A bicycle odometer (which measures distance traveled(jdc / ebi9djzn+-8 gb) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bijdn-gd( jz+9b ceb i/8cycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a poins5de2tmwvu3 a d7wv fu9l8+o(t on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential accelera(w+9 dfo3lvs 7uavwde25u8 mttion? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value ofc,z(g(oazs, 2 4giotjflfoe8m a; ;:u the angular velocity $\omega$ Explain.

Can a small force ever exert q dm17 hc jn: agpf6o1ul50n/oa greater 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 yos7( e uz,8gklrur hands behind your head than when your arms are stretchedl (7us 8ezk,rg out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheeqg7w73znymvj 9+p+r .g53wcuon1o qxl 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 sprocket or a lar935+.cy1rgj +37 nqw 7pvo mxzngu qwoge front sprocket? Why?

Mammals that depend ny l5y ufsezq5sd0o /.s/uh51on being able to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the bas5yh/.oeqssf0/1udzns l5u y5 is of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig.k1 hpd*q 4hdf. 8–35) carry a long, narrow beam?

If the net force on a system ikr(8bxy 9 n4 dp)xr9v)tqb7jn 0n p)wnts+t)cs zero, is the net torque also zero? If the net torque on a system is zero, is the net force zknqn()s cb7r d)4n)jrx9 tp)8pb9 0tty n wv+xero?

Two inclines have the same height but make different angles with the horizontal16e t rcvd82cq. The same steel ball is rolled down each incline. On which incline c6rect2 d1vq 8will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an ikjy3*cjfa- ujlabf (hj6fk h2;,uvx,.rz1b.ncline. One sphere has twice the radiusu6lkr-3.(habb 1 z,u cjvj.2;*jfj ahf xyf,k 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 bottom?

A sphere and a cylinder have the same radius and the same mass. They 1+x e4wkmdf,pls )t1lstart from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater to,lwt+xmp e 11)fls4kdtal kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are coemk1;2g/nj (,:9zt / hdetf yds;kpal nserved. Yet most moving or rotating ob dtytsmfl1k9 ;pge /a/kd2e( ;,jz:nhjects eventually slow down and stop. Explain.

If there were a great migration of peok7:9ji sze8 gdk coz2/ple toward the Earth’s equator, how would this affect the length o72kdzec:/ jkgz is98of the day?

Can the diver of Fig. 8–29 do a somersault wimvd c7y3;x(lnx xd a):f/v*mnthout having any initial rotation whe:;xnmvx)fd yl/n*vd mc37 x(an she leaves the board?

The moment of inertia of a rotating solid disk about an axis through ityrljtzu9udtt,t6a :55q-u 4 fs center of mass iaryq uz9j5 u:u 6-5 t4lf,ttdts $\frac{1}{2}WR^2$ (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 sto 1x hkgzln lzc-l4 fzy)zwq;6c9( o+b;ol holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will you c zbh+xcywlz 4l6kz9(qng);z fo l1-;r angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identic.*duetweio ap;l . uwcb292n 1al and have the same mass. However, one is hollow and the other isieau ww 2t.;1p.o2 *ldn9bu ec solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points we ,ia2u jgxuj16-u8pip st. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points 61uxpupu iaj ,-jig28 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 rotatinor 4l:jr;bp fvt 6fr8ewyb30(g turntable. What happens if you wo3b :jybp4tv608(;rlr ffrewalk toward the center?

A shortstop may leap into the;e 6 cafwa9fefrx.832qkp9yt6 x. jxj air to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If yoa28wya pc efexrt f.kq6 j9fx693.j;x u 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 d/7b/bc6pa gyconservation of angular momentum, discuss why a helicopter must h yb c7bp/g/da6ave 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 in8v89iu fae; 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 coinci.vh ,o79rz ugxdence. Calculate, using the information inside the Front Cover, the angular diameters (in ra.zug or97 x,hvdians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 6-vp ilzpot: tv17a::h-jcw 9i6kecs380,000 km from Earth. The beam diverges at an a7w:c ephzp-al96si:tk:c io j1v-tv6 ngle $\theta$ (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 b1.oip qdp6sb. : wf8g6500 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 down? bw. :iobqpf.s d6 8gp1   $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the ball makes dd)h wc-hnvo/yqx k 3v .6k3dv9+q i+r15.0 revolutions, what is its y+wr vhxq i+vkc) nkq96do 3h/3d-.v ddiameter?    m

  A bicycle with tires 68 cm in diameter travels 8f2mx6maea y9kd-9f j* .0 km. How many revolutions do the wheels m*9f9ayxjea f-kd2m 6make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rp 64gpwv vn/qbk rh2q yed+/(5fm. Calculate its angular velocity iqhv2r( +w//e6npbvf5 gyd q4kn $rad/s$ $\omega$ =    $rad/sec$
(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 revolucgk,cc k.4uf*i/9mn;6jmpatd:-f0 6q vud rqtion in 4.0 s (Fig. 8–38). (a) What is the . 9 qt/mjai; ,6fc c4fkv:qd0dkcngum6r* -pulinear speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orbit aroun1 sgdo4ew0f(wirk*0zrle ;) o, (qn jfd the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of ./b)g(dz mxhh 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 cmyg r(v8* ;05njx;gobig 8ft)nlz 1ruentrifuge rotate if a particle 7.0 cm from the axis of rotation is l* ; yng5rzojmu;nxt0gbi)8(8rfgv1 to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates ywxv+*ra/gx 4uniformly about its center from 130 rpm to 280 rpm in 4.0 s. Determ4ywax* gv+xr /ine
(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 radiumnn k *m s5yp-wyd wxgkc4:*7,s $R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboard the Apollo spacecraft put themse4k;y 8e q;v+l+kavy8vq-d mtclves into a slow rotation to distriv4c 8 q;mv8k;dktlqe ++-vyaybute 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. 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 uniau:ivh gn i)k t*-/t:bformly from rest to 15,000 rpm in 220 s. Through how many revolutions did it turn in this time?thgv):tbkiua i/- :n*    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 6g s9pk1ue. bw reu/vp3- s(xt2.5 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 ps++gn 5hmxv*;r*- wutd((twgm e3 qcfor the stresses of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before tpg( (hs+dqn rmvueww*x+5g3m* c;- t 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 /b d( k6v1v(clms sk c4plr +ie:s.z+qrpm in 6.5 s. i6lk+.(c /qzr+ vv(ssl sbp:1c4me dkHow far will a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It43i e2;7kiai esj lx6n turns 1500 revolutions before it come4nj e7i2isx3 ie l;k6as 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 spe( i+hg0jlsb v5ed uniformly from 95km/h to 45km/h The tires have a j+0gb5h vsi(ldiameter 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 ma.jmbi;wvgg : x(1av 8eke 65 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h:g;.m8(gvwaxvie 1 bj 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 hefy2,,w yhnj c1r weight on each pedal when climbing a hill. The pedals rotate i ,,jfw1n2cy hyn a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N b/+-tnxdqf 8kq;xt ;rvuoz19 on the end of a door 74 cm wide. What is the magnitude of the torque if 8+ku/bo x-n;xzr q1f9tv;q dtthe force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fig. 8–39. A-j f7rjg.eave1z(; p hssume that a friction torquj.rzv j-e 1pahf7g;e( e 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 xvcdcw p*, (p+qf;:k twhich pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the ma k*fvtd+ppx:;qcc,w( gnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engink m++kj9mwe jv-s9(+m 6s gwaie require tightening to a torque of 38 smm g- +kwj9w(mv9i +ajke+6s $m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of radius 0.648 m whe23 2xv iubx6p0ez1 nzcn the axis cu1e2 xz2in3bz0 x6v pof rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. The ri p zosy*(+6 2hrhqb;q adkxa6gs0feu(;n bh.1m and tire have a combined mass of 1.25 kg. The mass of the hub can be ignoreu1grpn aa s b+ohef*6z;0qk(;b.d syqh(hx62 d (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin,yo8bjcfd9jx *40l2wi 2k4na y light rod is rotated in a horizontal ci2yf2 ob 4j8lj49d0xywncaki* rcle of radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping cutt-i gq mh k/-ldf* +z557dea bowl on a potter’s wheel rotating at constant angular speed (Fig. 8–42). The fricti+d5/-zeimdc 5lk htg-fq 7ut *on force between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the ajwpe l-e3 eauud,: x62rray of point objects shown in Fig. 8–43 abou j3eu26:pa lux,ede-wt
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen atoms whose t: rnbb hg917yf k jk::6lcfsu3otal 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 ratoj* b .-ncjj5qe, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force o .*5j njcbqoj-f 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 radief7 k9a(rf4 a.s rhd9 hx +i;3calj8dpus of 8.50 cm and a mass of 0.580 kd4rad h ( kjx7+9l8;ea9cp3fsf.ir a hg. Calculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, acceleratpck0)q )y2xqu .p:kqs52txuc ing 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 mei 5sh8l77c9/xtczrk 4p6 u cbcrry-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 kg) sit opposite each other on the edge. Calculate the torque require6cuck5z/4rc 8s9tx7hp l bi7 cd 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 off and is4f(vz . (vr2hxqyc +nx eventually brought uniformly to rest by a frictional torqzh2qvf(.ncv (x+4ryxue 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. k0 as -hi.pvghtz60)n8–45 accelerates a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kzcvxb -49 .mv60o wye:f zsh( 3cbqlx/g ball is thrown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps mlfby 6:c0x . 3vo hq 4/zevzs(bw9-cxmuscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be coheslket8ykj2lpn 7 44v)p.g 4nsidered a long thin rod, as shown in Fig. 8–46l47etgl v2.4k pj h) 48eynkps.
(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 m2u14lya 6qqm5 ftj :txasses, $m_1$ 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 (reyn 1f0nlqsm4qb mz9, )volutions) and relbl9m4fqnzn s , m0q1)yeases 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 of ine)/iwn //vz(i)pkas4ee )u *i e (fpppsrtia 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 develoal un +xer/ o,1kjyct lh:3/*1wt isudd+(6kkps 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 )8s,9a sihr hu5glch:down a lane at :h8ar 9i5c) sguhls ,h3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to th6gq)f* qjbb 6.tetg-ge Sun as the sum of two terms,bbgg* -q.geqft6 tj6 )
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius of 7.50 m. How much netwa +,:odc:56r,sqjn7rxh2r e z:lhob work is required to accelerate it from rest to a rotation rate of 1.0 w5ho2x ,r7:h+rsrz:6nqa :o,e cbjl d0 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass zgs h; emx 51ks 3w)u)qjmol:91.80 kg starts from rest and rolls without slipping9 xe 1uo5hkm)qzl3mw)g;ss j: 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 vert 4xjge)k*yz8 xm :gq q8o3 o)niics71uically on its tip. It starts to fall and its lower end does not slip. What will be the speed o 3 mu:17x sqnicjkzgi*e8))4gox 8qyof the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball roto ))xqmi+9,kmb 4hguj ating on the end of a thin string in a h)4)gub + i9mkmq, joxcircle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform rnqqy173) vhs9jx+)7txikln cylindrical grinding wheel of radius 18 cm when rotating aji 93nq+ k1v)t7lxn 7rs) yqxht 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, h,fnp 6k2mzy7 on a platform that is rotating at a rate of 1.3rev/s If h m,76pz2ykf nhe raises his arms to a horizontal 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 moment of inertia d4;n 0 b1yjer5hjhn*p by a factor of about 3.5 when changing dr jynep1b5;4nhhj0* 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 al 5n ap:7c f(akcdnn-c*3hik1 n initial rate of 1.0 rev every 2.0 s to a final rak( nak13: clcpd-nn*57f haci te 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 center at a freq e caq:fo6z+3z;f7k tquency of 1.5revqt ze6;af:+ 7focz3kq /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 sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skatx9 g sabc 3;5ro i3.bcbc3q8luer spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum of th)pbjh( f xo.bty x1.ed+wu 4fpa0f*l+e 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 inertia I is dr; 40+-y0o -gia hqiwduudxp/eopped onto an identical y 0p/ hd- u04w+iqaxgei;-o uddisk 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 u+xq3i nn2vk iq n0fg*/;:xnf$rev/s$ 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 cente h,- ,nzt iq,*s.eyv8wsoh 2adm2+1 mndf sf4kr of a rotating merry-go-round platform of radius 3.0 m and momen-y,.mih qomefdna2s,s 8f4s t *1d2z wv+nkh, t 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 of 0l oelsvdqz(0e.5gi78 .8 7qd (8vz.l lse50go ie$rad/s$ 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 colu6j)f z.znll 2lapses 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 angulal. u2z)6lj znfr momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in excess ofyfya xac+j d* 5r*a89i 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 r*f5 9-1abbezzp j u,w$ 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 qx(wrhbxjlh88zvf1+ ,f 12fjwithout frictionvl8f,j fbwqx8h 1hrfj(1z 2+x. The moment of inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diameter merry-go-roun1rxa/ (jilwo4.dhzkr: ;dx 0a d turntable that is mounted on frictionless bearings and has a moment of inertia of r irwkzl:/; x0(a jdoa4hx.d1 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:u. mq2u kmi)bx1x gi2 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. Whatu bmq:1mki)2 gix.u2x length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that th dzcpzxp*i-f+shh6 66.oh bp1sw 8;lre same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (aboup f6 zc*w sr hzdbshi; .8-pl1+6opxh6t 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 fmhrtj 5,uv8m8 rom rest at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform lb lt)f*n6 kquh-v9p-cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the t)l9*h6-qkn ufpv- btl orque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of t+khq x,3/2xh afit 9umwo solid cylindrical disks, each of mass 0.050 kg and diamhx+kmti,ha u 2f qx93/eter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of th hikc/fy,91k- ( tsadre 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 wi .8z iec*i)qajth mass 8.0 times as 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 r 8*q.iaicez) jadius 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-au9 /zu9e3zm1vff-ac pollution 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 of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km wit-z1 m9u c3fezvuaa/f9hout needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates htz(qc q :(d) .zl4ia q6f:pviq;y0 mian $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal surface at speed vcl,3iuyv ;2w x=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 igmt+,b ;o3::dleg9f vz *dbt swnore 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 acceleradvt;f l+:wmteb ,9*g dzos :b3tion 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 radius R. It is standing vert r)chl9r da,;zically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where dr9l ,)zrhac ;h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making bl44qay p/ be4a turn with a radius The forces acting on the cyclist and cyclpq b4/y4eba 4le 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 rocn if53l j 48mbz.h3wbyk 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 torqueb m54lhn3j yiw.z3f8b required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cyliwc 6r 5w)bfc,8u, yucvnder and her arms as thin rods, making reasonable est ,,wu6 fry8wb5) vccucimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindrical plates, of 4bd /3 9f1vcpy3atckx 3pvb14x3cc af/ ty9kdmass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the looped rough track of Fig. 8–580yz8t x:p -b:tkxl3tu . What is the minimum value of the vertical height h that the marble must drop if it is to reax 8:uz30l:btp x tk-tych 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 o k njwhwwh)dpvn fz+dp/ acy-a4 6 04-4wv;r+assume $r\ll R$ h =    (R-r)

  The tires of a car makeg02(k 8rr i raw6cjg48-rbq hd 85 revolutions as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was tkg4-2(g6w8jqra 80 rircdhr bhe 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