from iyhl,5x ;5x id*x(pb crest 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 cylinde+ fo a8q5 *isisnz3;gxr of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval85 i *zs;+ noifqg3sxa 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, mm)71.k gvegof*tzk)+ziby2 fe/ 7g leach 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 calcula7gzz kevf +/m)kfymglt *o12. ) bgie7te 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 spewrihw8 ds ec*9k.vu 2, :g7bzmed 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 atj6hnp;tqyxi)n + jg + au-o0id+7 o;ns great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass ca0g ;in+6i+7n+ojhuy xdpattj-o ; nq)rries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use energy stored inss8w62 a s))vdhq dzc/ a heavy dw hs6 s)) /z2qsvdca8rotating 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 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 illustrateu v zr ul5h/efmnjk n5 c*;uca*,f0*;qs 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) ib:3akpyy z71ss rolling on a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can p9u5vullz5 gp8ivot 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, detel55g 9lpv8 zuurmine (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 radmypbo4e 5l:tk2d ii0 /ius R. It is standing vertically on the floor, and we want to exert a l/0 pkime24y:bd5io thorizontal force F at its axle so that it will climb a step against which it 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 turn 69y8gtsiyk-0 n +sgn twith a radius The force8yg9syt0t6 s-innkg +s 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-k mz.88jpya pq9g rock into a sling of length 1.5 m and begins whirling the rock in a nearly hor. a8y8mz p9pqjizontal 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 87fdh ;j;kfk ba solid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tij bh;; k7ffkd8ghtly against her body.   

  You are designing a m64ycsat 1/ymaxu*/ nclutch assembly which consists of two cylindrical plates, of mass6/m4 xyay1us/mn t *ca $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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. ( *vcizd sg0-:u wvw/z8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the w:/viz* gwc-0v(uzsdhighest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do nol(1tt pq(: grbt assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uniformly frjb j8l ui(/gx/om 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular accelerati8jul /bg /(ijxon 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