A bicycle odometer (which measures distance traveled) is attached near the ws; /xw+hyvk,e * 4-znevpr5ne heel hub and isv nsr,/ + ew5xekp;e4y-* nzhv 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 pogh 7iq*-ff ia)int on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point hav) -ah*7fq figie 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 i+,r 8-ga-.;ytxbzlxst5m k+ xx7y qtangular velocity $\omega$ Explain.

Can a small force ever exert aq-/sykt *mt*4n;k mau greater torque than a larger force? Explain.

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

Why is it more difficult to do a sit-up with your hands behind your head than w-1sj (xm8m9 op b;c j9b6dxnqmhen your arms are stretched out in frocdmq9 b( s jmo 9jp-8xx1;mnb6nt of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the a8wx,80 1et i(ixjqizxw+ ab0rear 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 sm0w ,bz xq1x+ia0 ti 88ex(iwajall front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run (k )6kdisuf,81 bo3ovr u/kvafast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotationa(srao3bu if k,8/d6)vk ov uk1l dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow be d7cfuo0a 4n*n8,ujiun5 zeu 0am?

If the net force on a system+lhj.ygm ,u / r rp5cfg;fn:e+ is zero, is the net torque also zero? If the net torque on a system is zero, is the net force zlgg,/p+.huf 5;ce r n +jr:fmyero?

Two inclines have theub5bm ud*80z m same height but make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball *0zu5mmbd 8buat the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an incls4r4 boa,ou gn;2jqzs3m( *ey ine. One sphere has twice the radius andu 3yaq,4bremn 2gozss j;(o*4 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 thepfg:yxmw j8c+dkxm:k;h r4 l4pt5vg,ave) 7( same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater spe,x7gydka)h:ervpm +gl(4 w 8j : fkx5 ;tm4cpved 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 moving or rowz6toauk0q-n(2g sf -tating objects eventually slow down and stop. Eq( -62nzfstkgu0 aw-o xplain.

If there were a great migration of people toward the Earth’s equator, hoq1d b4y;o( :z5at sgk51brbofw would this affect the length 5o tbf bka:1yobgr;z1( q45ds of the day?

Can the diver of Fig. 8–29 do a somersaub )cin1tr1o 7clt without having any initial rotation when st1c nr)o7 1biche leaves the board?

The moment of inertia of a rotating solid disk abou+.c/*fjvq ss bt an axis through its center ocfq v/jb.s *s+f mass 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 ;fij:cdh egx)1 f4x4ua:je z;each outstretched hand. If you suddenly drop the masses, will your angulzeej 4xx1 f;ijchadu;::4g )f ar velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, qelm 8s+47glg r8bbqgh b0m.9one is hollow and the other is r hbg+mm8b4 .b9glq7leqg s80solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizom1g03/v :l;/hy 4 u2p(kc ln w)vbjv9hqtqqv rntal 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 th rv l)hpcqqwgumq 9t4 (1bvjny:l;3/vk2v h/0e 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 edgewf vqt .wd /;1t4t-qqi of a large freely rotating turntable. What happens if yo tqwqt;1 d4 q.w/-vftiu walk toward the center?

A shortstop may leap into the air to catch a ball and throw it quickly. Aweib7 cx31(wf s he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in t fwiwc1e b(73xhe opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discu00rzm y e*zgo5ss why a helicopter must have more than5 m0e zro*g0yz one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (a)wm 0a8wj*8d/ldb3a sy 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 Ear ok6vj1kdm:ztxq sa-0fb. f -0th because of an amazing coincidence. Calculate, using the information inside qm 6xf sk-fokt d.0: j-0z1bavthe Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 3f /d6 z)x a0qvmxb)tqdr( d9 kr(pr3q;80,000 km from Earth. The beam diverges at an )p d9br/mz0qq)rv6kddr( xfq3t(ax;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 of 6500 rpm. When the motor is turned hiqv (i(-+gdm off during operation, the blades slow to rest in 3.0 s. What is -+hiv(di q (mgthe angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 mczv j,m3j+ml x33 bjt- to another child. If the ball makes 15.0 revolutijt-lzc jxmjmv +,b33 3ons, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many reo6,w)ow: cpazvolutions do the whea:w6)c oz pw,oels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter hzxm.54pyk*id3r ho; rotates at 2500 rpm. Calculate its angular velocitypr5z.hy4kodhxm;*i 3 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 (Fig. 8–380zcy6v 2,ixrxx1e j )0mq/uuu). (a) What is the linear speed of a child seated 1.2 m from tc1m0 uyiq 62,veuxx/ rzuj x)0he 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 a dqj 2hp;*+pl a w+urdeo1 g7p h*ooi0+e.g8teround the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a ai*s )uj vld f(r5j(*bpoint
(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 rot6wo* ilv0ogj*ynm)2l t5 eyu/ate if a particle 7.0 cm from the axis of rotation is to experience an acceleration of 1o )m v/2t*oyei50*llgj n6 wyu00,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel acceleratec w:y30hn mg;5nw/ wx cl3cnb.s uniformly about its center from 130 rpm to 280 rpm w ngw:. 53nyccwc0 nmblh; 3/xin 4.0 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 radiusyz m :0m;a.gbs $R_1$ 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 l if/fsn)3twar+yk2 4aboard 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 f)4/wksa+nilt r 2f 3yspacecraft 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 22tc,7b,zs cuek.6 b1yf0 s. Through how ma 1k.c6 sf,,b zuetc7ybny revolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calculay1regkia4-kr 9 q19h d w-i7mvte
(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 hi/ro, v up pqt66*au0ah gai;7hghspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revi6,/ o ga;uhap6h p0*tuqa r7volutions 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 acce-9)s.kbbnq hev qjgu 4j. u.u+lerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the w e qb)vjngkq 4u9 .-h.+usj.ubheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min +tb(c k2 iaiaqb*d3u)It turns 1500 revolutions beforeib 3 dc 2a(q)+b*kaiut 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 revolutions+8v0wmt. 7tano3 b jnm as the car reduces its speed uniformly from jm.ta8w+tv 370b onmn95km/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 uniformly kww:-0n x/a ugfrom 95km/h to 45km/h The tires have a diameter of 0/w g:wau 0x-nk.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 hg7h*hd ;fkd+ a bike puts all her weight on each pedal when climbing a hill. The pedals rotate in a circle of radius 17 cm.h;7khd fgh+ d*
(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 bayktd 4qet-x l:9/z,door 74 cm wide. What is the magnitude of the torque if th4yq9dt bek la/xt,- z:e 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 thikr 6/no*1h2*hasliwzoi +2c e wheel shown in Fig. 8–39. Assume th i1/io*hc*+ r hwa2ois62zklnat a friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass n;3i6m l *pfxup8vo z8m, are attached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magniu f6n; pi8pv*o8m zxl3tude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require twva/)8,djbxll. ) pnx ightening to a torque of 38 8 w)al,n ./plbxvjxd )$m \cdot N$ 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 jtd3c -(.tze w0e+to q53 -pc0szw*gsdtbfk 9 of a 10.8-kg sphere of radius 0.648 m when the axis of rotation -zs*f5tjeo te3w-dq9dctsg0bc3 w zt+ kp 0.(is through its center.    $kg \cdot m^2$

  Calculate the moment of inert r3;+opln qun,u5 sdq37max5uia of a bicycle wheel 66.7 cm in diameter. The rim and tire have a c5 7au;oqu3lx,rdpnu3 n s+ q5mombined 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 horizontal sun8hu 0+2tfshzfk+*ri x9y 9 circle of radius 1.2 m. Caiu*skf +rhnhzsuy09x9 f t28 +lculate
(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 potte k7n w7gf 6fv9cmdp0nikfk2; +r’s wheel rotating at constant angular speed (Fig. 8–42). The frict9pn7 2f7wvfk6 fdi0 gkcm+ n;kion force between 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 poif /y id2v)z4 bh,ak;kknt objects shown in Fig. 8–43 about k/z4d;ab ik)vf2hky,
(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+d x,1r9efcb)u he c h;y3a6tl 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 s1qt2h4h04n83 hqmi a* uy xfwhpinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite hau *mh4aq402h q1fhwy83th xn is a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a unil8uzt ba7,n:an cl 1;xform cylinder with a radius of 8.50 cm and a mass of 0.580 kg8nb;an :,a7z xtl1u lc. 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 k uci8nx /*lqp i35mz,rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on ap4q, a ytus;1h small hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-g 4uths ,yq1;apo-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 shuzj,tkswx+vg66, tw m5t off and is eventually brought uniformly to rest by a frictional jt5mz 6t,gwsx wv6k+,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-kgc-5m1r 9 dsigm 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 of8) 90idrbrqnm the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformlr9br8qin d)m 0y 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 bll d8)i +7i bsf-uzje.iade can be considered a long thin rod, as shown in Fig. 8–46. +fb uzi ili8es)j7d. -
(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 conidb9l*ctmoptn j o8, (e4jg) 8sists of 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 nz2hdkmuw1;- four full turns (revolutions) and releas ;zh muwn21-kdes 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 s*tg b.ln8fp6of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine dev 12p3ht n)reglk 0v:ahelops 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 uslq)ii(,3vh xn /6mkradius 9.0 cm rolls without slipping down a lane v 3nlix)s hk6im,q (u/at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the3ypqy r 3y*f:c/t jst7 sum of two3pytcj/:*7 yyqtr s3f 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 o*v.e/lj sj6n radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of6*/v.nl ojejs 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts 9 82e7qqhoa xuo4* a7z+r wtlvfrom rest and rolls without slipping do+twv4arhquo*xe z 2 a7olq987 wn 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 verti4 (i/oemr-d lb+oh gi-cally 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 omideorh4- ig+ -b(/ljust before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of t 9zr7vx9jf -ca 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an angular s97czrtjfx- 9 vpeed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grinding wh )d 8v5,iznpekeel of radius 18 cm when ripz , 58)dkvenotating 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 sideh5po.zeu q ;+w:kkc-pgt4 :x k, on a platform that is rotating at a rate of 1.;:k.o- kt:pux5ph cgqk+ezw43rev/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) can reduce heq y8h.0wfbvn+n 7x0gg(ie7 aui v3)fzr moment of inertia by a factor of about 3.5 when changing from the straight position to the nhyu 3f(+xaf7 .zwq0e 7) gin0vb8gvituck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial rate y9+d xy rf;; rv4nag,c,uza5 iof 1.0 rev every 2.0 s to a final rate of 3+ug4 rr ;yvnciy;fazadx,95, $rev/s$ 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 thro kpe 9bocrlw2 70ii0-blzkab1e7yp*9ugh its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What i 0p2kbi7a i01w9-e7bl zo9pkebrc l* ys the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a n5fj:gy xxcp: ec55dva8w 4u3figure 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 of q3*un39zs : hy+kgysqthe 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 onto an ide 1sn cqq0 h,by(nrabinj6bpnzvs ,:r;65q/v 6ntical disk rotating at angular r5b(b,svq hjnsnci1:a6 0qryqzp 6 n;b6 vn/,speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.pt z2ymgh/u+ )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-3a hb.txsu98gjh ov, 4round platform of radius 3.0 m and moment of iner, 48o3jxgh9vhtab. s utia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with qzd3d9+ r(x,doeu9mm an angular velocity of 0( mro,d3xeq+z9um dd 9.8 $rad/s$ 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 eventuallygg aj1 vox/0w7da ia hj,d9zc,+*b0nc collapses into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?n/ha z0d jb9o0ixw1,j*g aag+7d c,vc(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in excej6zkva 14( p.t;nmuctss 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 lhr+,k maf m 5bu*q/i7$ 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, initia9i,ebrny,g, be qm +b1lly at rest, that can rotate freely without friction. The moment of inertia of the person plus,,b1+rige9 enmb ,by q 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 diama. ior(rv 77te0f *m2shn 8eazq +svt8eter merry-go-round turntable that is mounted on frictionless beari+*7 ntfi om8vshs 072r tavr.8eqzae(ngs 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 the rope lying on tia pt0koic836dcm) n. he top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, d ica60cn )ipt3ok.8m 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 Eyfvg / ozsf m7*mjj:/0arth such that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular f7 :/zgofmy mj*0jv/s momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a r7m+, kqj8in )pmz-jnc ate 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 uniform cylinder of radius 0.20 m acqkqd pi2mw2bq*:7x j50f eb7esuires a rotational rate of from rest over a 6.0-s interval at constant angular accelerati2 pmb 7*2qfx0b 5kdsqe7 jwe:ion. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050eaah)v 5c3l ,hh82udsj ax9;n 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 it reaches the end of its 1.0-m-long string, if2)ahal dh ;au8s hc,jn 35e9vx it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of the rear wheel(z-bn gdpfi;3 ($\omega_R$) 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 wit s4ix718d mrhb5xpq1th 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 rotatior pqb1i8t1d54 hsx xm7n 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 energscbw, 2h, tu(zz jwv9-y stored in a heavy rotating flywheel. Suppose such a car has a total mass of s2h zj,,-9(tcw uzbvw1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates anmy/bvrltvti8 . ( nt9j:gl00 e $H_2O$ 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 subht*b- ,q6 gxurface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., wi+iso+ d ;mlf sr65.rk, p1fcfe 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, determinefsmor6 pif.f,d;+l1ic k sr +5 (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, and wi: dcnlu2w 8e5e want to exert a horizontal force F at its axle so that it will climb a step agai52le wi8n:cdu nst which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with spe/ m3e.:h2:nknmdlk keed v=4.2m/s on a flat road is making a turn with a radius Thelkmenhn.3:kkm2 e :/d 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 lkhv x)0:tzgf6ength 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the 6g :vtkh0)fzx torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her aj96 ol ip-j:wt8w5 o fdid03g fc1vp1srms as thin rods, making reasonable estimates for the d 6 sdl9:5 1fvcw1o-jwj30 tfippogid8imensions. 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 clutc yg5* ucepg-w toq7j/-h assembly which consists of two cylindrical plates, of mass 7-5 cj ypgtqe /*ugow-$M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 alovs) :r)ge9pkkng the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h ve ) s9krg:kp)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 n*g4es .vflv ,not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the 6sjzs 89qym r(car reduces its speed uniformly from 90km/h to 60km/h The tires have(yszs6rjm98 q 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