A bicycle odometer (which measuv/yc f6p;cq ahpkva 23mb28 vmil)/p0 res distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use ;f 6lhyp2 p cav p/qiv 2/8k30cvmb)mait on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on.t*dlu:2jb( )wh esm p the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential :d (j2h um .*tplsebw)acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single v awlyr- xv26g fw70)ipalue of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque6qoy9pq )kaamgi:5w ( 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 w*ogi( 8 :uzf8rzfq5o with your hands behind your head than when your arms are stretched ozzi:( 5o*f8rgof8q uw ut in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheelxp(9*pgmw xm :nr/dq* 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 harderwmdp:(np*/xq9rm xg * to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able t- 74lkipf: fzzo run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the ikl: f4zf-p7z basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narr.nu u;) q,xubfay- 68,vhwvo.p z)k kmow beam?

If the net force on a system is zero, is the net torque als)x,m9id c xg1h p5zx/zo zero? If the net torque on a system is zero, is t9 15z, hd/i mgxp)czxxhe net force zero?

Two inclines have the same height but make different angles with the hr hf+p+1t p1h8yr.ubuorizontal. The same steel ball is rh.y rp18+tfb pu+ur1h olled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an incli p.sj5z)s5at hne. 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 gss.htp j a55)zreater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius am(ba:c,n dmt7nd the same mass. 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 has the t am,(:7cnbdm greater rotational KE?

We claim that momentum and an)35ac j2 jjdhngular momentum are conserved. Yet most moving or rotatid jnajcj2h3 5)ng objects eventually slow down and stop. Explain.

If there were a great migration of people toward the Earth9ydnis37 )wiv ’s equator, how would this affect the length 3dwn7i 9v)is yof the day?

Can the diver of Fig. 8–29 do a somersaults:3.e.l3lapnwb)j lv6w5 e pb without having any initial rotation when she leaves the boallwv6 :)ej5w. s pa3enp. blb3rd?

The moment of inertia of a rotating solid disk about v3r2 fku irdsj84yjr+8fad :/an axis through its center o34d:fjk+8vfjd /82s rriya urf mass is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holdiyo 6iu**bug rs4(0fs-fkh yc-ng a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the sa sub h(f4usfg-0i o- rk*ycy*6me? Explain.

Two spheres look identical and have the same mass. a ntf uv8m+piw-leku8;:2 yj0However, one is hollow and the other is solid. Describe an experimentvny;08- uejlai:ku8 t2pw +m f to determine which is which.

The angular velocity of a whb z vbrz (q40/xs5e0jfeel rotating on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the vzx4 e0z sf0(rb5/ jbqwheel? 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 mv8a8ldg ;/q iedge of a large freely rotating turntable. What happens if you walk toward the center? /8vmi qa8;lgd

A shortstop may leap into /7zrymtzqd d(2st :(a 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 rotate in the opposite directio72tq:(szm( ta rdyz/ dn (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discaxvfmqr3k3b3t6 0y 2 zuss why a helic3q0 y avfb t3zkmx26r3opter must have 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 t0mcl)z 4d e2+( 9zz7hxee jmmin 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. Cbd57xq7 /ks m5bcq r*valculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon,7 *c5bvd brks 5/m7qxq as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The)st5bm9 fqpu5u pt35o beam diverges at5um ts5pfqt o3u9pb) 5 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 a rate of 6500 rpm. When the moto a((:t yxpjn+k2e.t szr is turned off during operation, the blades slow to rest in 3.0jtsek(x.atyz(2 +p : n 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 to another child. If the3uh 8 c8) d5gltz ntp503ufqjo ball makes 15.0 reopgzl30h 58c j5t uq)ndf 8u3tvolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. Howy2o.c+ z)mkkv many revolutions do the wheels my 2 )v+.ozkckmake?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculatnv/kz v)r:nw0j/ 4ga ee its angular veloc)wzg/ rv4 0/n :evnkjaity 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-ro5pr-bwta0 (*x lwo)7 yx4x,j tbhgp2y und makes one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from the cent5y(lg,bawpy tjw-)r 4pxxo* b 02ht7 xer?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of 3 j(dp2a;xio/t/rh eb 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 speed of ahx-:*vxu zgli6lw(o 5t.v* jj point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.0 cm from the axissn lm637lh4me* o*stc of rotation is t734s*lmn mlec* ho6s to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its centex28 2tyy9/vbvxdg-;6x uap v)prz -xc6 c3ywir from 130 rpm to 280 rpm in 4.0 szwup x3g y6 byvvp)6ayi;v/-22dxrt9cc8x -x . 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 radiuc5 jyoz2l w*m;nyu/t,u:7ucis $R_1$ 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, aow7fo s8 idp:3-7mx ppstronauts aboard the Apollo 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 revolution every minute during a 12-min time interval. The spacecraft c7pxow pfsom37 p8 d-i:an 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 rexc91ekt r6qiyi::w,v st to 15,000 rpm in 220 s. Through how many revolutions did it v96iw1rk: tixc :qye,turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calgv)q6 /f srr:rculate
(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 whirling 4 3-hpjyouo +w * pou.mbwk8f2“human centrifuge,” which takes 1.0 min to turn through 20 complete revolutik8oo . u+mpb3* w p42fwhoy-juons 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 rpm in p,39ec ; pf +sys3ztzh6.5 s. How far will a point on the edge of the why39,p zc sefhps +tz3;eel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 1500 revhet(gccwumya ) )9kag8xs m3y( 8qh1gw 0/2xi olutions befhy8t 012y g)mm9qwks()ug3x(8ge cc /iax awhore 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 red2v6cum +hmb5euces its speed uniformly from 95kmbvm5e6+m 2uch/h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the carxw 46pf( ) k2t4vj qucpqc)/yfam* 0 bzju1;es reduces its speed uniformly from 95km/h to 45km/h The tc)stea6)4bf / jf4p*02cx upu wq;vyqj( z1mkires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each po5;fbi3rwro4x- ru c3 edal when climbing a hill.c-woirf 35xuor b 43;r The 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 the end of a door 74 cm widerp f gc.j;yq ef+n-rd6bc 0n,f4.9 xvx. What is the magnitude of the torque . fv x jcpr-0 f;.6nx, c9nfyqbgde+r4if 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 Figoq 0lg8xz k+3n. 8–39. Assume that a x+kgqn038zlo friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a massless rod whichyxji /ard . 7eecqi)-) pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and directionx/j c7e-dy )qe.a)i ri of the net torque on this system.

  The bolts on the cylinder head of an k t1er k;o:9fnengine require tightening to a torque of 38 f t1e9:n;rk ok$m \cdot N$ 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 f4)c9 ivzv1d tsphere of radius 0.648 m when the axis of rotation is ) 4t 1f9vdvcizthrough its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle whe24ojbnz9 dsk *el 66.7 cm in diameter. The rim and tire have a k42s j*db 9zoncombined mass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on6zfp g65s 8ilh the end of a thin, light rod is rotated in a horizontal ci5lhps6zg6if 8rcle 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 a bowl on a potter’s wheel rotating at constan2py+kaq8 w ljjz3o*f+t angular speed (Fig. 8–42). The friction force between her hands and the clay is 1.k3jy8pajwflo* q+ +2z5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point objects showfid ,jwzj.q:;h7m iz ;n in Fig. 8–43 a7 iqw :zjj;,f;zimhd .bout
(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 total mass iss7za dw u1uk7+gou;t( $5.3 \times10^{ -26}$ 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 uye1wbnzb :7 ii49 h6psatellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady h94pb:ui7 y6n1wiebzforce 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 unif9qvr ,cd0hdv4q.zq / form cylinder with a radius of 8.50 cm and a mass of 0.580 kg. Calculatczd qh0d,4fv 9r.q q/ve
(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-wf:wn9a m8a sgib hkgc9m 4 lt:9cf9+ rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven merry-go-round and n mmi1-jpgfbqov ) 9 fti/xg)t5,1 th(is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acm1()bmf5ht,9 ig f -vt)itg/oxqnjp1celeration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotatingokdt(7n ojtkg x ;*;.9spq-nq at 10,300 rpm is shut off and is eventually brought uniformly to rest by a frictional torque of onk;d pj sqtxq*g; -ot.kn 7(91.2 $m \cdot N$ 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. 8–4xga p)ru-o5e, 5 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 thrown solef-+v pgb md(p5mm9yramd2(5 ily by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10r2 9mi5fdm gdpavb( m+-p5m y( $m/s$ 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 lo ocl lod*.9uq dp32yh/) *uibang thin rod, as shown in Fig. 8–469ul */2oc doqua*py3.dih) lb.
(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 masses0 sghvpq ye3j4j*ud 7lr m3s/0, $m_1$ 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 hammerl (g1kf ( m7n 5dlly,b- +gj0obzyumm1md +(ij from rest within four full turns (revolutions5(y l7fd(mj- b1mkz ui,+ old01jnylmmg + (gb) and releases 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 lijai xs4ut( o71s0wf *d4qpo(y((hxof 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 deu 6tdlc)d.(kt velops 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 -v4r dup,eo3h without slipping down a lane hdu4 ro,-v p3eat 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respec0afgs hznld-uf0p w2jv b)-..t to the Sun as the sum of tsflzbnjuh)-- fg2wpvd. 0.a0 wo terms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass 3ih4 vy -a: 9ed/dnhohof 1640 kg and a radius of 7.50 m. How much net work is required to accelerate it frohn:-ahdo eh3 diyv4 9/m rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls qr nfd1u:yd epn7*v(;o,p,a m without slipping don mope,n(pr;fq:7,yd duv1* a wn 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 verv1 tiop/b m8hrzo) h8*tically on its tip. It starts to fall and its lower end does not slip. What will be the spot v*b hh)1oz8ir/m8 peed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular moment ;; jw*p4w kanjwr7p7fum of a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an angular speed of 10p; pj*;frkajw77 wwn4.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grin9y-1qpmy;lr :qs 0ui )gcdme 0ding wheel of radius 18 crml-qmiy;y0epdu: cs) 9q 0g1m when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform thkuxipr f;lxv1k7f x + 7,glp;;at is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position7;7xur;kvp xpxilfl gf1,+ k ;, 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 he ;kgw-*w 0urfhr moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotatigh0 wkr* ;ufw-ons 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 increas3zu k,3w) 0vnre j2s9bsxca.r e her spin rotation rate from an initial rate of 1.0 rev every 2.0 s to a final rate ofc 93r, z2k.rexan uws) 30jvbs 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 x:nc /1d 4afaxits center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kgxf4d /caa1: xn 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 angula/)4bz62bhcel )ogk nzovi71 bc 8zki-ofb9m+ r momentum of a figure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum of the Ea b0ybw9w dl ncd9l+6hw3gyx3) rth
(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 r9l0qmm 8x+s xof inertia I is dropped onto an identical lm+ 9 xxs0qmr8disk 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 7a6gwp.9gvtc9mdia v ,n+cq :.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-gob k9p7j i,)qzhh o6ia/-round platform of hpo)h7ai,q/9 z ik6bj 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 uyi4h2 p;qoagz2vl,68 g yb*uan angular velocity of 0.8 guhiy2g8 yu2bplq,;v*z a46o $rad/s$ 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 about half itna1a t b3: w0rpqi;wrqe*8vl4 s mass in the process, and winding up with a radius 1.0% of its exvtqr q8 0r *bwpa:;ein3wa1 4listing 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 involve winds iis-md5mulec gcwp)8 ( ifuw0*e7 cg)o9m46y n n 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 9+f5k9 odlh zv$ 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, initial aqmh govs1dxe070,)lly at rest, that can rotate freely without friction. The moment of inertia of the xhvegm70d 0l,a1 osq)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 diameteqi hp/wxr- 37gr merry-go-round turntable that is mounted on frictionless beaw- 3 gqxrp7h/irings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rovb:j(s (6ufc -j0nadslls 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 rope unwinds from the spool? How far does the spool 0vsjsanf-(bj c:du 6(’s center of mass move?

  The Moon orbits the Earth s tp;-pfay7z6i znw*jcyv - os886 (kqsuch that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular mom8 ajn7vtoz z8sk6isp --*qypfy(c6;wentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a ratdeh0d: 4r ( xt2rnuam7w y-c9ve 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 shap)hf9gdqe41lg n )lvq .z-al k*e of a uniform cylinder of radius 0.20 m acquires agk nhg ze)f 4q-al9l *)1lvdq. rotational rate 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 massvote o0 ,fe: s9n8)qk4ryz l/w 0.050 kg and diameter 0.075 m, joined by a (concentric0f ,zs8l/ n4vote 9:yqk) owre) 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 angulazcu - v sv5v2bejje6uk,w)* h l:hi,4ir 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 wizu; -1x x6wiskth 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 radius 11 km, losing three-quarters of its mass in the pro1u-k6iw zxs;xcess, 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-pollution automobile is6nngzlkz7wa2b-/x s dbu h,91 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 needing a flywheel “spinup7z b2z9d 1sn,nwgxk-6h luab /.”
(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 illustratesjy g l/ h;v+6vpl,y:onco-m3 m an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal surfanxqzt+)eqh- 7 e /oa:mce 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 frcp5t+ p 6ornl7eely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. Thp n76+opr5ctle rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on the floor, and * wuwyl * sjzn35w x w(m+hrgl-q5d5*iwe want to exert a horizontal force Fj hm wi5ud**3g5n+lwlqy(zx sw-5 w*r 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 d 9rbf1 um)fj9g 17aaai- r,neflat road is making a turn with a radius The forces acting f71 d a9mu,r)njf1rbaa 9-egion the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.5 m ark2b 8uo c)a0 gsyeho9msq;): nd 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 to achieve this feat, anaur s:9h8;)mb0 sooeckyg2q)d where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thinq: lykmp ilz )-r v qnek-v0w1sf1/+b8 rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinninm1zv+q8)q: s lre 0l-1fy nbv/k kwp-ig skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which cogn xivirg/;+g, e(9cfnsists of two cylindrical plates, of mass fgegi ;i/rnxc9(, +vg$M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 l7fuv.tp y4+q1d ()umts qy3isv nu65 cooped rough track of Fig. 8–58. What is the minimum value of the vertical u s)q(smtuv+vui 74y.dt1 6qn3c5fyp height h that the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but0w :p6hrg vl8lgyj 6ze0)vgc: do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as th7c0aw9e tc1ovg;*byd e 21 ycbe car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angul ybe72cy c*ce0oa; w1v1bd 9tgar 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