A bicycle odometer (which measures distance tt/k fyt pu7f,+raveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on af kpfu+t 7yt/, bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity.y.o j:q ldosdhg0/*f c)qdc1) Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases wo)hfljqgoy):d0do c .1*d c/squld the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the e-b -nn 5qg0ppangular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force?k;i;gp*zl iw, 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 b*9 pw -rdsc0hani- khmju+b69 ehind your head than when your arms are stretched out in front of you?h9 bn*ura-kcmjwhp-09i6+ds A diagram may help you to answer this.

A 21-speed bicycle has di1 */4x8wr,1w d 8xyt rmr o7ydn 51qmsme+iwseven sprockets at the rear wheel and three at the pedal cranks. In which gear is it harder mx4di/yd ryrewdom w ir1xs1 8 ,187nm5q+wt*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 able to run fast have slender lower legs wzbumn5 :u;2ax ith flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of5m uab2u;xz: n rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, nkz kiwb 444a:darrow beam?

If the net force on a systeqfe8ik.d 26qz2m vp ,vm is zero, is the net torque also zero? If the net torque onvzkq 2d 6,p v8eq.fi2m a system is zero, is the net force zero?

Two inclines have the same height but make differetopt)tye7m 8 1nt angles with the horizontal. The same steel ball is rolled down each iny1ptmte )78o tcline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (frmqhc-kab7k4,sqqm3 v;x .))p2 tnsi t om rest) down an incline. One sphere has twice the r mq )x2aqb pq4vk.) ntmi, hc;3tssk7-adius 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 energy at the bottom?

A sphere and a cylinder have the same radius and the same7is.t v 3n+i rg6n555tugpmti mass. They start from rest at the top of an incline. Which reaches the bottom first? Which hsi3. numg+r5ngtt 6i7p5v5t ias 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. Yet most movingc s6zm7xj75z7 c lg*er or rotating objects eventually slow down andz*rgzc767 5mx s7eljc stop. Explain.

If there were a great migrati4qe n1z /st (ernot;t:on of people toward the Earth’s equator, how would this affect :t ne(tozsnq /4et;1r the length of the day?

Can the diver of Fig. 8–29 do a somersault without having any initial gd95nh x8j nd+ 2a/yubrotation when she leaven 2+ xga/d5bu jny8d9hs the board?

The moment of inertia of a rotating solid disk about an axis throuoxhwc98r6e , mqm: e )neg*vwo1.qkh+gh its center of mass i*nch :wm 6qo8) 9wqmxg,e1keer.hov +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 4 bp;srqi ,fzfed z0sc..4zx, stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Exdefcqf.b,zpz x;isr,4 s4 0z.plain.

Two spheres look identical and have the s03xotkx h jra2t3- f(yame mass. However, one is hollow and the other is solid. Describe an experiment to determine which is whichtjox3a2 f0h x (kt3-yr.

The angular velocity of a wheel rotating on a hori)j))17ix: g /g du p6bgtiokguzontal 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 tangential linear acceleration of this point at thdto1gg:ipu/x6u7 bg)i k j g))e top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating tu u(qgx/tdje/jn/)u;grntable. What happens if you walk toward the cenej g/xu)j/(nd/t ; qugter?

A shortstop may leap into the air to catch a ball and throw it qui( fxnvriv xmf4;.d :)pckly. As he throws the ball, the upper part of his body rotates. If you look quickly p)dr xvn (: ;vmifx.f4you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of consercd +,xr*ecg 4cvation of angular momentum, 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 helicopter stabcc,4erc* x+dg le.

  Express the following ann +v 7:m;yu gtzse s uj1:i8j7p0v1pnbgles 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 bec+ofun:bu h2 pqe;j8/q ause of an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radia q+8nb feh2ju/ou:p;qns) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beam dx9u*lbuy4.4 ipitz3d x wq bv-7zx)+tiverges at an a 3u4qbivxyl z.pb)xi +dw-7ut*tz9 4 xngle $\theta$ (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 turned +2 lc8cea-ez koff during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blek+ac2 z-c 8elades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the ballmts 1g;8xoc9jw;z; fl makes 15.0 revolu gc8z;;jft;1 x9slow mtions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travn8owl): +raplxc7n* v els 8.0 km. How many revolutions do the whee:nran)wc pv*+x7llo 8 ls make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate its 5 2fg2yopx3op(h ,obfangular velocihg25f3,o x2f poob(ypty 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 complete revolution in v3+m he(b b ey.sn4f+t4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from them h4fn3b(b.+t+y seev 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 its or-;k,;a(d /hnhrfa uurbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speecf ,p4g rmy,5cd 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 centrifuge rotate ifsy yco: y(r 7clmjqb:7jrt6(: a particle 7.0 cm from the axis of rotation is to experience an a:rjlb7 tsyoj7y6:yc(rq c(:m cceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 13u hj. zwa),,jtw 3u*rx0 rpm to 280 rpm in 4.0 s. Determinw ,3*)zjw hxjuaru.,t e
(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 radiul0gba2pnt( a0 s $R_1$ 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 spacecraft put themser w9)iq p.y6ublves into a slow rotation to distribute the Sun’s energy evenly. At the start of their tru.q9rbypi)6 wip, 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+syizxl: ; s64peeni( 15,000 rpm in 220 s. Through how many revolut+i:6x(y4 snez;i es plions did it turn in this time?    $rev$

  An automobile engine slows down f;fc cu-:,divirom 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 for h7y1z /gb,zgythe stresses of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions gzh 1y/b gz7y,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 3gfb0 njox 40jin 6.5 s. How far will a point on the edge of the wheel 0 j4jn0 b3xfgohave traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev it+9h.fv1 wdt/min It turns 1500 revolutions before it comes to a stov1t9f+di htw .p.
(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 spe )24fiotjdzj8:* oq3w w3bl bded uniformly from 95km/h to 45km/h The tires have a )j 4d23z:lqb b doiofww j8t*3diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reducuvci5 wrp/ hv*v49mzt7 2tf2q +pke z+es its speed uniformly from 95km/h to 45km/h The tiresp 2+mk tzr2+wvtu7 /pqceh4 5f*iv9vz 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 aom (e2fh: hx*5 fb/lvs bike puts all her weight on each pedal when climbing a hill:(5fvelf *obm/h2hsx . 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 ofg9q h) pjf7z,4vtft 6m 55 N on the end of a door 74 cm wide. What is the magnitude of vfq 7p4g69jt,zth m)f 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 t 9b;qhniug,1mhe wheel shown in Fig. 8–39. Assume that a friction torqhugi,m9 nbq1;ue of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attachl k sb4/u5qcji*t:-gged to the ends of a massless rod which pivot-/5j*ik 4t ul:gbqscgs as shown in Fig. 8–40. Initially 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 cylinder head of an engine requdtpq,nj7 m j0.ire tightening to a torque of 38qnj t7.m p,j0d $m \cdot N$ 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 ofo)p oljzn+wv e ly(01: inertia of a 10.8-kg sphere of radius 0.648 m when the axis of rotation is through its center. j onl zyl0e:w+pvo ()1   $kg \cdot m^2$

  Calculate the moment ye 41rb2 ;)ji wg,wrvmof inertia of a bicycle wheel 66.7 cm in diameter. The rim and tire ;,2)rwme1jb i vr ygw4have 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 e/6f 0m6n h 2a9g;7mhj sugeh8dik(kvjnd of a thin, light rod is rotated in a horizontal circle of radius 1.2 m. Calculat9vi;0 ahuegj 6 hms md6f2g jh/(kk78ne
(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 angular l 8ujgq 3pme , xhtd:8px)bb)3speed (Fig. 8–42). The friction force betb38xlp )eu,p 8xbhj d:g)3q mtween 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 objects dqp33l5lqq6hv (e7 bl shown in Fig. 8–43 aboblh d l65e3p(l3qv 7qqut
(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 b 7m+q8g-fix,1 lkhxe of two oxygen atoms whose 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 satellite spinning at the cpaly+ n+hq9/forrect 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+hpf/ y9lna+q m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder wi y 5 .,8 ehpyj0+uz4atav;nkc .npu)jwth a radius of 8.50 cm and a mass of 0.5)aue+ ck,.ppn8 utzwv j5hja.n40yy ;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, accelerating it from remdxwo 0q*j*vs 8an8wo3* vm6qst 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 mernom204do ug tgwh :c9kssg31 r cs7j;)ry-go-round and is able to accelerate it from rest to a frequency of 15 sojd;msuogw1rh c 342 g 7k0c9 gs:t)nrpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and oq5 bb*gi.)ltmj wq33o i;zdnu*3md :is eventually brought uniformly to rest d.b uo33q q:in )bzwo;gm35mji*lt *dby 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.zbl+nq8c- +lj 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 solely by the action of the forearm, e ml9hlen-gm 94-5y3jxaa0 iowhich rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uxya5h9lm93 laon 4 me-gei- 0jniformly 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 ce 6f4/nng83 z gd(mhfdonsidered a long thin rod, as shown in Fig. 8–436 dg48 fngf mzeh(d/n6.
(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 masseyi1qwfz ) j2oz )u2rf(8cz) zks, $m_1$ 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 haq1h(uxu0)kpx ixz 87emmer from rest within four full turns (revolutions) and releases it at a speed of 2p x qeuh0 uix1x87)(kz8 $m/s$ 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 mopx/uph, 9 8kmement 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 develops a torqu ch)2 yf8sockbc5,)w ie 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 withouv.b(w: 0 em4v ehsld9gt slipping down a lane a m4v 9ge(l hsevw.b:0dt 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of thexm*ukb v(nvl g)+s n6q-5kq ml/ )mbe+ Earth with respect to the Sun as the sum of twql v)smg*- ux (vkb+)m+5b/ eq6knnm lo 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 )wv1n-; krmgs m. How much net work is requir 1 nvks;)wmg-red to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm j00q ne0y ahy8and mass 1.80 kg starts from rest and rolls without slipping down a 30.0yh0q8yajne 0 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 verticallk5-0w -(ff daikiimz,+ a+lyry 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 i,-+akyd(wfim+ f0ziia l- k r5t hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular m-8 /vyxr7kzlg)q59qfitz6ykomentum of a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an angulklqg/9qx 7zyrzvi-5 yt)k8f6 ar speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg unifmttx 8ixo/albk 8890 + 8omoezorm cylindrical grinding wheel of radiatb8xikm+ooxm 8e8 t09 / 8olzus 18 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotating at a ra j( d9rf1(mpx,9 vbwfote of 1.3od1f vfb,wjr(9p m 9x(rev/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 one shown in Fig. 8–29) c s5x(gjlujd y1 j.k0d3an reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in0ludj x1 k3.j(5sjdg y the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spxfipa ,u1b 3y:in rotation rate from an initial rate of 1.0 rev every 2.0 s to a final r:1 iua3yfb px,ate 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 center at a f6ukwk)/iyd8i m/ /oadtlk7 5 prequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the law5/)ddy /p uit mk6/ o8k7ikclay sticks to it?    $rev/s$

  (a) What is the angular momentum of a tjc:;yv*: iul figure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum oft o)+jc /y 7jarb1fs8y 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 4 xai*p3gz z8dof inertia I is dropped onto an identical disk rotatiniz*dzp3xg a4 8g at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns anul h3.tlus ;.t 2.4 $rev/s$ 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 p3a +q-f ce4vepot ,+qstands at the center of a rotating merry-go-round platform of radius 3.0 m and moment of inertiape q 3pao,4vtecq++f - 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 merr ,pr; ov poj-aj66be:u vh/fl, jg/w-p*ege j5y-go-round is rotating freely with an angular velocity of 0.f,6ovj/ber;pg:* gp je, u-5-ja/h lp wejvo 68 $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, losing aboutf5:1; /0 df9n4tq fp:lkle 2kg qiheaf half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mas lf2f hi1kfp0 e9 gaq:nk4qt;d/e5l :fs 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 win rzo ).k(-*cn*gi rg(ek0my(a1v psznds 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 4el :cpx v w:5pux*3aq$ 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, thatpijro3 j80v0ums( q x+x++ooy can rotate freely without friction. The moment of inertia of the +3 s j+u8yx0r(o0vojioqm+p x 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 edgetak*8 hxmq8q4fbjqjon4i; yx;7tw 76 of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a ;;wq8 y7f h4jkt xn8 txqjq6mi*a7o4bmoment 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 rolld)zpz 8 )ic7iu34gwva s on the ground with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slp3iad wczi))8uz7vg 4ipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Enpw9h tdft-4 u xy5w-+-adi1 xarth such that the same side always faces the Earth. Determine th5 +--w hni-tux y9xwaddftp41e ratio of the Moon’s spin 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 rest at a rate oftqd c 0rru (amrq;a+bc0x6 v05qz:3j4wy;f lh 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 radiux fqpyw+xk+ ())b .wvfs 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Cx.bpfw )wk +)xq yfv+(alculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.05a adczv *r)z730 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 aa3d*7 r zczv)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 speeu88rf h/mxxsk) kbo+ 0d 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 fdof0b7n /7suwith mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in 7s/uod f fb07nthe 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 energrsbm7-. eo q h+fsu0/gy 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 shfh-ssrb / .oe7q+ 0ugmould 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 illustrates an*t kecx*d3+h q $H_2O$ 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 )aq h(octv j//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 frefh s 0kg:o n2qtu*yy56ely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assk2 otu0syh:5 gqfy*6n ume 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 ver zq(x p.)mvat xvj2t:m,hfzf8- 9 cm)mtically on the floor, and we want to exert a horizontal force F at m 9-t8v)mct xvh,qxjfma 2 f pzzm().:its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making g-o:agpc/ 4 qza turn with a radius The fq:ggc op z4/a-orces 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 length 1.5 m and beginb 5a isg)-oxuh3/,k)slz; dkys whirling the rock in a nearly horizontal circle above his head, accelerating it from ) a3)k-/ydzbs, lx 5kiugsh;orest 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 c2dtatk q19 e/;q2deg3k8 zbk6n, pslsylinder and her arms as thin rods, making reasonable estimates for the dimension t gqz3,9 ;lk2 asp6bdqeed/k8 2ts1nks. 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 tuu0 .p0he; vvj13qeg xfe.vy:5uqb g6 wo cylindrical plates, of massevu0v.uy5;1bv 3 g ueg. 0fhpxq:6 ejq $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 track of Fig.o98k2 ,vnnc(cb ne 6ej 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 thb6cnk v9en jno2 ,e8(ce loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do 7 kzuv (e24jydnot assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolut)x4/q2xpj3 k v,vb 9y3b3uaj nxghwp:ions as the car reduces its speed uniformly from 90km/h vkv j)pwbhgpx,: /3axun243b39jyxqto 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s