A bicycle odometer (which measures distance traveled) is m139z* j x:5x( yqxzkt6 siwzvattached near the wheel hub and is designed for 27-inch wheels. What hzxvj z9tqs3:kxm1 5yxw *zi (6appens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velo nzjs+:d hpo-5city. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity i+hjd-pzo5: snncreases 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 sinszp9 vg8ebqs()6 ( uwnkt ru17gle value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? E6ee 1c/gkp u5xxplain.

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 behc qf9i*z-p,3yqdjbjz( v-6mkind your head than when your arms are stretched ozm( y9kvc*,pbj d --q 36zfijqut in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven52m f2 rzpekke q wx 7zmg7g+h1(76jim sprockets 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 reagm72(6k + xg5r ewemhmz7qpf1 z72jikr sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend u19k8optwmz 3 on being able to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distributi t1 pwm3uoz89kon of mass is advantageous.

Why do tightrope walkers ejv*.9.pect y(Fig. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the net torque alsoq8 q2r32rag gvsve38q zero? If the net torque on a system is zero, is the net forc82qs3g ev qq32av8rg re zero?

Two inclines have the same height b+ 7g8f,fqhpd-x5;,axup5kk jt h +f 3yq)ysxrut make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball sq;+85tdx-akf ry)3jp5qg y7xf kx h,fpuh+ ,at the bottom be greater? Explain.

Two solid spheres simultan*w h ,n/xxpd.)k ,hwmuv8pa /cyb j a0v;(x/ikeously start rolling (from 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 has the greater speed there? Which hadcwkpm)a/ 8 pu0ih/;w, x(x.bv x*jn/kh vy a,s the greater total kinetic energy at the bottom?

A sphere and a cylinder have the ktdb1 qg3 d;9e4,zaxfsame radius and the same mass. They start from rest at the top of an incline. Which reachesx; 4kafdq9g,td3bz1 e the bottom first? Which has the greater 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 momentum are conserved. Yethbfvw u 4.x .u 49im07c6ji/zqveu5ti most moving or rotating objects eventually slx/bwz7f06t .u vcuih94m i4iu.j v5eqow down and stop. Explain.

If there were a great migratiot)1mz-vgh8ieaue (me kcrpw5 2r25 v +n of people toward the Earth’s equator, how would this affect the length of+hare8 uw)-5m p vrckt1 (ei2m 2eg5vz the day?

Can the diver of Fig. 8–29 do a somersault without having any inh-obgh1i 8 c7hhb9dw-itial rotation when she leaves the boa ih-d 8gcb1bh-79ohhwrd?

The moment of inertia of a rotating2(g,ku7z jb sp solid disk about an axis through its center of mass isbupkg ,zj(2s 7 $\frac{1}{2}WR^2$ (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-:zfqr j r1ow q7eh2f5)kg mass in each outstretched hand. If you suddenly drop trhqfof2:)rjeq5 7 z1whe masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. Howevet mj; c+tvx.p,r, one is hollow and the other i, v.mt+ tpjx;cs solid. Describe an experiment to determine which is which.

The angular velocityt+3;mni,m/ v) ig*gansnrro * of a wheel rotating on 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 points east, describe the gmogti3r msi nar)*vnn;*+/, 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 rotating turntable. Whateq-(gx4wt9gg5zpdv ,n 1lb )n happens if you walk toward v9p4 5dgxb)tlnq z1,e ng-w( gthe center?

A shortstop may leap into the air to catch a ball and throw it quickl b)2 *xz :puso.zewn sjm u+9b9 q2yi6voc,:ory. 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 z, b u:s2 oesvx+j* 6ronpc9 :9q iwu2zomb.y)direction (Fig. 8–36). Explain.

On the basis of the law of cos0fd lvj00smz-m, m-wnservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller)0m-mj mlsdw-v0s, 0 fz. Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the followingu7aocy2, b: ml angles in radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing c fr u)fvlz-.:- lt-pkxoincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as kz xvu -rtpl-f). -lf:seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beam divergesys81.ppf :nvy*5odon at an a1 vyfo8opn5*:np.s yd ngle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. When the mot 8-+1fwdn1*zw arlpodor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slo1dr l p -*8fw1nwaod+zw down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If , 15.n6ge 9w l0ak5fcolfvucdthe ball makes udgw6 ,a9 e cff lko1vn550c.l15.0 revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How ma t+,x6en ii76ee 2ejf9s+l(ti eme.e any revolutions do the itnj6+mfe26ei. +ee lea9iexs(,e 7twheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter r .e0kt*f (0rdpv o, 4b lizn2yd4pequ3otates at 2500 rpm. Calculate its angulavp4,oey q0dd2.rknbip3u f* tl(ze40 r 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 one j*dam:8gc uhvg4r:b3) ybdagp s h;80 complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child uc8v d3:bh:g j;b sgp4g *ydam0hr)a 8seated 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 v7 : n+/awg9+zx vkkwdEarth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speq 3ynfci1w 6*zed 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 cenon ))-x -eiadbofas+ ,trifuge rotate if a particle 7.0 cm from the axis of rotation is to experience an acceleration e, xa)) -fo-obs+andi of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpm to g m2peuhl n+abq07zk2q54g )at j.2i a280 rpm in 4. mjngi7a52ql.+a)b24a z0tuq ep2kgh 0 s. 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 radius k+5lv3quqyj;g bq-1-d yf;1 pl to kj:$R_1$ 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 spacecrq7s-mtg k8apn o)) kh8aft 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-mipt nk o87-qh) )g8skman 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 1589kty:ew4tk ogzb a*i 1r--cw,000 rpm in 220 s. Through how many revolac b:y*ez19ti 8o k-gtk-4wr wutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1k-jmje h iaf /(tq8u+4200 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 str+xfd9w .ktxa./7,lvvvim: w aduu41eesses of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete dkv.v.iuluvewa t 41+ :x,fax9w 7/dmrevolutions 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 uz5xy-:8t3 ckok2q mw ox(7gdin 6.5 s. Ho oo:xdcymt u3kw8qz572x-(kg w 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 rwhdl+5ku8. jl unning at 850rev/min It turns 1500 revolutions before it come+h.lk8 udjwl5s to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces uv t 6f*m9bs8 i6 ybfm3mq3x)nits speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.80 6i)839mb6 snv b3qmft *ymfx um.
(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 carpdz6:j)g1 e9mssmz-m reduces its speed uniformly from 95km/h tos1 )s-p6zezgm:9mj dm 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

  A 55-kg person riding a bike puts all her weight on each pedal when5 k gz(8qcd4qj climbing a hill 4cqzj5d8kq(g. 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 end1sk/ hl-e xwl3 of a door 74 cm wide. What is the magnitude of the torquxl /ewl 1s-h3ke 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 j k2a())g e5p ;tfys4l re;jwgwheel shown in Fig. 8–39. Assume that2)yjgsk5 ; eew(tlpfg a4j;)r a friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, a kzo;gxwx)w, r.mm d jx62mj92re attached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod isd2mkgwm9o.); z xxjj,x6rm w2 held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder hea3 3j7g:rg bf3qlpa ry6:bshj5d of an engine require tightening to a torque of 38 7hag l33q r3 bp:g:brj5jyf 6s$m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment otatxfy76lb95lv s(m 0 f inertia of a 10.8-kg sphere of radius 0.648 m when the axis of rotat (xvtt756mll0y bf a9sion is through its center.    $kg \cdot m^2$

  Calculate the moment of iner p-2pqi6.qg5ujg-ev yf9xd ;otia of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignored (why?.;xqig ejf9p- 6ydqgv up5-o 2).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in a horizok er: n +-/q9ta(cvrhfntal circle of radius 1.2 m. Calcula- q(+9hrkec a:f/r ntvte
(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 pott acpm,n(xc 6c2yv9j//e s 2xjcer’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and the clay is 1.5 N tota2p v,/9jcycemsj/2x n 6cx( cal.
(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 edhg1t3: pq 5vof inertia of the array of point objects shown in Fig. 8–43 abou3vd:eq1p h5gtt
(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 whosevq xf9 d*:7hzt total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satelli0:hr56 g6ctl3 ahfyepte 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 each6at6 flgh3hc y5e pr:0 rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.wer6sj b ,j b*vyb2g2050 cm and a mass of 0.580 kg.6b2 r*b,js wey0 gv2jb 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, accelerating it qq wds04;xr: g*jx0zsfrom 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(ezjzf/pf.e bp 89n ;j( bqs.m 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 requiredffen sbj jp.bp(e; z8/9m(q.z 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 k70 n7s js +ihm)w(r:q6k z r)witb3haand is eventually brought uniformly t 7)jk0kzq 7h3wwt (ihsain)r+s m:6 rbo 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 biv*: 9f grgqv0e;xm; zall 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 solely by the w,/eharft yd7mbw p)ch (,9r 5qo.uu1 ( hgd:waction of the forearm, which rotates about the elbow joint under the acti .b9u/(,mw q ygh1awo)rf ,h ( duwd75:rectphon of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be conside + xp90nbxx-fored a long thin rod, as shown in Fig. 8–4x -+n pofxbx096.
(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 mass,t*4 geeahk7uvs.xk40;et w pes, $m_1$ 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 accelrfigfox6psl.y 8/h.f l -y+ ,r*eoaz 6erates the hammer from rest within four full turns (revolutoz6ayi *r8 - p./ olel+xsfhy.ff6,rgions) 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 of inertia ofdj2-9h6)r ,e8rr wcaua s.h rf $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque of 280 glzc/+m.mo( f8+y p-gk gpbo/$m \cdot N$ 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 witho wyx sh6s1fi)0ut slipping down a lane at 3.3 iyfx1 0 h)w6ss$m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the yk/ 7qviu9a5pw(wdc)8.w jfbh e+1 lx Sun as the sum of t 8dc5j+9 xww7ih/e wvua.p y)l (kfq1bwo 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 v-a(/c jd(od)7antqdf 1u: gt much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 )q: gajcnav u( -of(t17d/ddts? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls wiw +9j fwyh yguq q(j1 ks.bc;n)6vq)p,thout slippin y1v+y(g9pwsjw;6jq b,)f hq.)ukcqng down a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

  Two masses, $m_1$ = 18 kg and $m_2$ = 26.5 kg are connected by a rope that hangs over a pulley (as in Fig. 8–47). The pulley is a uniform cylinder of radius 0.260 m and mass 7.50 kg. Initially, is on the ground and $m_2$ rests 3.00 m above the ground. If the system is now released, use conservation of energy to determine the speed of $m_2$ just before it strikes the ground. Assume the pulley is frictionless.    $m/s$

  A 2.30-m-long pole is balanced vertically on its tip. It starts to fall and its zb9)e1 jntl*ve4( qxuloq xn4uvbz1() el j9*etwer end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating o7ms))/afmi*lz5(fi n2u pwwen the end of a thin string in a circle of radius 1.10 m at an angul7fa pml*e/imunws )5w z)(2fiar speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8- 09tct)m3yr +r2xr hadkg uniform cylindrical grinding wheel of radius 18 cm when rotatra3r t )xc+hd0rmy92 ting 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 that is rotating at a 7tlydy )b4a:ad03tvq rate of 1.3ry3) 0 bd7tlat4qyvda: ev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the ongjqz*7gqe(3 ne shown in Fig. 8–29) can reduce her moment of inertia by a factor of abo(eq3zjq7 g* gnut 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase 7li/))nbz owl60z8 hmfqrz.jd k4w0mher spin rotation rate from an initial rate of 1.0 rev everhzb6r08 ozj 4.im l)kzqdn/l w7)mfw0y 2.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotx d41itvg4z6ps,zt ;: c-yp ahating 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 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto pa4;st, hzvdt gz -6p:14yicx 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 aq(cs9u mjx35m 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 m31k/zz ;gmx piomentum 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/b38wgkc 3qi y onto an identical disk wq y3bk/c8g3irotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.43 z:kxk0iizddrduncmc hz f-d)-5/ ** $rev/s$ 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 es89 q)e7zq7 arngjl,4i -03vacmll: z(mb o drotating merry-go-round platform of radius 3.0 m aia7a8ge(),7m :b l olrnqlz39c q0j -4mdsez vnd 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-goptbr 2 9ysu+/6jhg0m f-round is rotating freely with an angular velocity of 0.8 26s0 9br/pym h+uftjg$rad/s$ 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 i.hny 2yrc9/hxti8q 1into 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 masrqiy 2ctx8ih 19/ynh.s 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 9zdt:f(dx41lh no-wfz l .b/+2fa zca 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 k9(gp 7 (* ojozoskzz($ 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 rotatexnn *c8bjr7 k :6u*mon freely without friction. The moment of inertia of the person plus7rbj*n 8:6nnuocx km* 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;e3oj2t(tgaake 2 s-l stands at the edge of a 6.5-m diameter merry-go-round turntable that is m (sag3l2;et kae2jt o-ounted on frictionless bearings 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 rolls on the ground with t2*krfd k53ywr he end of the rope lying on the top edge of the spool. A person grabs the end of the rope and wa rk3f5kwyd*r2 lks 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’s center of mass move?

  The Moon orbits the Earth such that the same side always faces the Earth. Detr*v zl7eqyn y ikya seg1a+7p4,ck 9k/q0y:7eermine the ratio of the Moon’s spin angular momentum (about its own axis) 1p eezrqnq0yagy/k9*e ic477:,kv+y7skl ya to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from 2xsus-;8jjc h rest at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shay4 lh7/n*o mqspe of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque o /4l7*y hmnsqdelivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, /:cnpxl ( fcn*1.s bejeach of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0..(cp/:*xfbn ec1s nlj0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of the rear whem:fk 5q( l3o j,sqg(+a cql,.mvl2m exel ($\omega_R$) 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 8.0 times as great, we0ktd/uyl2r( en3 63g yahn f5jre rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what wou 0rjy3h6g5u3 (2a nn ylt/edfkld 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 automodmcym5m: ve,s-qwkqq8 ; b)-gbile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without ; g qdms e:ybq8--mm5,kvq)c wneeding 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 illustracoapg -k lh/0 h22kc5ftes 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 d (afqvh) k0dq(tl6ekc3g7a/ 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 friction)60sg o+fhb8 cn about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizo0n8ob6c ghf +sntally 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, andtfwct1 hc5. *v we want to exert a horizontal force F at its axle so v 5 t*c1wthfc.that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is mhv p2ku+,ry0yn qc :-laking a turn with a radius The forces acting on the cycli0vu c,l:hyr2-yqpk +nst 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 and begins ttl m3 9z ;zh4j-vay0hwhirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque comeltmh0-y 49z ;va3hjtz from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thin rods,jtpx zq+, 3wz: making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with zxpzq,w:jt3+ outstretched arms, and with arms held tightly against her body.   

  You are designing a c l7to x,aab 9 ml (ky-d7gz3vh+)leww+lutch assembly which consists of two cylindrical plates, of m+7kx9,w 73lyval-od g) e(a +lmwtbzh ass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 alo cl/lnistv.cv)7j9 z 6ng the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it i9zt lns) v6/clcv7j.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 ad*p53y 5tjm8hnm4dcgc7 -zub ssume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the: - uwd-(3i.r f83caffmjq1:hu id zvj car reduces its speed uniformly from 90km/h to 60km/h The tj qd3ru-m c a(z-8fu:1wf:jii hd.v3fires 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