A bicycle odometer (which measures distanc-6 pw)h/fb.3i 2yja-ngothfc 3fllt 3e traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-incbt-yc 6h3 .tln2) fh o/gff-ja 3il3wph wheels?

Suppose a disk rotates at constant angular velocity. Dn -0g.79j taosm2 es6 igtui4eoes a point on the rim have radialmin0 ge7e46. tugi9 tsj-a 2so and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single valgm68py cx4r*;x dcdwew4r1k( hbl h//ue of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater t8c; e 1uxr m iy3zvc-9)xup.uborque than a larger force? Explain.

If a force $\vec{F}$ acts on an object such that its lever arm is zero, does it have any effect on the object’s motion? Explain.

Why is it more difficult to do a sit-up with your hands behind yopu;sf2x x:y(l ur head than when your arms are stretched out in front of you? A diagram may ux2xs:f; ply( help you to answer this.

A 21-speed bicycle has seven sprowdf .m p65lb,/ dpo)ovckets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprock5.6pdpb l m vfd,)/owoet? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slenscte2 :i3 wd7qder lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of masswsqtce73 2i d: is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long4t7sg(3jgb 4+tv joto2 eq. av atx2fbg1 2s:r, narrow beam?

If the net force on a system is zero, is the n 1/ffs0p5xb 4y3)mjzwa hsc;het torque also zero? If the net torque on a system is zero,as/ bs)y j;zf 0x53hcfh14mwp is the net force zero?

Two inclines have the same height but make different at0s 4gg 0u,maongles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Expltm4u0sag0, g oain.

Two solid spheres simultaneously start rolling (fr*n+ .zkmo0 bjgom rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which hasob.+zj g0*kmn the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have tougit:6j e twu1/ r1l. i9)qeahx7*yvhe same radius and 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 botttwi7je . raxu u*v1i1qog6:t/h) ely 9om? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserve,0kz5 mlbz -ens9)ty ld. Yet most moving or rotating objects eventually slo5-y9 ,knbze)mzl l0st w down and stop. Explain.

If there were a great migration of people toward the Earth’s equatorm: 38kslzmg 7m, how would thism ls zmm83g:k7 affect the length of the day?

Can the diver of Fig. 8–29 mrho9+uvhp n66a 1s 3ydo a somersault without having any initial rotation when she leaves the boroyma hv9p661ns+3 uh ard?

The moment of inertia of a rotating solid disk anm;oqhnfdu)nk/ga;x2 2)6 y pbout an axis through its center of mass is uaq )fp /y;mnd 6nx;kh2g)o2 n$\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a.c,mbgpg n0k1 2-kg mass in each outstretched hand. If you sudd1p 0.ggck,m nbenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However,*/ 48tjnf)zy hn- qvvj one is hollow and the other is solid. Describe an experiment to determine which is whicqnn jf h4j8/zty-* vv)h.

The angular velocity of a wheel rotating q pf+f57l mg-s am37zdon a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration popm+afflq7dm z- 735gs ints 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 rotatinh y jg+rx25ee6g turntable. What happens e+rg65e 2hyjxif you walk toward the center?

A shortstop may leap into the air to catch a ball and throw it quickly. As h+y(5zyw xxrr 0e 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 z y5x+w(rryx0 opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation v u-i7*sumy3 oof angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second 7u m*si-yv o3upropeller can operate to keep the helicopter stable.

  Express the following angles in radians: (a) 30 r j h(18vwqxtadg r2s4wbx9;b/:0z et$^{\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, usinxpbs5eo6 oxs1 t1.rbwsd j) 79g the information inside the Front Cover, the angular diameters (in radians) of the Su xobsp rot) jbsx57s9 1w.6d1en and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed rf5cu- 58snt lat the Moon, 380,000 km from Earth. The beam diverges atr u55nfsc-8 lt 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 motor is turne*y7+l8welum8f 1 v ql;u+malqd off during operation, th8myv q+l*u+qfl1w8 a;ueml l 7e blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the ba17( 9gyvb1g:f5lounbikx 5b v ll makes 15.0 revolutions, what is its d9kbn5: f xu (b51givvolgy71b iameter?    m

  A bicycle with tires 68 cm in diamete6zxde6 c-nk v6r travels 8.0 km. How many revolutions do the w-x66c k6edvnzheels make?    $rev$

  (a) A grinding wheel 0.35 mn,. zp.q hn6 ud 8fvn9fesh2.c in diameter rotates at 2500 rpm. Calculate its angular velocity is6.n8 v.h,dc 9uhe2nfq fpz.nn $rad/s$ $\omega$ =    $rad/sec$
(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 in 4.0 s (F5.0bux3pa n1he pg g:9/1)q smqodgl wig. 8–38). (a) What is the linear speedm)aebqo3x9 1 5.lgwg /dp:gun pqh0s 1 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 i z e:hrys5 25/qzm,cjnts orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speea(;k deka9ofv;h)ebos) w wiogn( /32 d 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 a*wj m2nd3p2s yl6/h1 rke2a ccentrifuge rotate if a particle 7.0 cm from the axis of rotation is to experience an acceleration of 100,000 /sa2k h1ane *m6c 2jw lr3dp2y$g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates unifcdfe j g4tb;i7z7n+y )ormly about its center from 130 rpm to 280 rpm in 4.0 s. Dg;n7 7jdt+iey4z) cfb etermine
(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 radiuw c/iv8q vc ) 00oros:gkqqj+8s $R_1$ 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 spacecrafe:r+r 4xus+vz v(.zgu t put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of theirz+ u rx+zr( s4e:.uvvg 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 rest to 15,000 rpm in 2r0 zfqbps*rk * 86(qcz6wsg3n20 s. Through how many revolutions did it turn in tckw6r 3rbp *g0z8 q(fn6*ssqzhis time?    $rev$

  An automobile engine slows down f)b f(em/6qqifrom 4500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested f hiw( 3x8y7i9cj,b2-pb qz xhyor the stresses of flying highspeed jets in a whirling “hubxby3j9y(h, w hic i8qz7-x p2man centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diaqwyffpf,d e h,;cq-w0n2 j+-v meter accelerates uniformly from 240 rpm to 360 rpm in 6.5 sfwvf0dp+ qf;-q2-cjwe y,h ,n. How 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 It 9 z2kqhh:wb nf )(up/tturns 1500 revolutions before it comes tot n2:f/buk wph h)9zq( a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed uniformly fr*wo d4df0s+ lq:r ob:vom f+ld:b dv:4w o*rqo s095km/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 bmxv -j+ uiz;t4qzz5h+sipt5 3hh-+ r65 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h Thezz+v p5himh b5 3+t qrx--izs4hu;tj+ tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each pedu f*i0 nzq88na) 9nqteal when climbing a hill. The pedals rotatuz enn8 0*q)f qni98ate 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*khi1wni p3w 3 N on the end of a door 74 cm wide. What is the magnw*w3ik1hip n3itude of the torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fig. 8–39. Assumezkgz7b11p k 3w that a friction torque of 0.1b7 3w1zz kkpg4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a mr +zm 4v2fc/ky rwq-1qassless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the h y2wqcm rv-/qfkzr1 +4orizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cyligv pu:g0v 34c1ykn lo6nder head of an engine require tightening to a torque of 38g yp4:uno 3kl10vgvc6 $m \cdot N$ 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 007ees*ki; x067 w 5jn,qcshxm n-rztl.648 m when the axis of rotation ie0i e7kz;0 w6hx5 jsnlstc *7 m-rxnq,s through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bi-.+ ghhg zsmi0cycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.20sm-.ghi zg+h5 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end l 5:mn ksueria(),n j:of a thin, light rod is rotated in a horizontal circle of radius 1.2 m. Calc( :rusn,:5jean)lki mulate
(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 constant -thys2r;iakm ;1v 8 w+z,ud9b r0 ygbwangular speed;wb98 gtizh wym0,+suy;ra v r1d-k 2b (Fig. 8–42). The friction 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 array of point o(n3.v7p1f + 4bomlfgfq aiby7bjects shown in Fig. 8–43 aboutl7 fv1f+7fgq. 4nai(bb 3pym o
(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 massfd,:yq el+s-i 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 cox(+ce,56 0joua/c cc(dee* vuvv vzk(rrect 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 ste((vzv+, 5/ (k6eo jeuvcc au e*cvdxc0ady 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 un z7mbny1*r12m ek:yrniform cylinder with a radius of 8.50 cm and a mass of 0.51*b:7y1rr2mnnk e zym 80 kg. 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, acceleraqph 7j 8s9 4r88 upsi;gr3qhjhting 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 merry-go-round and is a(5lpw2k(t9 imcqwu/u : pzxh7ble 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 lm9z/cu:qxh7(ti2pk wu5 p (w of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 gt q h21hh/xne;t 0exr9yu +n6rpm is shut off and is eventually brought uniformly to rest by a f9uxe+ ;et/ hh h61g n20nqxtryrictional 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 acsx +sbrx n6fp9e-,(sc celerates 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 s7o-;tg/vc q)+ta- eumz e5ocgolely 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 mttq ;v/-7o uge- 5cagoe+zc)to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, as sh0ycir8 h0g /pxown in Fig. 8–4h/gcpy80x i 0r6.
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine consists of two masse mqa: *ho(hbo;r(n-kvs, $m_1$ 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 turnstgkmw :f)pfl9 6 lnk*5 (revolutions) and releases it at a speed of :*fg)9t5wk n llkfm6p 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 inerti*m)j a 3kxann4a 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 )iyj- d ;bcht*.nw:inrque 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 withou+o 7 ; glmbma(1/wz9;lkyc ldmt slipping down a lane at 3l /; +dylz; cwkog1ml ab(m97m.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic enei8xy;egf wx3c .1a xq9l*h.aq rgy of the Earth with respect to the Sun as the sum of twf 1. gx xw9h8qa.yi;lc 3xq*aeo 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 of 1640 kg and a radius of 7.50 m. How much 2omdsvn9 e:pe 1 e2kcoqf28c/q ;f6cjnet 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 cylin;esck18q2o /o cmq vj 26:fcfnep2de9der.    J

  A sphere of radius 20.0 cm and mass 1.80 kg r)e jfg y6i)vb7s8m5cx.(yn /6tuutt starts from rest and rolls without slnsgjtt5e)u.trvxb)(8myi7u/ c y6f 6 ipping 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 balancem(iriy; 7. ef2h4zcn rd vertically on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ; he4r yn2rz.(ficmi 7ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum o,n6v9 e;j ajy ;3)igf1sbod badn:k 5df a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an ang n9,; js)d;bk1ey ib5d3ajvdn:f6 a ogular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8 7 -r--cda(qjx5iqgb n-kg uniform cylindrical grinding wheel of radius 18 cm when rotating at 1500 dx7 q 5qab-cgijr--n( rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotati d -zadn: gdyw:c9yy: o.+qb8zng at a rate of 1.3rev/s If he raises his arms to zdqdw d +: zng .yay:8b:o9y-ca 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–f5kr f4/ p fxdgliy*ur)-6w (l29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to-kfi/p xlyd5u fl*rr6fg)w 4( the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an ini5e;h q wyyk73en;ida8 cvm1c/tial rate of 1.0 rev every 2.0 s to a final rate of wn71emqya ;c vy h8c5 edi3k/;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 rg dx4sc tp(wip2q 0+g+otating around a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 + 0gcptiw4qsx( pg2+d m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinning a90)zqolp 1p u oxz8+dnt 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 m sg0 ;utmn 2apyl5ic1 9jp1v7pomentum 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 inertia I is dropped ontw1kk +vm1m8pdo an identical disk rotat18w kvm+dpmk1 ing at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at l.4f6jv7( n ggu,gv*etw ou/ t2.4 $rev/s$ 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 ce8skqm7:6 wq4vaok) o wnter of a rotating merry-go-round platform of radius 3.0 m and moment 8qvk647so )ma:o kwqwof 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 w8zr rs1g7 c 0zpjjuqhdv 5/h7)ith an angular velocity of 0.8s vz 1rc/jrujd)58g 0h q77phz $rad/s$ 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 colla azo(psde3 nt(n1tk11pses 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 angu otk d1tnsa3 (z1(ep1nlar 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 involvjp :elgsmui00t ;j7t*e 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 : xy36+.bc oco2qas zb$ 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(;62n3bp/:rl stnm m7j )u* srjcoudf without friction. T2f mouj t:nnl;(/jb cpur3md6r)*s s7he 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 dia32;lumue brj0xpmr 2rz(* -ermeter merry-go-round turntable that is mounted on frictionless bearings and has a mr0eurub-le;j32 r*p(z mx 2mroment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end o*mrf w-2q7w n2u4vkamf 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 spoolnr w f2umqa 47k*w-2vm? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always- gmsd2*pxn kfqz2c(7 faces the Earth. Determine the ratio of the Moon’s s*pd 2mzx-n g(qk2 fc7spin angular momentum (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 res5w p u)tawv+t/t 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 cyliocu.q.jz08cz ar if 66)gim:sbu wl/4nder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular a:ga8sb uf i46) mc.6ulzq 0cjz.owi/rcceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of maw a. fndz*ztovs b--g+qrw;j/gr 0 4p5ss 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from wr/a*4+ pzbf s-no-r5tggjz.;wvq0drest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speehlcd)e ug.-gd tk(f.(84 1jtoyzlc b1d 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 sla8gb uk7;;k1n8m fy iize of our Sun, but with 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 to78;lakiu f k ngb;18ym a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for1xnttu .ndd *b3a,gtf c ,kj2, it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 14 gct2,1n tk3*x,f btjd,du. na00 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustra93g9 viw97 ria8;t0g btacpth tes an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal surface at speed v=3 ,ivqvml909ja .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 freem-vsi.5-gd6 hkew1s /xk4 dftkz jxk.ei: 33h ly (i.e., we ignore friction) about a hinf e-d kh mg3w5.3 jiidxke41k kx-6/.hs:vsztge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing verticabzh; 9/ent8n+s-yejt r/p) d jlly on the floor, and we wat+ )hnpjzrj/d;yt9n 8 /es be-nt 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 flyxciil .f0zh5) m8,w iat road is making a turn with a radius The forces acting on the cyclist and cycle are the normal.)xiy mc5zf ,hiilw 80 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 dpt.a1r( k5t uk-nsc(length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it fr1t-k( cspun5 ( .tardkom 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 come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her g9o0mx)4 bkfvxhn .5sdvc9+ barms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with ox 4)90gfh9kxnvc+bb. v d5mosutstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindrical plates,jco3h+n;bncc6 z)s*a of mj )3cnnzco*sca h6;+bass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 tvb)l5bgt oy0-rack 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 the track? Assume tyg 0lo) b-5bv$r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assume z6 nz) zo0vxh+$r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as thpkx wmc;it9x-(oakq8 h)9 s2h e car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each ti9ot mc)ih-x;9skah(x2 8w q kpre? $\approx$    $rad/s^2$(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