A bicycle odometer (which measures distan 0,9scz u9dncmg2ckmg4 *f3rxce traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inchgmc2f,dguzs3mx 9kc0 4r c9n * wheels?

Suppose a disk rotat(rq( + shrj4hmj04hf kes at constant angular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular veloci hh4+(qs jrrm (hk40fjty 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 angular m 9f-ud9* ivkrvelocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger fo+.p.i;9kf a)l r)tx ntbserd5 rce? 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 fp.fd 5d)jvc7 your hands behind your head than when your arms are dc57dfjv) p. fstretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the pe0mnnhp ,j t 80disw-:gof*nz ,dal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front*j, his ofn8t w0g:-,nnmp z0d sprocket? Why?

Mammals that depend on being able to run fast have slen 9vht)sac::b0w4wck cab +qkh1w/ag 1der lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotationbcgh bta:10cv:/)14q k csa+akw w hw9al dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig 4,(g,3p ytrdxdim/ (tu:ee 8cfqoza.. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is8 il/w1q0obl 0/m pp o.vrfy,i the net torque also zero? If the net torque on a system is b/0rqip,./f8 wmo y i0l lvop1zero, is the net force zero?

Two inclines have the same height but / k *c0nv hfn1rd5)dahphr8r9make different angles with the horizontal. The same steel ball is rcn1 /5af d)nk r9 r8*0phvdhhrolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rollingksw sn jdxa6k4* bl;6+ (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*n;b s4d6kwsj l 6kax+ bottom?

A sphere and a cylinder have the same radius and the same mass. Theu+1ym04 jpkjpy start from rest at the top of an incline. Which reaches the bottom fi p4pm+1 0yjukjrst? 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 5s *3qclwgg kk.ie*a-are conserved. Yet most moving or rotatins. 5gqke3 a*cw-*l ikgg objects eventually slow down and stop. Explain.

If there were a great migration of people / bteo /(m8cw1m ahb5utoward the Earth’s equator, how would this affect the length of t5/1(o aec8 bm bmwt/huhe day?

Can the diver of Fig. 8–29 do a somersault without a- -w :qe:1o6gvfrke q5itup (1an2nwhaving any initial rotation when she leaveg-t:1f na-51 pweqvo(qw6k :r2n eiaus the board?

The moment of inertia of a rotating solid disk about an axis through its center fe+m3ofmx-nbyxs b w3* - +e,cofb3o+x3bn-y+ms cfx*e -w m ,ef 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-kg mass iop bzv( h2y-9sn each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decreaspbo9sy2(-vz he, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one i4f)j wq,wn u b;6fdg vw,r;b 7utjhn,(s hollow and the other is solid. Describe an experifdgtr jun)4q fw; ,bjbnw( 7h6uv,w ,;ment to determine which is which.

The angular velocity of a wheel rotating on a horizonta.(/ama /s r,zxlnekf0l axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points ea arz.en/ a(/s klm0xf,st, 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 rotatii;ps u7hga)uk- dtq 13ej*i* x03oqeung turntable. What hap d hp;e3)xq0j sqget o1*3i7a iuuu*-kpens if you walk toward the center?

A shortstop may leap intoc0a , )xvlo7tj*ze-ve the air to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you loav )0,c-*x l7joeveztok 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 angular momentum, discuss why a helic )qu1mk+f8ktb 2 +d,r (r8qi.nlx qbu(b.zd vlopter must have more than one rotor (or propeller). Discuss one or more ways the secon v+x) 8rb i,8m(u.+. q( bl1urknlzd2bdtkfqqd propeller can operate to keep the helicopter stable.

  Express the following-ogo.sr39dqygg0) bm angles in 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. Calculate,haa3pn-h0-y p using the information inside the Front Cover, the angular diah3hpna y-0-p ameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moo; ; /ruin9 7/nbjjbjbpn, 380,000 km from Earth. The beam diverges at an b7 ;jn9b jj/ /pn;briuangle $\theta$ (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 motor n5 m.(sm xe:e0hcav p:is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the:ca emv snm5p(x0 .:eh blades slow down?    $rad/s^2$

  A child rolls a ball on a lev qm)7i)q88gprm- s/ nhjm8 jdeel floor 3.5 m to another child. If the ball makes 15.0 revolutions, what is its diamete )erdmj 8-iqq8m ghp/j78)snm r?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. Ho3bav.eqs )cuf(jb .(ow many revolutions do the 3(u.( .ceo) bbafjsqvwheels make?    $rev$

  (a) A grinding wheel 0jovn.rhp1h mfea3ww:1xe*w) us 37 o8 .35 m in diameter rotates at 2500 rpm. Calculate its angular velocitu 31v.8nejo)war3mh*w1e fwxhpo7 : s y 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 revolutu)2c28uejd livw y0 +mion in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m fro2uwe ycmu +2jd0li)8vm 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 (wgo46 0fo0poh3rm 69/my qwqlj)m c i3a) 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 p gm:n,zg+jqe 5oint
(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 fromy j fir:xtj)7aim7t91 the axis of rota9 7 a:j fi1i7xtjmyt)rtion is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformoi8 k3htj- (x gwp(m jh.72rzlly about its center from 130 rpm to 280 rpm zpjt.h3il(8gwh 2jr - kmox(7in 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 radiue( g1h r pc9y5x om/a4gpovk)2s $R_1$ 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 themskfuy 5v 8l -rl+p5;; rhsh1yczelves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. r81vy;s5fhl +h rlp-; k zyc5uDetermine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpm 5 ps8acgxu a-7+) lbyein 220 s. Through how many revolutions did it turn in this time? s pc)ge8yl7x-+a bua5    $rev$

  An automobile engine slows down from 4500 rpm to 1200 0qz)pmh+fj8(w8 lr sg 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 ws jx3wp n:a7m 9+yih.thirling “human centrifuge,” which takes 1.0 min to turn through 20 ws37n9xay p .:thji+ mcomplete 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 240 lo*c .yx0 .oc(ip)vk+hwxz /;gzgmb6rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have g 6xh+.okp0.lwm(xz )ig/o c*c;byvz traveled in this time?    m

  A cooling fan is turned off when it is runqxhd4, ab h27:cb ,c82-bte as.scvnvning at 850rev/min It turns 1500 revolutions before it comes to a stopevd.h8247:vss bn,x t hccq2aa-cb b,.
(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 6 8.frc1;wykxdxy4: rt0f yoj65 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires 8wf;4r6: rk 10cxofyxyy.dt jhave 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 rmng.:a+twlp0cd wh3 q svj (c+ d);oqw6+wdi1evolutions as the car reduces its speed uniformly from 95km/h to 4h+ddg6:qdivja;1t c nc.+ wwls+ )0qmw w(p o35km/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

  A 55-kg person riding a bike puts all her weight on eachi8facr;xn; 1tmd+g6hwx +(l9sf pll3 pedal when climbing a hill. The pedals rotate in a circle6nf(8lpx; m1wt3 l++ sx ir;agl9dhcf 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 t))hhq 19jd yzvhe end of a door 74 cm wide. What is the magnitude of the torque if the force id hqy jz9h))v1s 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.ihiur*.q0 d -e 8–39. Assume that a friction .urq0 e d-ih*itorque 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 massleri66bb zh ctd) a)1t+wss 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 net torque on t 1iw6zh)tb) rcb+t a6dhis system.

  The bolts on the cylindeh0 p*ul78h ti+ufno3jr head of an engine require tightening to a torque of 38 lf7i o083jth+h*n pu u$m \cdot N$ 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 ine bsybp 9pnh)(c26q)zrrtia of a 10.8-kg sphere of radius 0.648 m when the axis of rotation is through its cecy )bpp2z(b69 qsn) rhnter.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter)z 2cbff:wsyc tb273v. The rim and tire have a combined mass of 1.25 kg. The bwf b3s zf)t2 cc2y:7vmass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of t,soy-mvf09pl+ p 0fp a thin, light rod is rotated in a horizontal circle ly0 f+v m9p ,0ptpos-fof 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 a potter’s wheel rota3gdf f9* 4 eb2(rwad:meo r0wvting at constant angular speed (Fig. 8–42). The friction force betweef0r(mrd*4 e3w:wbge9dof av 2n 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 shown in Fig. 8/q;zo sexya) dc. 2i1c–43 about qda ;isx.c )z2 ce1oy/
(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 atxpwm2 z*e pg *ci,km0)oms 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 , yf6o xa lkc2xe:s-;kat 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 forcf xo ec akxylk6-:2s,;e 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 8c)ajc + :l,cbzeb p20r.50 cm and a mass of 0.580 kg. Calculate 0b2)e:j + cb,cczplra
(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,i/ss5hdmh). j+(rp/8qvmu iu 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 tansq5f+1 y7 bedggentially on a small hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 1s+ qbfey 7gd5kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at b)8 zx7xxe4:vz hm vz)10,300 rpm is shut off and is eventually brought uniformly tov )8xbv :x)7z4zhze mx rest by a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a 3.6-kg bakmu x 3oq:0z7fgj lcz6x,z1brkd8z26ws r66 rll 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 forearmtdzw+u:yq68a r q vso/;z(f+c, which rotates about the elbow joint under the action of the triceps muscle, Fig. y+8;udt+rco(zqa/6zs: q vfw8–45. 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 bei8 y/+w adc)tv considered a long thin rod, as shown in Fig. 8–46.v )/ 8tyic+adw
(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 xy* zucs9b3.bs, $m_1$ 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 turns (resms od5s h,p4,g-/j,wr dds4qvolutions) and releases it at a spssjsddqpo454, ,-,r dg/hm sweed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a moment of inertia ofyfaj9y /nw 7g.*s8z dn $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque of n;fuk* 1amyp)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 rol) hebqpqg+(mou u)e/loc/ uh72:/d do ls without slipping down a lane at h oo/bcd2m)(u qu/legphe:7) uo+d q/3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic th (9o4et pv2menergy of the Earth with respect to the Sun as the sum of two terms 9(2evtoh pt4m,
(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 e7lk: y.ecf j-mass of 1640 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotatikfe. 7l:e cjy-on rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest,z)t+u:eaol gib5 y:mba3 pe6 and rolls without slipping down ao aetbeu,y:)p6lbz3+img 5: 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 vertix,y;0.+cwb yrzb5 h )930 rg lybvdbmkcally on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the grouhc ydz wv k),3xrm5rb.+;b bb0 lgyy09nd? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on tvh)ck +sx +ol 11woqo3he end of a thin string in a circle of radius 1.10 m at an angular speed ofsoch +vk lo1o+)x3qw 1 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momes+km ;a33-8z;x iduo8gydqtzntum of a 2.8-kg uniform cylindrical grinding wheel of radazo;zs-3x;ddq u8m8i tky + g3ius 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 that is rotating at a rate of 3iy r0-q1 q0hqccf -y;mcrx1t 1.3rev/s If he raises his arms to a horizontal r- yhqfix -3cqcc0;t1q y0r1m position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce her mo8 e4ytd(e8ic k*a ulc qlzhk p0-kb)66ment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she make 8aktu6-k6)hey kcdc qlbipz4*e(0 l8s 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 rate from an in q*v-ltkmx q/0itial rate of 1.0 rev every 2.0 s to avq0k- tq/xl*m final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical ax2ajlp tgv 7: (n-qf6kais 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 q:t gpaa7 -fnlv6(jk2wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum )z4r v ofx6dws*7 dnx9of 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 ro(a);)+xfsq6f y rxvmomentum 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 momeir(,9p+o:kj i 80wrwc prwqb4 nt of inertia I is dropped onto an identical di wb(orqj+k:pw0wrcpi 8 i94r,sk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns atn g1qr 7/(hbm7sfb rgj*/s; i2p jq+pg 2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the cen i68g ijf-)f1:qphep ster of a rotating merry-go-round platform oi6ejgh:p s 8f-iqp1f)f radius 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 rotating freely with an angular velocity o dh,2 4 6zmtmlkplep2crd23jp( 6czb7f 0.8dk2r 32pj,bl26 (cepzhp lt4z7cd m6 m $rad/s$ 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.63sko elhhr( a(kjcnx8l i+ : into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what j(r:kn+k(l.ciah o6 e8sx3hlwould 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 in exce eyhtw:,p h/pws ej.7:8pfzl,ss 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 ;t127kc 0nctwzo-omf $ 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 platfo* .baid1jlal51 s6t;u e /alalrm, initially at rest, that can rotate freely without friction. The moment of inertia of the person plus the platform jl1;.aa utbe*5l/ 6i sla1 dalis $I_P$ 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 ofhv;59 plm ctxhv5 jrzrja1,-/ a 6.5-m diameter merry-go-round turntable that is mounted cmj5vx1l -varj5r,hh/tp; 9zon 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 on9cro/x+f b8jx the ground with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a disx8o+9fj crb/x tance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces the Ear)wpv2swjy4 idpa1tox1rw* 5, th. Determine the ratio of the Moon’s spin angular momentum (twpa)ypj xi 5 wrvo1w2s*4d1 ,about 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 k3k.kvsy ( ta+from 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 ohfj 6gy nw zh6bb oya/(u.p1+6f a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculath ba 6h.y/uyn1 j(6pzwb+6gfoe the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cyli: 3ekukl*c .lindrical 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 :.l*3kelk ciu 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 anguleip0ljq0) ( * hw)ocvpar speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but wibkz+4h88oma: e7mo3fv o7sxq th mass 8.0 times as great, were rotatio sovhamf3:z8q48bo me77 +kxng 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 no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-polluti1 b,xsygw2j* gon automobile is for it to use energy stored in a heavy rotating flywheel. Suppose gyxb2*sj 1gw, 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 illustra(1bkn z;n/nci tes an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal ss;id/hf7 s blyk12,iwurface 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 ign nn1c:n( :tm-b2uh rd r.bn8srore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–2n2rd:.n8n rn thb1(cs:r bu m-1g.]

A wheel of mass M has radius R. It is standivf7fw zu0/d v1ng vertically on the floor, and we want to exert a horizontal forcevd7 z1/wu0fv f 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 vcr1yj9c1l wsfl20 1 yyl0ivo- e3ekjga4)k)v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist and cycle y40)yck1-k2f )vresj1joe3 0 wl 9gl1ivlyacare 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 lga;p*vzcj (g )b qi-l7ength 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is tq v*ipj;-c(gb a g)7zlhe 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 hervi:xc 37codvn7h -tz0xe+e,r 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 tii ,czev+x:77nhrdcov- 3ext0ghtly against her body.   

  You are designing a clutch assembly which consists of two cylitihdnl7p-xdhb: 0y. +ndrical plates, of - dh 7+.lnt:bxdhpy0imass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 rolllhl nc;k)w+8u 8;qljgs along the looped rough track of Fig. 8–58. What is the minimum value of the 8ul ;g; j) qllw+c8knhvertical height h that the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assum.l- kvt4 u2 sonh7v;eo yckehzd ,,*.ge $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutionp q3cs-90ia4 qzd/1/n i pyxols as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m./ qp0soia/iy-cx 4qznd3l9 1p (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