A bicycle odometer (which measures distance traveled) is attached near the wh .-y56y7 ;.d*nao x5jb/xycs cftfcj seel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?f*-/t6do xynccjy5f.bx .;7sjs5 ay c

Suppose a disk rotates at constant angular velocity. Does a point on the rim(f ojvhi-eh/s5uv k., have radial and/or tangential acceleratioe .u,jhhiv(- s/fkv5on? 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 des0k v7wc.fx0oo8r2y vk4 rtnu1 cribed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greate)h c n;nm*qb1ar torque than a larger force? Explain.

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

Why is it more difficult to do a sit-up with your hands behind your head xw* bkske;1 a3than when your arms are stretched out in front of you? A diagram may help you to answer this*;s1ek3wb xk a.

A 21-speed bicycle has seven sprockets at the rear wheel and three at t2lhrl 2at 2 hec9blx+nav*y2 3he pedal cranks. 2hyla+2clv2xl r t 3n29h ba*eIn 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?

Mammals that depend on being abl jqb+,-ujji32a sa4xo e to run fast have slender lower legs with flesh and muscle concentrated high, cl-4xj a3ji2 jb,so+auq 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) carrgrq.bl82 2jep nu*e( fy a long, narrow beam?

If the net force on a system is zero, is the net torque alsovkxkq2):q. /e 07ajpb yqnuf0 zero? If the net torque on a system is zero, is the net force zero? vqe k007/.x)n fqkauj:bp2qy

Two inclines have the same height but make different angles wiwazy5cs*ux 3 wfa dp10x9j: g0th the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom bj0spda 3w *a g czx1xwu09f5y:e greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an izh5ataqsf .rp re:,7. ncline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic ene.f s.5parr,qezh 7 :targy at the bottom?

A sphere and a cylinder have n 4d-q(-ngh ena5: lvt.g wb5+zlvi; vthe same radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the grea.l5gz t ;he nqn-gvw+ :(nvbaiv5dl4- ter speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular m8xnxr 5al.:q6kg 5vb gua pee0/g:e,somentum are conserved. Yet most moving or rotating objects eventually slow down and gv5./qxgx6s58le:a: e,kgran up be0 stop. Explain.

If there were a great migration of people toward the Earth’s;xey.w7x l y2w equator, how would this affect the lenxy7y. ;xe2wlwgth of the day?

Can the diver of Fig. 8–29 dcapnde7.1i bprol 8+aqxu x-i74 wh;8o a somersault without having any initial rotation when she lea;owp8.ca1d8i 77rxn- +eqb lxh ua p4ives the board?

The moment of inertia of a rotating solid disk about an axis thjzcu6c j(-op/ma; p8 ioh0av (rough its center of mass ipo(uam0 i cvazpj8/j c6;-(ho s $\frac{1}{2}WR^2$ (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 2-kg mass in each outstre eu;y68 8ckn-0jsloc j 3q,lglot,l a/tched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Expl-6l,8 gyltancje jouskoc30;l/,l8 q ain.

Two spheres look identical and have the engwpow3 3ka,c+)q0k3s , n rbsame mass. However, one is hollow and the other is solid. Describe an be g, npw3, qs3)+no0a3rkkwcexperiment to determine which is which.

The angular velocity of a wheel rotati:eqx9zqs(ug of;qp ) ,ifm*1 ong on a horizontal axle points west. In what direction is the lin z(eqofu ifqo)mp 9g*s,q:;x1ear 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 standing on the edge of aov sy9p s300bz large freely rotating turntable. What happens if you walk towa9svs bo0z 0yp3rd the center?

A shortstop may leap into the air to catch a ball and throw it quicklqeery; , 07f)oyfj7g*iw fb-ky. As he throws the ball, the upper part of his body rotates. If you look qug7y*7ey-rwif)0b q;kof,ef j ickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular mon**3wbdnfz nqg ,r6 ti pk(q+.mentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopkwn nzb (+*ptr,qg*iq. f6nd3ter stable.

  Express the following angles in radians: e q s /.cwvtca0b7c23y(a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincidence. Calculate, u8atv ojnr o9ycjelz5df92903sing the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earthz f9tno9je or8vl2d03 9c jya5.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000h dcu*+0r )rxn km from Earth. The beam diverges at an angl*)h0 +rcxud nre $\theta$ (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 mo eo44f y+eoe y+eud:)ygim*- ztor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blaieo e+e4dy4yuy:e*zm o -)+gfdes slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the ball makes q8 n6c hd ,+2shbe8nr 6b*v9yswd 0lum15.0 revolutions, what is itsqhb8s 8vdn0sdl cr *eub66 2wym9+,nh diameter?    m

  A bicycle with tires 68 cm fwbns14 d+3 zm(ds1gby7(dgz in diameter travels 8.0 km. How many revolutions do thesg fd(wzd1b+d 3g1m 4yz(7b ns wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculagifw+;rm*r * v,0 cs0me tjy)qte its angularc )*q0 jwmryif v;, m+0grts*e velocity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes lq-oy:i6ij976 cqt wmone complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from the centi-qly9it joc6:7 6mwqer?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular veloup8j 9t/ v:bnqcity 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 speed of a poin2 02n jlkx.n9 t/lcpjat
(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 centrifug 9ugz mt4:3etbe rotate if a particle 7.0 cm from the axis of rotation is t eumg 43bt9z:to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel abp zn02sauwc.x:l o +9px87v qccelerates uniformly about its center from 130 rpm to 280 rpm in 4.0 s. Deterwsx 2 zpvc:.7ao 9upn80 +xblqmine
(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 radius xwslkv*70d,8 1joqezbz2i4m $R_1$ 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 spacecra6 qwh:yswxqo)3u : stra8,l3fft 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 18tas wh, lfqr: y3wx6)sq o:3u2-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 uniform1mgg(yv xi5esevc4v :b/,ml 7 ly from rest to 15,000 rpm in 220 s. Through how myvm ixgb7gv e/sc4:e ,m15 (vlany revolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to ; bzb4.mrm:wret8o,e 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeed i+6si a a7x0 a(qdkt1djets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before react sakidq(a0+xd a71i6hing 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 diamet(c-tz 7+xntw 7x(ub yfer accelerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled in 7nx7(+fzbtt xyw(c u -this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 1500 rey)brc)p 3g o1hvolutions before it comes to 3o)gcr)yp1h ba 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 s;g)w8dalbiom ,o38ty peed uniformly from 95km/h to 45km/h The tires have a diameter ofail ,otb ;g)38dyw8mo 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 asxl3rhcfd a/;9 sw1y6t the car reduces its speed uniformly from 95km/h to 45km/h The tires have a dhr63x9adl1/yt ;swc fiameter 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 h;j o/ jt07qj) np;j arjg;egm6er weight on each pedal when climbing a hill. The pedals rotate in a circle of radiu;pjatj ;mgjjr) no0j7g6 ;q/es 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 thio,bp+z6vo sykl1p 7l03 zvo:e end of a door 74 cm wide. What is the magnitude of the torque if the force ik7spzo,v p 30v zlol:i1y6+bo s 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 torq +j7sq+9(kad tqr g-jgue about the axle of the wheel shown in Fig. 8–39. Assume that ks( jj79+tg da q+g-qra 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 en7 g3h:t6uy+ earwecg 1ds of a massless rod which pivots as shown in Fig. 8–40. Inu+ h16ct7ryw a:ee3gg itially the rod 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 cylindeks7acpeuy 1w)..dc5tq gb; )zr head of an engine require tightening to a torque of 38 uqp5 .w cteky 7dsa1).)zgbc; $m \cdot N$ 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 *i:-f)b ec im)exr5pukkf3w a vs8-7tof a 10.8-kg sphere of radius 0.648 m when the axis of rotation : kmfia)bw7f-3 *kpex e -vits5u8)cris through its center.    $kg \cdot m^2$

  Calculate the moment of inptar52tr jn4b2a .sd g;r p5k,ertia of a bicycle wheel 66.7 cm in diameter. The rim and tire havep4 b.g5 tpj,rarr2ank 5;sdt2 a combined mass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in ;qfqn 1;.t:i el/ohvk a horizontal circle oq fniv:t e1/kl;o;.qhf 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 constant angul8dvr/m, 8e: gsroxyg:ar speed (Fig. 8–42). The friction force between her hands ,xd8:gyge/vo8s: rmr 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 inertiiehvb):l (a w2a of the array of point objects shown in Fig. 8–43 ha :)bvew 2(ilabout
(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 ma9zt5j.iz.tu fss 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 satellin/exzxi lw m ba/zaz119ze 08g+bo4zqdv 3.w0te 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 force of each rocket if the satellite is to r3x/ zm .wbdq+ieo91/e1wavlz 8g zn4xb00z za each 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with acbr 4n53bq+1i.p dqmo radius of 8.50 cm and a mass of 0.580 kgbm.14b+ qqdo5rcp3 in. 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, acevxy0rdo6pdd sr8xo60: km/5celerating 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-xan5e mi s w1ou6)e4j- b(ppp+driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Cal-5(p opse1 b)mixpue6na4j + wculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off an +5znq 3/dwzl:3 pqi7fsxmth( d is eventually brought uniformlyz3(wf5 qt:qnh spz/xld 73 +im to rest by a frictional 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 accelerates a 3.6-kg ball at k72f,. anf nhz7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the5g*oxuo c825uidlj d wu35 k+p-vlw4y action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniwyd55gc8uulv- uwo*d2k4x p +o5 il3jformly 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 c /yh v0ychhme/0e c/ 9r3flju28omgr3an be considered a long thin rod, as shown in Fig. 8–46m/ 0ry/3he3/ l02jeho9ucfmy c vr8hg.
(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 +lelzhd a3:v-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 from4ld(hy kavg *( rest within four full turns (revolutions) and releases it at a speed (vk4l ah(y dg*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 whm m232zdem*of 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 devel v :le 4myv,ytqz,a1(kops 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. pxz;1j8vknqq at 5)iiok8+d ;3 kg and radius 9.0 cm rolls without slipping down a lane at 3.3vq;1i 58p +kkqnx z)oa 8jdt;i $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energku;ou ;p oi+d7y of the Earth with respect to the Sun as the sum of two terms,;7 p+;okiouud
(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 y,6ye3yf/g gxkg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.xggey6/3 yy f,00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0t*uhkh2qho (:yg6 o4n cm and mass 1.80 kg starts from rest and rolls without slipping down a 30 ko2y6h4*:tgo(q nhuh.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 vertically on its tip. It starts to fall and(qqjnd 2vlk:b z+4oz9n il /9y its lower end does not slip. What will be the speed of the upper end of the pole jusz 9nbj9:o lqlzdiny(q k+v/24t before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular c( ;a,pa2fwab momentum of a 0.210-kg ball rotating on the end of a thin string in a circle of radia ca(bp;f2wa,us 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindricalu7o6w3 b0 bg jp yd8ke+)c.loi grinding wheel of radius 18 cm when rotating at ygl uc6)+db0p e8bow3kj7o .i1500 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 8 (8 1yep pmv/m;b:cztehja,w-a:asgng at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreasemga:jwm1: ze/h eas-8 b p;y cp,tv(8as 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 Fige fyi .56rjkce 3g68qp. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight posit3i85e y6ke6frjgq .cpion 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 rotat9lu;zeav jv-xv :s.vev;y4vece+0lg-(mh )eion rate from an initial rate of 1.0 rev every 2.0 s to a f4e).+evhy09xe :j;mgv(alevc-u lvz e; sv -v inal 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 cen ht2q0jm;y 7d2o7eos rter at a frequency of 1.5rev/s The wheel can be considered a uniform diry77om 0 edsj2q2h o;tsk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinning at 3.5 btosit (t/9j1i kv8rc..up06ysf q0kg1n.au$rev/s$ 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 oo06 duqtw*u e:f 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 c20koa b r2-c:-qsbominertia I is dropped onto an identical disk rot qa-km- oc o:sb0r2bc2ating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4 xvq44ej28kk;u n 3leu$rev/s$ 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 (s1pcmq1vu+t :ui i34qyn5g kqu:- mtcenter of a rotating merry-go-round platform of radius 3.m:vu 1g(:-u5ttp1qn+ 4q cuiiqk3ym s0 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 angutg: f o0m0w2aolar velocity of 0.0w oma2 f:g0to8 $rad/s$ 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 an1ow 9ys2r- x 2unm26jkdyl/n white dwarf, losing about half its mass in the process, and winding up /ur9y nxyj12n2 6 wn mldso-2kwith 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 involve winds in excesst gv*l;mmg:h2 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 f1/ qi2p*fw1dyn ofl,$ 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 freelyd 9 hmqw awgl1nuvcf8x1n *8c8a zi..1 without friction. Th8wmf*qg c.9 8 a.d81 nv1xu nalw1czhie 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 ob(gw 53lg o8-h.a)w- j o6xp2 qpztlnhf a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia olwhnb o 5phjgqap3g.w x- 6)oz(t8l2-f 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 ro ghus bs)*e:-rxh 7d.ppe lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Firbp-shh .7:gd *uexs) g. 8–50. The spool rolls behind 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-osh wfih; uw,z f.)1r side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to ih; i-)wz.hswfou rf 1,ts 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 rate of zw(6la v2nkd1 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 cylinder of radius 0.20h0 fs297xru6eqn v.c dm a4yzo, v3b(t m acquires a rotational rabq 76 .x vvyna4rehsd90o fz(u t2m3c,te 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 mass 0.050 kg aswkd36 gf ffg;wv i369nd diameter 0.075 m, joined by a (concentr ff6 936gf;wid vgkws3ic) 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 theqcmy;-j)0sl q angular speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.lz+(ds b 6o10z(gsj+ir v0rap0 times as great, were rotating at a speed of 1.0 revdjzg0 0boi( l z(rsrp a6s+v1+olution 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 carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use energy stored iku+gu6t8dha01c9 o2huq4 y m kn 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, umu 42o0qc8dh 1h9+6gu t akkyand 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 illustran(m1d f:,7 ylrzrtjen/ u7j5q 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 + 7x3 sve xz6k+o+fiogrolling on a 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 ignore 6i xp,ysi)be:;l ojf2 friction) about a hinge attached to a wall, as in Fig. 8–5ji ;fy, 2p)6l:sbox ie4. 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 vertico ogndrhie9/*y -2j -cally on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, -/ecoi9 ydho njr *-2gwhere h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a turn with hsq4gauj ; 8:a t2zezc1+gbn; a radhg4:asc 1q; ; 82az ng+zbtjueius The forces acting on 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 lehz h8+/ip9-ki4md i1,*iz s.spe zowzngth 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, ake*8z1.sz4/i zih9p,+ ms - wpiidzhond where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder anzr *;, ,t ei0zj:5jps,ifig iwd her arms as thin rods, making reasonable estimatespjzrz* ;ii 5gw, it,,0fi:jes for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindriq5vram2ba)s0- +ew :w*he gducal plates, of mase 5gvqeb u smrw:a*a-hwd20)+s $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 a.b mu*b7c57do2powe rough track of Fig. 8–58. What is the minimum value of the vertical heigh7d72cmube*w a .oobp5t 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, but do not assum pal11m.q*v bre $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions cw ( 4al5yc8v+8 ppovnas the car reduces its speed uniformly from 90km/h to 60km/wc yvn(o4vc 8lp8+ 5aph 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