A bicycle odometer (which measures distance traveled) is attached nearg s.fobduz0 /ru 3-0jx the wheel hub and is designed for 27-inch wheels. What happens if you use it grjxu3 .fubz-d/s0o 0 on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angulaelz)m 6e;2zc0i+vi hnr velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocnhl0ze+2 v)izi6ecm ;ity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the angular velocity -hppqfi6 r .lg 1va3o*$\omega$ Explain.

Can a small force ever exert a g/kkr5 jodxhh b331g4rdh s*m7 reater 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 witx ) /*bqyhih: u;*xmkth your hands behind your head than when your arms are stretched out in front of you? A dx: */tmkq* hhu yb;ix)iagram may help you to answer this.

A 21-speed bicycle has seven sewgl ,pok qbr:kc 654(prockets at the rear wheel and three at the pedal cranks. Ig6(r:,cbo k qe5kl4wpn 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 able to run fast have slender lower ly a/6uivkmp-890qhvregs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advhi q/k rm6pv u-a908yvantageous.

Why do tightrope walked bdx4-i- yqr(-lieh y4dt30brs (Fig. 8–35) carry a long, narrow beam?

If the net force on a system is zero,j )xhx i+iaymc0sq7 6) is the net torque also zero? If the net torque on a system is zero, is the net force zerxim xqy si)ch7+)j0a6o?

Two inclines have the same height but make different angles wip)* js h2 c+m8n9eoflath the horizontal. Tsc8hj9+aopf ) 2m ne*lhe same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling w 8s xus8, ebx/:qo0ol(o8vg:zx k) sj(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 xqsguozo 8 k0 x/:se8(l:),x 8bsjvwofirst? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the zu3-e13v qf. wqypp )msame radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which haq )-y33qewpv1.muz fps 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 a ,( xogp;mjjs7re conserved. Yet most moving or rotating objects eventually slowp7,(sgjmj x o; down and stop. Explain.

If there were a great migration of pe/ ,g f wxeqog3f;n-6gwg2oo x9ople toward the Earth’s equator, how would this affect gexwo,o9fgn qfxg/63o2; wg -the length of the day?

Can the diver of Fig. 8–29 do a somersault without having any initial rotation wqk4yzd6d/ *.icy ;f18fsie9jrmz d4) :n igv mhendcs1r/jiin;myf9fdmi84k.dzeg z* 4 q : )6yv she leaves the board?

The moment of inertia of a rotating solid disk about apc3 3b+;uopwq n axis through its center of m w3q3po cub+;pass is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2-kg mass in eacppyn.0 6 p1 7s aenrkfll 86cfjtukp3x5s5g7:h outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or 7s3e67ua:5fptnc0x1skp 6 kpp fjy.g8 l5rnl stay the same? Explain.

Two spheres look identical and have the samtwr q qc, 6p-o0sncx 7+ttrw-;vbk7.ye mass. However, one is hollow and the other is solid. Describe antc w7n6bpt k-r;ryq sto,qx-w70 v .c+ experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points west.)dm7y;ysvb l, 1vin3n In what direction is the linear velocity of yni,vl17bs)ny 3dvm; 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 ou:q ,xyc*1e na large freely rotating turntable. What happens if you walk towar1y, u e*x:noqcd the center?

A shortstop may leap into the air to catch a ball and throw it quickly. Asob 86lkdjgo8* he throws the ball, the upper part of his body roto*8okb6lj d 8gates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law ofu;xc s;w2 iq.d conservation of angular momentum, discuss why a helicopter must have more than one rotor (orduwi;sc. x;2q propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles io qn1 h8 /2qy9yu2kreyn 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 (vz1g7cpw ox 2 v+hzu0an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radianszpv0zo 71huw cv+ 2(gx) 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 diverges apwx q49r,:rk6hsu hq1 2wjk5rt r4wkhsx w 6,ru15k29pjr q hq:an angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate oflz6t.5h fw g7/u2 ued9ajojji +ohp67 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 365 fe69.wp 277 /jjhziato jhgoul+du.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m t 3 zpyrq5sddx ;(e,xa*.g,wsho another child. If the ball makes 15; erz,hwqs*g(d x35p.sx a dy,.0 revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. H-kz(d8x c xi:aqerr7/7, qiwu ow many revolutions do thwz/-( iq u,rxqake7ir8:x7cd e wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diametsxmgx jh:7f3tk5lz :m j, s*dp 7i9ju:er rotates at 2500 rpm. Calculate its angular vepu7 5fj*s ,j9x mxdl7k :gz:tj:shm3ilocity 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 mak:1asl )z)hctv(im. adwwn4/8r b,ro u es one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.rcvwrl/))iudams .:tw1nb a,zo( h84 2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in itqctqzgd e x,2i*4jz *-i:2i, ixy5 otxs orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed nw -a2i0n+y0x l4 hx7*plaxnq 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 if a particle 7.0 cm from the axis ofyf 8 xsceew3671 5 -hr/mltgof rotation is to experience an act l7-8f 3who1gxmr5y/s6eecf celeration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly aboi s7g 0bqm41wkut its center from 130 rpm to 280 rpm in 4.0 470igm w1q bkss. 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 radiu97 crmyc.7ewas $R_1$ 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, astronaut3,-x)(/ah qp rgd9sdasqa ( xqs aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecrafa q(hd-3, rqq9 x ssadxg/(a)pt can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpm in 220 s..ef7uqyt*8z 18 ukdd0cy iw/c Through how many revolutions did it te8*wy /8 kt0dccqfd uyiz.u17urn in this time?    $rev$

  An automobile engine slows down ophr 9rki;0v2 from 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 the stresses of flyinu8tp pfg*/)7i /c iwplg highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching p c/tpiwl/fup*ig8)7 its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 360 rpm in 6.5 s4 foc;mg0oijt7tlf9 rw:3*gvx ,k,r dqp*k4c . How far will a point on the edge of the wheel ha7jtr,gwoocrk,mdf0q l4 p:9*3xc;4gtif k *vve traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 1500 rev6i+n7oh3osj oolutions before it co o3o7jnis ho+6mes 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 redu)2hsh 1u u9sjuces its speed uniformly from 95km/h to 45km/h The tuujs2su h h19)ires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reddyoj z:nilr1d,:, 2ytuces its speed uniformly from 95km/h to 4z:dd :,n, lrty oi21yj5km/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 mggdh0bjc0 7gy 7s o//all her weight on each pedal when climbing a hill. The pedals rotate in a circhm/b/7d yggoj0g7s0 cle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a do4fll menxd7/;or 74 cm wide. What is the magnitude of the tom4 lldef n7/x;rque 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 wheelmgf)/bh4m rcaf/v5j7zsia :90ic1sl shown in Fig. 8–39. Assume thaij )cl54 1fv fsa0/ g9h:m bzm/cs7irat 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, are attached to the ends of a massless rod whichuxvgz-iqrf 2(,r 8 h5q pivots as shown in Fig. 8–40. Initially the rod iq5,x8g hrri(v uq-z 2fs 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 require tightening to a torque of 38cjd46q5up+t j u5pcj+q jt46d $m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of radius 0.648v,j fsgo,bc;y b:2hlj :-hmz- m when the axis of rotation is through its cenh;jlf ,b,zh:yb g-sjv2 c-o:mter.    $kg \cdot m^2$

  Calculate the moment of ine2i +vula4)n mppcli qn+.cb91 rtia of a bicycle wheel 66.7 cm in diameter. The rim and tirv l1 anc+9qi. u)pip4c +bnm2le have 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 a horipxd*z+y;+yfh zontal circle of r*dh+f ;z+xy ypadius 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 plxnr n/5 u7q* eftu7q/otter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her haq* fu nutr/7qxe75n/lnds 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 obje:+4-qed ry tz tr/dt4s*dt1 sects shown in Fig. 8–43 about q1ssy /dzt-e4e+4ddt r*t r:t
(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 ovgkbzs,r9e3;1; rl nyau)v,g xygen 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 cylindrical6rdqd( z)hv3)fcr k c- satellite 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 steadyfzdcv c-r3))rhq (d6k 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 cyli la msen q,vy2g9i4 )lb1-bds7nder with a radius of 8.50 cm and a mass of 0.580 kg. Calculalgv12q)bs a lde,sib4nym-79 te
(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 rest to 3 /tnsvcw 2 j*t auubzd 1xtl *17w)7;vz$rev/s$ 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 heav4* wjw7k: 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 o* eakwwhj:7v 4f 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 i,epvyo3w mg z 9+9xzuvk.r2;shut off and is eventually brought un imerug 2z9x9;kvv,pw3.+zyo iformly 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 bal:m960gga ohqm(ifl 4y l 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 forepk+ 0eusl4))i/. eiac xht7ijarm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformcki)/u+ )ep het4js 7iiaxl.0 ly 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 b 4r 5s vd3hurvnxv: ,h,q2ll n:d8vlm)e considered a long thin rod, as shown in Fig. 8–48hvl,vsr ddm2nv,hrn l4 5vx)ul :: q36.
(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 ma0repss7m.: dmlri a, .sses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest within four full tur6ip6utaic n hx s;83)dj,j; wpns (revolutions) and releases it at a speed of 28 u ijin3 d)w hp;p;casxjt866,$m/s$ 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 iner0kusmsf-e fq.b8i3 z2 tia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a to 2wsd5 mm zz0cnm;h,cw ,7wrn:rque 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 without slipping +emg6o:d8d p5ox:lcksd5. +ut9dbj e down a ladb5e xt.c e5duod pd:ksjo ++8lm:6 g9ne at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as t w2nxui 8(nax.eg*c c:he sum of two twcu2g :*a. (xenx c8inerms,
(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)9b*j kr2nos f kg and a radius of 7.50 m. How much net work is required to accelerate it from re2ojn rk9) *bfsst to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.8msb,kzzn:l* k(3n 91lkog.vs2 o tz4h0 kg starts from rest and rolls without slipping *om 3o:klvnkst9 1l2sk(nb 4 hgz.zz,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 iv 3jtj2f e0/ozts lower end does not slip. What will be the speed of the upper end of the pole just before it hitsv2 zf/3ej0otj the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end of a thifgj cx.mcn8 ahi)63nn dpg ,-api2;h 1n string in a circl1pgc2p,mg 8i3;.nahn n jhx6dafic -)e of radius 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 uniformy41gr(r us4r(ad gx2 epj(7o n cylindrical grinding wheel of radius 18 cm when rotating at 1500 rpm44jgeayg((o1 d rr 7(spxu2 rn?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands aqt s)gsq +1fe qf0w5)fu5b8lmt his side, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a hob ffe )5f8tqglsm0qu1)+qs5w rizontal 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) can rex3v 9jh3oalkx9jvz, q;u y 4(fduce 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 in the tuck position, what q ,9u;hl943xjz3v f o kyv(ajxis her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial rate . /ywj2-brcix6tvj9c of 1.0 rev every 2.0 s to a fc/i 9ytwbxjc2 j6v.-rinal 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 arout.l2h cf79yq. xuft y5l-ppc ,nd a vertical axis through its center at a frequency oft-l x ch p9.l.,7fy5fq yc2utp 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 clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater t3)hw - p 9ke;mgdw7fwspinning 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 kip+byxvo/- :cw p,s l59ua. omomentum 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 cylindr 8d ongyt6n5/fical disk of moment of inertia I is dropped onto an identical disk rotating at angn8/td ofng65y ular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns ;z,eaisp 12rn3,e* lrecl7z jat 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 stands at the center of a rotating merry-go-round platfor jn;l3tl6ga5fm of radius 3.0 m and moment ogj lfl5na3; t6f 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-round7da7*i js+fat is rotating freely with an angular velocity of 0ifs7 d*7+ajat .8 $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, losing about halv ml m 85:.dz.ugihk1mf its mass in the process, az i.lmdvkgm m1:h5.u 8nd winding up with 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 i zj9h nb**p6sk prkz3jbxe/ 0;n excess of 120 $km/h$ at the outer edge. Make a crude estimate of
(a) the energy,    $ \times10^{16 }$ J
(b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 $kg \cdot m^2$) of radius 100 km and height 4.0 km.    $ \times10^{20 }$ $kg \cdot m^2$

  An asteroid of mass d s /i;c0h:mhj$ 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 re -czijaf0w454jnxb v 6va6l6est, that can rotate freely without friction. The moment of inertia o 4 ixbvef n4jvcw56a- 6alzj06f the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diameter merry-go-round turu1se1ob2h7fo e sv9 c3 3i7qxdrq2k. cntable that is mounted on frictionless bearing q ku 9beorc3f2eo cq2 3ds7x1h7.ivs1s 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 the end of thblp;31am /dn0d9 fio d*l :ipqqy8 r/ne rope lying on the top edge of the spool. A person grabs the end of the rope and walks ao ;8n d/y*lpp:q dan10qfim39 /idr lb 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 sucq byq+hcpe4dw84 l3+qh that the same side always faces the Earth. Determq+h 8p dcq4qyel+4bw3ine the 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 of 1 k 6xp8ui/*gk (/mstpm 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 0.20 m u : 1ln8ra tm5l( jex/a:cc6qlacquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the 8:l a /cjla61q 5ec(x mtunlr:torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0s gegtkuj zc6e2 xm2u. l,l s-1f9j81u.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mal1zu1l .-ucg6j2e 8 ,et9 xsugfks2m jss 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angularlfd:2:uli0kup8)ig0 il. 0 vig8m ga gdo-b 8i 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, bu:xcb0ma kf c4+t with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse 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: b0f+amcckx 4 the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a lo(u .b6s;pndtiz+ j7 baw-pollution automobile 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 without7+ p;isauj z.tdn(bb6 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 au1+6zywvxzc 0q9 u2hpn $H_2O$ 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) i.n:xvp x.n38xt,tv g es rolling 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 freelyi v4otok/jsmce11 3 d; (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, and1me3id41koc s ;v jt/o (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 is6i.c( skntfp)95-ee f; qqz jp standing vertically 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, where hp k6; tcfqjz (qesfi9)e .5-pn

  A bicyclist traveling with speed v=4.2m/s on a flat road is mb 2be-l,e z5* + -upskjvkbly5aking a turn with a radius The forces acting on the cyclist and cycle are the normal fok5 ysbk,ulb ee2jb-v*p+z5l-rce $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ 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-gn(myyz n5d 7+oo;ky26 y3 ghfkg rock into a sling of length 1.5 m and begins whirling the rocmfyzny 3o;yd +(oghk7ny g652k 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 come from?    $m \cdot N$

  Model a figure skater’s body as a solid cy v/3g :wleyho- m)r(tylinder and her arms as thin rods, making reasonable estimates 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. m er 3 v)-:y(gthw/lyo  

  You are designing a clutch asse;bfwdoqy+e8 qvi6(fng g 6 g*-b(dq+p mbly which consists of two cylindrical plates, of mass +qbw y8i6d6 g -(f ofbdqgnegp;q+v(*$M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 xv6gt /pajf:5 the looped rough track of Fig. 8–58. What is the minimum v ft5p/g xa6v:jalue of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assuvrwy m due 1rqg w34.1q*ts90vme $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions aiw9 veul )n)- ;u7xrors the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0vurnix ew;lo- 7u9))r.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