A bicycle odometer (which measures distance traveled) is at; w+*fq g)aa3ilw e mr,)k7meh4i;p jftached near the wheel hub and is designed for 27-inch wheels. What happens q+)ej;k far) m 4w7,a lii w;3p*fgmehif you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at cons7rgm4y -lrie /tant angular velocity. Does a point on the rim have radiair-le7 4mr/g yl and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the angul8u)pa. tfuid1i jv(9psze9jlq 7g-i 0ar velocity $\omega$ Explain.

Can a small force ever exert a greate1difovc*1 oqv.75 liy r torque 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 with your hands behind your head than wh),ttwr8)w kws;l r 4yqen yo wq )yrlts)k8 r;,tw4wur arms are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear whycoql)7b,;l- wi/e bs9jdlcfo5) ji5eel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front -dcy)blo5 c;7ij)el9j5/lbi,sqwfosprocket or a large front sprocket? Why?

Mammals that depend on bez;np3rax 78m. bws1 w5vpyw -oing able to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. mwnyzp8vp.;bw 5w -a3xrso 178–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–4 fi0c9 tr;myz35) carry a long, narrow beam?

If the net force on a system is zero, is the net torque also z2 g.dnzxj su1zlu -:p*ero? If the net torque on a system is zero, is the nep:1*n .zxgjsdlu 2-uzt force zero?

Two inclines have the same height but make different angles with to;b n7ne (dr0 o)z)uoshe horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at t;)) sondoor(nuz0 7behe bottom be greater? Explain.

Two solid spheres simultaneously stark0xc :v5-ecnlv y w6k,t 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 kine0v c klnk,wy-5 :vec6xtic energy at the bottom?

A sphere and a cylinder have the same radius and the same mas3/ ydcc :uc ahy)l5se5s. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bot/ccud53s5 hcyl ay): etom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are consz)o ,-lzmq m p-ahc5v4erved. Yet most moving or rotating objects eventually slow down and stop. Explain.m4p oh5a)q m z-l-z,vc

If there were a great migration of people toward the Earth’s equator, how would 0 f4gb cho5d5zthhgd 4zb5 0cf5ois affect the length of the day?

Can the diver of Fig. 8–29 do a somersault without having any initial rotdxbo;obftajz.jz6n 7vd)m2 g86m 1g9ation when s o dznt8gb)vj1 m6og6d jm.29azbf;x7he leaves the board?

The moment of inertia of a rotating solid disk about an axis through its ce9, /nqp5l5dpwlah; fis1sqidn-4a- unter of 1pi-alh n,dsa5 5 dufis-;l4q pq/wn9mass 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 j/y5ad3czu c :on a rotating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay zcucy: /3j5adthe same? Explain.

Two spheres look identicaq5xeo) 0uox v+vd*o,xl and have the same mass. However, one is hollow and the other is solid. Describe an experiment to determine wed0 v)oqxvxx 5+ o,uo*hich is which.

The angular velocity of a wheel rotating on a9-e:yln y4err-e*wih 9av(a x horizontal axle points west. In what direction is the linear velocity of a point on the -l :ee*nay-wa yrvh9x(9i4 retop 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 ed+(n ag2k d8wah;fr*r yge of a large freely rotating turntable. What happens if you walk toward the crr;aadwkn 8yg f h2+*(enter?

A shortstop may leap into the air to catch a ball anc1l9o lbz ee2q2(4p jy aut,-gd throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his qo ul9t2(-,cy1 bale4j2gzpe hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discusspjge*b- )u-tw/j(l-i + z qsgd why a helicopter must hau td+gjleip(--- /)*jq z gbwsve more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles i*3lff +du2rxmn 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, ntmt6 :sk08 5gr/lroh using the:lo 60nt /hkr5t 8mgrs information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380 nh l2hly+/s0 l-pir4a,000 km from Earth. The beam diverges at an anglilpr4+/n0lshy -a hl 2e $\theta$ (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 acs;t/7 5vm-gt zv+rzqt a rate of 6500 rpm. When the motor is turned off d7rsq 5 tvctv-zmgz/;+uring operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m hw1)0s ao phral- f*/vto another child. If the ball makes 15.0 revolutions, what is irh0-a *1a pfolh vw)/sts diameter?    m

  A bicycle with tires 68 cm in diame9luyr:i84;0 dl5yzi qznscg4ter travels 8.0 km. How many revolutions do the wheeyl 54nqs;8uigc zilyd0r49z : ls make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Cal)g il7n (nqo(lww4::gwk zf,dculate its angular velocit:g(kfz):qnw4 low7 (dglin, wy 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 comwj+,r cgr sf,j4v oe/ p-lu,zk/jie06plete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of +/zek ovp f0 wsrl6-j,u ,jr, g/cjei4a 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 orbit a6v* hdk bihn:8round the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear ss1b nb4 h a,lnzrtgf*2f:-pm 0peed of a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.0 cm *x2khk(sb y9g1l r.axidq5 p* from the axis of rotation is to experience an acceleration of 100,yhsqr1xi*xd5(ab. lpg 9k2* k000 $g’s$?    $rpm$

  A 70-cm-diameter whe5scblask. wpe ce1, +us9tk/+ el accelerates uniformly about its center from 130 rpm to 280 rpm in 4.0 k,csp5a cw+9us.sl1 tb e/ e+ks. 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 radiusipppqm33t k1,5/au /fekd m9j $R_1$ 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)ti.rg7 ,dhic spacecraft put themselves 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 revolutionh)ir7 i.d tg,c 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 unif4)q+ 3ufoei,g0q:z ffuh8uk oormly from rest to 15,000 rpm in 220 s. Through how many q0o3uufffe:,o4z hi8+kq g)u revolutions did it turn in this time?    $rev$

  An automobile engine slows dowj 6x*17 ja 2msqld+jbxxc j,z6n 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 whirlingl,48x yamwy jq-1v rp8 “human ce81rya-,qwvyjx ml8p4ntrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 360 yd,vqnwa h ;7:rpm in 6.5 s. How far q:n;dhvayw ,7will a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off wh l h +mmd9tmk /af.0(i+wik5qaen it is running at 850rev/min It turns 1500 revolutions be+i5 tmh0 .mwamkq(9l+k f ia/dfore 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 uniformly fr/ lsfs/3ic(cwvpg0 b,i rq4 4aom 95km/h to 45km/h The tires have a,/qfpiialc(0 4s sb c3v/g4w r 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 65dkmslx96z;* ;3m( z qe bd g/xzb70fyv revolutions as the car reduces its speed uniformly from 95km/h to 45kl xzm*qkvxs(mg ;d;fz6 y b3b7z9/de0m/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 rctzt7r9.)f .sxk;ej all her weight on each pedal when climbing a hill. The pedals rotate in a circle of radius 17 zfrt 9jr) s7e.c x.;ktcm.
(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 en zb+go5y 7ytk3b7ycy; d of a door 74 cm wide. What is the magnitudz o+;t3by775ykbycgy e 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 a2xjgo63 a *c wyfx+.no. 7qjlzxle of the wheel shown in Fig. 8–39. Assume that a 6a.x3 lxz 2*ygcjfq +wo.njo7 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 aj+ebgg,46y ph ttached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this s,ye4g6b+g j hpystem.

  The bolts on the cylinder head of an engine require tightenin7 lge* cltlriuxbc8++g3nl 9a0k i9ic ;x6oz-g to a torque of 38k*zgi i u c 6+ inoclx38ebt0;gl9cr+x l9la7- $m \cdot N$ 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 ofr+ g 0jxyf;aw3 f.d tn;8;zp*ccet)p x a 10.8-kg sphere of radius 0.648 m when the axis of rotaz;tge8 p3trd; x)+ x;.j *cwypacfn0ftion is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in d,h jti rpz mam5p sjv8.-so2+:iameter. The rim and tire have a cosh8t5-pzrjv+,2soj :pm a m.imbined 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 thin, light rod is rotated in a horixa.cmvjm(u mlw2: e3/ zontal circle of radius 1.2 m. Calcu.le2x3mjmc / mvau( :wlate
(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 at constantbwb+dcm-6a2l 41svq)kvsm;a angular speed (Fig. 8–42). The friction force between her hands and the clay is cqlm1-a6+ ;2k )vvws4dba msb1.5 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 object6uvme 2o./*l qks7era s shown in Fig. 8–43 abo.m* /sr2elv7au6eq k out
(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 whos evg,vc*5u2dzs *he-de 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 cozc8 ( n gnd9ww5t18zzvrrect 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,g8nz(czwn5 z t8w19 vd 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 cylinder with a radius of 8. 88a jjvc( .q )8ykg;io3sewvn50 cm and a mass of 0.580 kg.wy8sv8qkvin oj ae; g (c8j.)3 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 batx l.rwbd,dh) (, 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-drive5yx x +q5 7 enr4ghru/x1i+ ue0+kdnken 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 o+150u ixe457yu e+rkkx+qdx engrh/n f 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut offne kk.2g2we bqo s+(w; and is eventually brought uniformly to rest by a frictionan 2g we2sw(o.bek;+qk l 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 accelerav,,nvj;u (roates 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 th8pt;;piz5u6 fy 0hb-al*mm y ie action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. tz;f;68yli* bpm-0 5 umya hip8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, as show4k+sh4zbbdqf;j4 4(g hfr :ugn in Fig. 8–46sj;bz hbr4(qd4g 4 4ukfgh:f+.
(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 ce 4uo s;5hm9hgqr) rvd v;1ki1onsists 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 hammer from rest ioe 5nb p 0vjvy8p90j9within four full turns (revolutions) and e8 vj0p90pvyo j9bni5releases it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a moment ofa; gln8a nd43:9 8x4 ook2 iyz1bfjkri inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops:0 mqxwc yoj458c ah y7)ytl*u 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 down ztxm+ gt*q t+d6 36vlva la+x*vd 3+6tlt zmqv6t gne at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy ofsx(* xd:r oibp0w5)fe the Earth with respect to the Sun as the sum of two tewo d(fierxbp*s : x50)rms,
(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 zih d7u, ;o :h/djbh:qof 1640 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution pedhhdouh,bzj ;q 7::i /r 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 and rolls withlzvor1, g:sb+ out slipping bl, r1v:+so zgdown 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 velxg:v :w. +t sgl+8zexrtically on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end ofv:gl+z l sw+:xg. 8etx the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a ,vtv o:5y1 te h -drzy1nnf(1x0.210-kg ball rotating on the end of a thin stro1t zv,hndyte5 f n(1yxr: 1v-ing in a circle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular /y apg z7m: +la63ts11 tmlihqg59thbmomentum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotating at 1500 rpmtl1 pl9i hgmg5q6m tz /1t+aabs73h y:?    $kg \cdot m^2$
(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 :py b(bs(wex/j 8;xtuetry 4 1rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–4ty b 1x;8t(puew/(esy4:bx jr8, 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 c 66amnsz;x qla,r)s5 shown in Fig. 8–29) can reduce her moment of inertia by a factor of abc6 m,ss5lrqa) 6za;xnout 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 speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an inio e2jv u/dleafy 1qvv:6sm -59tial rate of 1.0 rev every 2.0 s to av sv1l:j5ou2- de6y9 vmeafq/ final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis through its )f1l;sd1fg oa/j(zc i lcym((center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of m(a )f 1d(oy1cl;mjzfcgls(/ iass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinning atb,( qsvjw06cfgc41kb 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 owx*h:,:itsu9 q0 d zqxf 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 cylindricasl( fs5 6/kmc3y+n:8y w3cj8 jukbmyxl disk of moment of inertia I is dropped onto an identical diyx fk 83cjs 853 y( :mbywk/n6m+ulscjsk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turnseyds-fk je--2 at 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 ats84s hx+dmg -zboy9 i4 the center of a rotating merry-go-round platform -do9b s+ zxms hig4y48of 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 a0 q+1bxiby,0lcklf -tj6ez; l ngular velocity of 0;lb k t0eclq ,lb0zji f+1xy-6.8 $rad/s$ 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, losing * fpu4nai*99ja0alan 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 ratla4aapn9in* *0jf au 9e 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 involv2:a) 1d aj4rv*tpfhof 8si i;fe winds in excess of 120 $km/h$ at the outer edge. Make a crude estimate of
(a) the energy,    $ \times10^{16 }$ J
(b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 $kg \cdot m^2$) of radius 100 km and height 4.0 km.    $ \times10^{20 }$ $kg \cdot m^2$

  An asteroid of mass 3 u zo4b9ctt9h$ 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 pl,fg ,pg 0pou; s1a7swqatform, initially at rest, that can rotate freely without friction. The moment of inertia of0ups,sgq7apw o, fg1; the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diameter merry-go-xn( :11xd*x f4fzcjfg round turntable that is mounted on fri 4n:ff(cdgz 1 *jfxxx1ctionless 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 rope7rs01jn c(j jnd ;g1 /av,jvv+gzrz (p rolls on 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 distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of ro 1raz(0cdngz j7gpvjr+n;1 ( v j,jv/spe unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces the Earth. Determ ooo i-pg7vd e5unnrnf4k. p-)b 2zz(,ine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the lud.nokrngivo25-pzz7 f )bp en,-4 o( atter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a ratq3kvw+8 cfh((6b gccpe 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 s:wyod8als5 z 2hape of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular accelerationz y5sa:w2dl8o. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of to9 g+xysi9 gd;wo solid 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 relea9g; +9i gysxdosed from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, h t7q-jr2 *pp+iw8 eobyow is the angular speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 time 0kf sibbp8-k4z+ yo+es as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutroei-b0k8 4pykofs+z b+n 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-polluthqiv1gx f-8 l3ion automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel off1 i-ql8vxgh 3 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 illustrate 2v1eo git6+fds 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 horizontalfpq ,f7;gt pqi4t v:-yrc6xmi j a*1-7 ohu0ne 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 pivot freely (ice msw2 vfs6f. 6ku79y.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, determine (a) the angular acceleration of the rod, and (b) the line mu7kfs2cy.s9 w fv66ear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on the floor, and wex l:.:3drk:v9pzj)8uc wpcj40foh sk want to exert a horizontal force F at l pr9 .xjjf s08hpuc::kvzc dk4)o3 w: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 i(fq/ny: zrnzd x79s59 *opz purts78m s making a turn with a radius The forces acting on the cyclist z87n(r7:/zyumr5n*9tdox pq9ss zp f and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock 2b 5ceh-nz9mgac1p7i into a sling of length 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 the torque required gh259bme 1api-zn7ccto 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 gqg/+keat+ 7ch-tg d+arms as thin rods, making reasonable estimates for the dimensions. Then calculattg++at7kqge d+ g-/che the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindrical plates qey* r (vu90 is.au5)nm3ukyg, of muukn (0r 3q sga59iyv*e) yu.mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 loo 9ngg8y6r: wvyped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach6 :gnv8g9ryyw 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 not5za *t b 0 fvrm0o3.zzeon2hl:o8 e)ur assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uniformi)x;s3 yqkv f ip/ze0+avx82sly from 90km/h to 60kme3q)/+ssz if i 2yk xv;vp0xa8/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s