A bicycle odometer (which measures distance traveled) is attal:uc5pwl8ni* b qz1)nched near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wulnlc5i1)zw q n*:pb8heels?

Suppose a disk rotates at constant angular v c:off(k ,y)f mpi6d.j/3htbvelocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular vef./ ff, )iodvtymhb 36p(j:k clocity 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 m02 4hncbzyt de,2:lr value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a m,,1gabz g 9qp7fd r9klarger 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 tjjf :46; ygy+wywnyt,gwj9 k;o do a sit-up with your hands behind your head than when your arms are stretched out in fr:wjtgj4;jw,9;kn yywyy fg6 +ont of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wm*;b hwl5gw2 q a2ra ,g7v rxl654opcwheel 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 large front sprocket? Why*lr amo 2, vr2q5g bw4wlgcha;p56x7w?

Mammals that depend on being able to run fast have sloymj i pgf4d46s-nre: ;kic( tlw7z65ender lower legs with flesh and muscle concentrated high, cl 6jcm4 754dlposwi(- :eyin6r ktfzg; ose to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) cas;6ojsdd+5 nw*d ssmyw n-.7(vr yx.brry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero? If the net to(gyy qoerh:2 heiqh72cg);q1rque on a system is zero, is the net force hg g:yyo( c2217q;q)ierh ehq zero?

Two inclines have the sjmn-ceh 3 w0 t5fa /+i)f2ripgame height but make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed0h/ aiwp ritm+c53)fjn e-f g2 of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from restn/9.dl1 mnz .f a*znrj) down an incline. One sphere has twice the radius and twice the mass of the other. Which reachm jlzn*9n/z .ad.rn1fes 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 mas i wbzyfm+xv7cgfd0( 53qr a54s. They start 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 total kinetic energy at the bottom? Which hasa z(35 vxmrd0igc4 y+wqf5fb7 the greater rotational KE?

We claim that momentum and ixb(w( ri,n djxxzy - 7tm2+th1 xd(b3angular momentum are conserved. Yet most moving or rotating objects eventim 3x( twx,n(b yhz7 ixx+1(t2jd brd-ually slow down and stop. Explain.

If there were a great migration of people toward the Earth’soiww3x;z zs 68 equator, how would zosx8 wi6z;3wthis affect the length of the day?

Can the diver of Fig. 8–29 do a somersault without having an r;co .rd g(b;fa:)s1ozru-jv y initial rotation wu;cr.oaf;-or jr 1v ds)(zg:b hen she leaves the board?

The moment of inertia of a rotating soli,l* cq8hbyw0/awvta v0g:bl + .xux 7bd disk about an axis through its center of mat+wyb8w,cbxlu. bq/a *avh 0 07lg v:xss 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 rotating0l dksbjx e(;bb 3/x.d stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angulaj.0xl(d xed3bk b;/s br velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the h,0l:xxpk* j cdq w3:zsame mass. However, one is hollow and the other is solid. Describe anqpld3z0wc:k xxjh :*, experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points westq gurs*ad*3iq bb2u*8vv df7co1 sh26. In what direction is the liubsiqvaq2dgdh6c71os*f v8 ur 3*2b*near velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are stand4smruu q: bv1 q 8ufc,;ugh+c8ing on the edge of a large freely rotating turntable. What happens if you walk q:fquru8muc8,c+gb vu;h 1 4stoward the center?

A shortstop may leap iamt.8m prdw) (nto the air to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotatr w.tmpma8)(d e in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of cd5 fybi27f 73spli qdgc..da9-jkj n+onservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second pf -7acj3jd9.7.lbsnfgd2i 5kpd iq +yropeller can operate to keep the helicopter stable.

  Express the following angles in radians: (xcoy )nl+be19r gf5um qc2-;sa) 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, using th,ycnue3*t*b3m6y 2vv a83bibsbo1 vm e information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen oe,*ob2a*68yc3mbsvyt 1b v 33imvbu nn Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 k: ;s :pg,38i:xbkln t4sgqofim from Earth. The beam diverges a:oq3gnfs; bpix g4:t,k 8sli:t an angle $\theta$ (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 a4qt/m d ;u8xjgc.9e8 qcjxh5f rate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is eg m 8qtcq/f.8hj9u;dx4 5cjx the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the ballp )d-yf7mik/x3.jzi t makes 15.0 revolud)iyz f -j.pk7imxt3/tions, what is its diameter?    m

  A bicycle with tires 68 c s ni) qorjth,5h;8lk,m in diameter travels 8.0 km. How many revolutions do the wheelso5jr)htnh,8q i,l ;sk make?    $rev$

  (a) A grinding wheel 0.35 m q35pc0 8/ lv wvbjl.bzzs)m :din diameter rotates at 2500 rpm. Calculate its angular velocity in )3v z5dsl08vwzqbm l:j c./bp $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution xk54u/*rimh.l- udxw oj1-p9g dd n2sin 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from the center?u ljin p4g*ddkx-/.9-w5osurd1 m2hx    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angulawy 5 qb0zx+d-bqgl5 rr1pa -lb.v (qc,r velocity of the Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speewgujq 5q;m.3g.tb + l x6.ayqkd 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 centrit+ k.da,;ce *jbyij 7sfuge rotate if a particle 7.0 cm from the axis of rotation is to experiencet, jkci y*+d7e.;sjb a an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center 0z* ah + yy5kci.3raepfrom 130 rpm to 280 rpm in 4.0 s. Determiney *5rp ya0a3ikzehc. +
(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 radiuq:gry;umt-ib,9la7 ns $R_1$ 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 My.vy)e lb)u9foon, astronauts aboard the Apollo spacecraft put themselves into a slyu)f .)eb vly9ow 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. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from a.v8t,x6hyh29fat -pelr/h+ygq ( qf rest to 15,000 rpm in 220 s. Through how many revolutions did it turn i yah./8(6 v +9fyxerq2 ,qa -hpgtfhtln this time?    $rev$

  An automobile engine slows down from 4500 r5d nix***q0cwdv rt0o pm 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 flyinh/tlbm,v 8cgz9 *gshos2i b 89g highspeed jets in a whirling “human centrifuge,” which takes 1.0 m *c z 82gobh9m sbvi9ts8/hg,lin 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 f4td2p mdzjtf2/j+zss (wn 5 rqr-4fg :rom 240 rpm to 360 rpm in 6.5 s. How far wil5sfg2fd( ztszt2-q n4d 4j:j pr+ /rmwl 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 It turns 150/mzka*(l2is*7 h bx jnl )lx(v0 revolutions m* i( /hns)lavxk z7b *ljl2(xbefore 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 revolutionsb/u v,5 w(fwsg*n5bot as the car reduces its speed uniformly from 95km/h to 45km/h Theu/5bwg(,o5 twvsbf* n tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car 7;734u/ v jva gt+ xs2;epm mpfi(g n9xwdp7zyreduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.80 nx eaxpj w(yuvm+smp4vzg3pf g;/729 ; td77im.
(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 m49ru/wxiv ;aix b-,5u owus1weight on each pedal when climbing a hill. Ther ubvwio,u- xxms5ui; w91 a4/ pedals rotate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on w9 mg;ko8g+lzm3trx / ((ioot the end of a door 74 cm wide. What is the magnitude of((rl ; iggwm+83otmoot k/x9z the torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fig. 8–39. A3w43 uih.j ixlssume that a friction torque ox 3l.wi4j 3huif 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are att66r sh207+jy x0b5exk wggzhzached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod 6 7+yz05hr6bjegzwxhx g2k0s is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an en/p+:xnf0 m dhcgine require tightening to a torque of 38f+: hp/cd0mn x $m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of radius 0.648 m when .ot-e4y6phni the axis of rotation ytien-. 64p ohis through its center.    $kg \cdot m^2$

  Calculate the moment of inertia ofw7 h p3t)7ubiimr ;ot. a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub cwt)rpbi 7ot h;i.3um7 an be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in a 9 2jq r/hncnneybbw+ad9u;6(c)-, mjdi 5 yfc horizontal circle of radius 1.2 m. qau,n6hwcfj )5ij2cb+ r;y d -9( yb/ne 9nmdcCalculate
(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 2p9maq/qs iey17eta( constant angular speed (Fig. 8–42). The friction force between her hands and the clay is 1.5 N totae9 yqpam7/2e 1 (tqasil.
(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– shec61d8x/ ; xm8watp43 abhdeac1xxwm s ;tp6/88out
(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 -9nl rj6wwtb:+ne0 qpotal 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 cylitpyu3 .a-g cl:ejh;h0 wypy8txtg ;. /ndrical satellite spinning at the correct rate, engineers fire four tangential rockets as shaw0 3 jpyye .pg;t -;./cxtgl h:ut8yhown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each 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 8.50 cm and8;6uvv eg5mp ej v(6m5 xl.hfdm.hwm ( a mass of 0.580 kg. Cm mwvv5xle5hmj( 6medu(p..; h6fg v8alculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerating it from rest j hhem;)8oxjjpo x4;(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 o 2 jrfo)gse/gpm d,*2un a small hand-driven merry-go-round and is able to accelerate it from rest t2rp )de g*go,fus2j/ mo a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,30oic t*ne*3*t qi.,*sdgfaf6(q5cmmi 0 rpm is shut off and is eventually brought uniformly to rest by a frictio ai.6s*tmcmi,n5*3*dffq g qe *(io tcnal 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 o/d4elj* nve ,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-kg ball is vbr-z63fxb 87hwx2;m u5zli r thrown solely by the action of the forearm, which rotates about the elbow joint under the action orbrh 6f5m ;7w uzv8z 3-libx2xf the triceps muscle, 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 con d*8jg5q)-kmiuetaav ubh azi5*7z/0:s7 wj nsidered a long thin rod, as shown in Fig. 8–46.s 75tvabd k niih z:aj8-u*um )ejq7agw0*z/5
(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 co nym jbhj3e3:f0bhno6 r+u7nmop8:)q nsists 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 fry ;.3hdfrvyq-om rest within four full turns (revolutions) and re-yy.v3 qh;fdr leases 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 iht3 /;ctn 3cuby.5dqm nertia 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 a to-a 3gxqzop gdj-)(cue2, p0f drque 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 kgro o82)nju)c-vyg .sh and radius 9.0 cm rolls without slipping down a lan))s-8nr.gyj ohcu 2vo e at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Su+zd9 qk3)bp vh9bj ;fpn as the sum of two terms9 3) khp;fvbbp9+ qdzj,
(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.g5pa0qgu 1:j qr )mpy/ How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cq:5g0m) qa/ppy r1uj gylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls withou jen 3j)+92hajftp-ys/ kz- qft slipping spj2tff)+z hjnjy 9q/ ake-3-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 verticallybmkkofk7svpx, dl4 .m3s-92 u on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of th2.x4 v b3m,kkfs ml-p7k9os due 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 rotating on the end ofwj lb(x-bw .;c4vmk9f a thin string iw-v9jc4x.flm ;bwk( bn a circle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angu a5wcem,y w/5wls1 ,aalar momentum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotating atec,a1/yl amw5 asw ,w5 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 a:. 4 fnuo;s) 4-nog,mmenuao at his side, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases to 4g mn; a,u-om u):fsoo4.enna 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) ,ugw:7zdv;wtqs v8wko f96 4jcan reduce her moment of inertia by a factor of about 3.5 when chk;vu8s6vjf zg d7o4qt,ww:9 w anging 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 rate2-tiwy1o4cz. l xdp.v6l 3cuzux d1n* from an initial rate of 1.0 re1yzpcl v .u3n6 1dz24 *tiw-o. uxdlxcv every 2.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis through its cenntq qh-ut7+6 t7o xqw,2j5vsd ter 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 wheeo6qd u5wt-2jt+ ,vx7qst7q n hl. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater5wih7yplearz 80 0knhn6v n.; 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 momentubu/j/ y57lslum of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inehk s 6 xegz4-ti(vg5 ;7koo(orrtia I is dropped onto an identical disk rotating at anguoi ze -kxggsv6h(o5or t(47k ;lar speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at ts2 .vm2.yh:x n; afemn+meb92.4 $rev/s$ 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 center of a rotating merry-go-round platfooyn*y 2uux8q4 rm of rad82 xqn*o yuyu4ius 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 of0n aikad4z3i6 0.z4d 3a i6na0ki8 $rad/s$ 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 eventuallp*8n,v(hzl 5/3lsbmvm ak8vf ;m 8kgiy collapses into a white dwarf, losing about half its mass in the pk5 ah v p;nvm lmm,l(zs8vgf/8b*i 38krocess, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (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 involva uurd zoc4c99v0xde +oyb* 4g28pq1ue 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 lqgm -cuu .81d$ 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 ro7ye0 ;ck irw;9s zlpa k)o;0zbtate freely without friction. The moment of inertia of the )9oez;sp kzla ki c;;07rby0w 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 diamet; ljfn7 xl t;v)apm7.eq6fkl7er merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia;mj;x )fk. tf7p a76lne qllv7 of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of the rope lyingwwl1q7h vqsuk-/v1 8x 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 behindswu h x 7qkvqv8lw1-/1 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 alwafvj zqtd9+ t*txal,( 8ys faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, tr*ttvq+d8zjf 9ax(t, l eat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 1oo iqe r9t:1 pyn/*k.(lr*i1s s l3kvjah/+ze 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 cylinder of i4: t6p9- fms4dplnhuk g i5d9radius 0.20 m acquires a rotational rate:i n-pt f9d 64hm5d9p4su ilgk of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mo b3t:b x4kbykc73i2cass 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 conser2ik:33xcb ykot4cb7 b vation 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 t7c/vwb p03w qi ub (u0r(tan:nhe 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.u h)zp:7b9 q i2h/bmbt0 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 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 cp2z9ibuhb7/ h)qm:t barries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollut x, .jmqn-gl8gion 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 without needigqjx -lg 8m.n,ng 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 sa8 h,(c*rxz fan $H_2O$ 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 nw esgo4(6 ad*2k6xu ka horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., we ignorebyg3 5t5w/3e ulw1m ck 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. Assumek3 cm55g wwu31 yet/bl 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 standingarp+8h1, js1 crcchlr,yc*e) vertically on the floor, and we wan)rch*8ch p, s cc+yrare 1,lj1t 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 h

  A bicyclist traveling with speed v=4.2m/s on a flat roa(c,um (egm6 tsd is making a turn with a radius The forces acting on the cyclist a,( gces( umm6tnd cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.5 m and bez0 .syq1 o5 o1tsi0fctgins 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 to achieve this feat, and where does the torque comtc t01ofys1os . 5qzi0e from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thin rods, mart)u 9d5qs/ q+v hiw0lking reasona) hsdl+/i qr0vtqw9 u5ble estimates 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 assemi4.wtw9w9/iseorci) f*yf r.bly which consists of two cylindrical plates, ofi 9f) sw /9yf w..*iewrrtoci4 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 rh0 v 8r w:wrwreu*s-4bolls along the looped 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 reach the highest point of the loop without leaving hub0*4vrw wr-:e w8r sthe track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not x s7w:kdt;c,tassume $r\ll R$ h =    (R-r)

  The tires of a car mak.himp8q1(g os1s 4lzt e 85 revolutions as the car reduces its speed uniformly from 90kmizsqg o1.1pls48h( tm /h to 60km/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