A bicycle odometer (which measures distance traveled) is attached near the 4(vgpe5z2 u*assoet 6wheel hub agpuevs56easot ( 2*z4 nd is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a coy+3e k2na/tpoint on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point havo 32a t/keyn+ce 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 j(rds-v n3t +u5tvw4y$\omega$ Explain.

Can a small force ever exert a greater torque t 833jmrf ;2pwzvo6q xcw.iq/nhan a larger force? Explain.

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

Why is it more difficult to do a sit-up with your hands vwbk+(sgcd5zm40cmzp i,7 v 0behind your head than when your arms are stretched out in front of you? A diagram may help you to answer this.wvv+ 75,4s(c gkbzdcpmm00 zi

A 21-speed bicycle has seven sprocketsnr h7f,v ho/w)-6cnfeycy66 o at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Whv7 f hh/6no6wfc n yec-y)or6,y?

Mammals that depend on being able 8e,k vg e k4/-d6swowdto 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 distribution of mass is advantagev g wke-kd/wdse64, o8ous.

Why do tightrope walkers (Fig. 8–35) carry a long, .j6p/mk m 1jsmnarrow beam?

If the net force on a system wka5pklto p4528rnglx *5h1 xis zero, is the net torque also zero? If the net torquhrp25*l tkxl k554 1wanp 8goxe on a system is zero, is the net force zero?

Two inclines have the same height but make different anglfrc8 /sxik4 y ;nry9lqb jp/)9es with the horizontal. The ;9 9 pcrbx/ 8nqslkfyi /)4ryjsame 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 (from rest) dbx - ttwoku)(dsc(+u d2quz*4 own 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 has the greater tkc stw4)b ou2q+(*ux-tz(ud dotal kinetic energy at the bottom?

A sphere and a cylinder have the sax5 +ej- g 4t*pz7fhwhkme radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater spe w-*fhe p+k4 h7gx5jzted 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 con:peh42l, zpop served. Yet most moving or rotating objects eventually slow dowpop:pl,e 4zh2n and stop. Explain.

If there were a great migration of people towarnl80dt l7 zowa/.a-l2jij6y yd the Earth’s equator, how would this affect the length of th.6- zl djiw ylo/y2ta78jaln0 e day?

Can the diver of Fig. pg/(w,n( s*vbkz d/f03bxcbe 8–29 do a somersault without having any initial rotation when she leaves the bg,v e(pd*nbsc0zwxb/b k(3 f/ oard?

The moment of inertia of rk+5 ar;w j-+t8/eua sc;njt17fk vt ya rotating solid disk about an axis through its center of masy+ crk7-e/1w ut;n rv8 t5a +kjfas;jts 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 ,7isvqxy ;/ qayg 3;sxin each outstretched hand. If you suddenly drop the masses, wil i /3sqyqsv x,;xya7g;l your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look idenrvm.iz w6y,m.i b (+fktical and have the same mass. However, one is hollow and th6b, f ry.(vizim+w m.ke other is solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle sk vk-w5.u*g+(dy cjb.n g9tspoints west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, desctbnk+ku. s(.vgdwgj*5-y sc 9ribe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating ojxru**y dr6i2 d+d0i)o qhb ;turntable. What happens if you walk toward thrbuh) 6x0d*i dyrdoi*o 2+ q;je center?

A shortstop may leap into the air to catch a balmu 7ns. -s9od6 x5yufal and throw it quickly. As he throws the ball, the upper part of his body rotates. nsaf9so-7u mxd56.u yIf 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 of conseru6 :4;xiimn, amvl a op+ziiyr3:7gxm w3 :ih/vation of angular momentum, discuss why a helicoptp:3ii a/:m4ar 7 m+nv; w,6l3ziyixg im x:houer must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radiansiw 6buq(v.m5 6cemlg2ko43zuiz -2 v j: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincidence. Calculate, using th47w bifd - ksbovp0j*m ao jdx:;.gq74e information inside the Front Cover, the angular diameterq o7vbgfm04;w -b *x4 o :ajd.7pjkdiss (in radians) 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. T +(26quasvbazq5 +6sgiu azs/s0xn8fhe beam diverges at an sqszg60qzuv6 aaa5 sx+ (fs2i8nu/b+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 *q+ )io qihn(hat a rate of 6500 rpm. When the motor is turned of +h(hqq) i*inof during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. I yk ,sjr0u2bf4f the ball makes 15.0 revolutiu, 4s0kryfbj2 ons, what is its diameter?    m

  A bicycle with tires 68 cm in dk qgrzk38wm;t d0f ob*b /mn)7iameter travels 8.0 km. How many revolutions do the wheel/wzknoq 0*;kb dg)3b tmm7r8 fs make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate its angu 8ghbh 8j8)725rk z fhl(2qpy;axa 43aay dpiflab78kx884d;p2 iz gfqh) yal py2ra5j fahh(a3r 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 complete revolution in 4.0 s ( m5(yh, xmkfqpz:.gl 1Fig. 8–38). (a) What k,xf.q1 : lmh(yp5g mzis the linear speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orbit around p701lwl5w2qoabqs lz i 4 a:l7the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a r-* xg(f 6.k :eeajnagpoint
(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 centr nsh+pg. u*fxs (ax4s-ifuge rotate if a particle 7.0 cm from the axis of rotatisugs s(.- pnx 4afx*+hon is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 137p bnf6 u9w7b l)rn)au0 rpm to 280 rpm in 4 ub)6wblaurnn 7f7)p9.0 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 radiuyw- m3tly q4q6s $R_1$ 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 fxs7apq4 bk:tu5j j rb 9:ohe3+mr-y; 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 spacecraft can be thought jafqjym-: ;to rerbh34x+57:b9 sk up 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. Through h bq+dbi51)irbd,1 by bow many revolutions did it turnb, brd1bd+)yq i1b5bi in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calculat po2*i hrmk:-pe
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeed jets in a whirling “.uum2yr yuu:+obk ;ige/et1 . human centrifuge,” which takes 1.1er u/uyue..;imu : yko2b+tg0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 360 rpm in 6gq47pf 5rbg wj5:ez(q .5 s. How far will a point on the edge o grb(j 45w7p:fzeqqg5f the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/mih ; ygdc ba-sk*7,ot9n -*bhqdn It turns 1500 revolutions beforeg bao--hh d; d sbqy,t9*n*7kc it comes 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 revolut+lol;.oul nhrenfr/ 9-,)n yn ions as the car reduces its speed uniformly from 95k)h +oly9nl;nu-/,ol.n nferrm/h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed uniu;x hw3u q+,qvformly from 95km/h to 45km/h The uqv uh;,x3qw+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 when climbing a ayaj ql 93;3gqhill. The pedals rotate in a circle of radiuaa3jl93;gq y qs 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 door 74 cm wide. What is t;xd2h 6 c;/whq,ckr ishe magnitude of the torque ifi;h6dc/; qxrh,skw c2 the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fig. 8–39+s+s6 tosa-c 811oki.1v nj+eti )srk e6ms af. Assume that a frifs k k c 1en6ss+8si) aos1+m.rv+iateo1-jt6ction 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 wxvn:ghb3ih.r k -hhx*f), ; cchich pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate t ;kch)vg,:h 3cbxih- nf.hrx*he magnitude and direction of the net torque on this system.

  The bolts on the cylinder h(mgung. ,m s ipczw;lu9+h: 51ugbb2 head of an engine require tightening to a torque of 38 ,ig(hmgu5ucn2bgzl+ ws.hpb 1 u;:9m $m \cdot N$ 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 10.85xfj+;j z rp/ 7bmekn ua9ak*-,w(xvg -kg sphere of radius 0.648 m when the axis of rotation is thgk +/muaz7*vx 9e5x( ,-wb jknarp j;frough its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a b:g o)klnkkfw2m c: .y/v68zy0kj xb 6aicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub canfy2vkb xa/m)80gy:lj .:6 kw zok6nkc be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in a hor0,,uz jr,gny:til4:le+c1o s6zbz1 t 9 w kexnizontal circle of radius 1e 1k:z9rnwctg ill:, e,, obx sny6+4 zj1u0tz.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s hu( 64wac8psn-84 uai6znek a3i d*j hwheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and the clay is 1.5 N8eih auh4 n-pkaznd6a s4(*ji3 c wu68 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 showm)q9q- ssoc-qse pt-;n in Fig. 8–43 abouttq-;9q-cpe-q os) mss
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen atoms whose total mass is ik(b2r mojiq:6;. nk r$5.3 \times10^{ -26}$ 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 correct ra- +hembca(3ebgyp 09bte, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rockyb 0ec+baem (gbh3-9 pet 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 q ,4 rnq, ua4dcbv1hc1a radius of 8.50 cm and a mass of 0.580 kq4 cc1q 1d,vuhan ,b4rg. 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 rest tofhm,s z,-m/l:u 9ffhj 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 mk4zvr+mr6a 9:yl6b 4xe x6 zjzerry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleraraxlx6ryb j466z+9 e mzvk z:4tion, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rog idvvu16+kw 9 b8ad1z1u)jsztating at 10,300 rpm is shut off and is eventually brought uniformly 1zu s v 1dkuz+a8gbj9iw)v61d 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 aco: 5nejlzs:3+o1nix1 gpih c/ 0cfs+iefr. -bcelerates a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the action of the forearm, whuf0)f hc9bdz3ich rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uf3uh0 dz b9)cfniformly 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 considered a long thin cb( a;l;3; tpnmxg;qp9zyrk0bpc:1krod, as shown in Fig. 8–46pa g:kx;zbkc ; tq 9301;y(;cbpp rmln.
(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 oq3;p5 n 3 8t*dz83u0xtnbg zys6nsesqf two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest within four ful ooj1ywh *1l:wz 5vqt1l turns (revolutions1qy5j wv*z twhlo:1 1o) 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 of9(h 0wqy wz,-k .nfkzgdx6 hi; $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine developfft8).0kr;wa3/ ai-s ccbm;rj .drj r5.nrgms a torque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rofjps(i 0lg l5ne6xl vm) +:2iells without slipping down a lane6ejl(:ix)l5ev +gm0 lf isn2p at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the mw+0q )3mvm gq5m*sto sum of twot mq owmmm)* g+530qvs 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 op1l4xwvo4 vz5i1 ro9j+c ohg5f 7.50 m. How much net work is required top511o4o5v r +ihowgxjvc4 z9l 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 and mass 1.802lfgrmhqk 4+y26j 1esux c)6a xuc8x+ kg starts from rest and rolls without slifx+ms+kegr 1u)l2h6cu4 2xx aqjcy8 6pping 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 viq)dn;5 ,x m lgt uiqnkt-kw4)*7s m0pertically on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservatiopt q wqiu57;)*-x)gd kktis0m4m nln, n of energy.]    $m/s$

  What is the angular momentum of a 0));kgnj st 9xea2+ixcwac;.p0ahj f /.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an angular speed of 10 f.ia/)ex xahjp0c2 jw)sca k;;9 +tgn.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg unifor0h:rht tcxs5q9r1w ,b1rlgtlr:g; ) sru4 s 2dm cylindrical grinding wheel of radius 18htgx:q,c:l1rs ut52thr r1)wg ;9sr0dlb4sr 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 fs:j:+lh a3j mthat is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, thjaj+h:l:m 3sfe 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 redoph8qi/j+pua81a 0zfuce 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 wo/afh8qpu paz +i0 1j8hen in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increri 4- 8m)wtzol /i+mg i5tthzjt268 qcase her spin rotation rate from an initial rate of 1.0 r8oz6z52wmi hmg+qti4 c -ltt/ i trj)8ev every 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 rotating around a vertical axis through its cem7g4okv6a p6ccrr)2q 8; bpv gnter at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 72 g arc6kopq)gv6cpvmr48;b 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 fic6k8eljk ntrzt/ wsz/) ;9bg0gure 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 momentupuz9j7q r 7,p0 dc;dhem 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 ontutuu;vtkxo * z3(vm,7 gkgq3 3o an identical diskg*3vg , ttkm7zv3x u;oq 3uku( rotating 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;if :fz fcgc8+.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 stands2wr5* cpg-hbu at the center of a rotating merry-go-round platform of radius 3.0 m and mom2uhg p-5cwrb *ent of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is roa1a k(9 ua7zqn8kzlx0tating freely with an angular velocity of 0.8kaq( ua17n l89zk x0az $rad/s$ 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 eventuae; m85eznafe 2lly collapses into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0 z;mene5e 2fa8% 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 invol5msk qxz 6x2d5 (b-gofve 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 f1;x v oz9ngk;$ 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 pllnp4 5w*htaw2d;o(w x2nh w:c:l qo3 vatform, initially at rest, that can rotate freely without friction. The moment of inertia of the person plus the p ;w:wh2l5c3qld owahx:tnw( 2 nv4*polatform 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 jg;a6yse.m8( kmueg. -m diameter merry-go-round turntable that is mounted on frictionless bearings and has a momen8am 6m y;keg (e.u.gjst 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 t617fwwli1g0chwqc0 d6c ,bdc he 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 slipping. What length of rope unwinldq6f1cd0cb,w h1g cc06 7wwids from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side alwt)vv m1n67nu/vu -erh ays faces the Earth. Determine the ratio of the Moonvv7rt -h6en 1mnvu/u )’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 ob7 t5bcq2u: ybo:kv(rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a unif*rq xt8xf1 4hos)ym0form cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constaytxohfxm )4s01qf 8* rnt angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each oca 9r)aw1 r19ypv-ixbf mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when c-wr1ira9 )b p yav1x9it 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 spe1(l 8bfl:),yx ns ja73*p jpoorr;ayced 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 si+sdiotmgn+xhi ,7 g67 *luhl0hl quo1f: wa7 +ze of our Sun, but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost lig+6lho*sx 7nh,w uom : qgd0a1 t7+i7hf+lu mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a g rzc56339i;kja pxcery6/gi low-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 without ig/ey; 6 ajrp9 3grkx6z3ic c5needing 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 uoqbu0; c,whsb2p8z7 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 od0eam8lq. dk7n 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 Mj/f kynb6)tha,o7r z; and length L can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then zft/7;oa,h j)6k rnbyreleased. 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, an.l7rbv og0: xkd 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 l 7vx.g kb:ro0step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat ez zr 2nm-5+vq*-gqy croad is making a turn with a 5zc* z-evgqnrq2m-+y radius The forces acting on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.5 m andfpp ./b:g ic-dv+wy 0,vqte u. begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?y-tdu.pg v,/ +pw qvb0cef. i:    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thin rods, makis e e16/bqu6nrng reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, an1urb/n6q see 6d with arms held tightly against her body.   

  You are designing a clutch tu1q*ile .a4 iu;jv6fassembly which consists of two cylindrical plates, of masa;uvitle*iu61j fq4.s $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along 6:gargh (n qt-the looped rough track of Fig. 8–58. What is the minimum value of a: n-r(gt6qhgthe 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 assx (6cn)*q0co k9dr4u8;, abigf ygxzq ume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uniformlyd m74t j n+ow a01mxw(ku5:asn from 90km/h to 1x 4adonjtw+0a7m n(:5skwum 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