A bicycle odometer (which measures distance traveled) is,3k zy6qyk-e ikfwa9) ( ydwz6qac(h+ attached near the wheel hub and is designed w9zki(dayz- q6 ,fa)ekyk6hq(w3yc+for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angulgrd2 l 4qv32bhar velocity. Does a point on the rim have radial and/or tangential acceleratiodvr lq42g23h bn? If the disk’s angular velocity 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 ang1s ubw6 7*dlnjular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger forceo e)zz; ec q9-8wp*vl qr8)wvg? 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 c(itha7q6 1yqmk: x1 chead than when your arms are stret ymakqc6ti 7q 1:1hc(xched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at thax w09 vt0 rmn nl)yk66s8 llg*h9mt;be rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large tt;w * b06l6nharn0xsg v )ymm89ll9krear 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 ablel4/fli-d wa5) o57qmz :zpgel to run fast have slender lower legs with flesh and muscle conc7i/qp5wzl )lz -4ogefmla :5dentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow byt:jf;k ft/4y eam?

If the net force on a system is zero, is the net torque also zer4.tl jgp1 jl:se;l(zyo? If the net torque on a syst:.l l ejpyt4g1(lzj; sem is zero, is the net force zero?

Two inclines have the same hpx)x e snqnx yn1-))f2eight but make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline wi)-nx fqsy1xxn2n p))e ll the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an inclin 4vdz5,w6n gjbe. One sphere has twice the radius and twice the mass of the other. Which reaches thbn 6g5wzd ,j4ve bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. They nth);f+r ez1vgz)eki/9fhsq/b a80 zstart from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Whicbsf/n 10qrf+vkez)ith)9g; /zha z 8eh has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yetjj2o(/ ps3wi c most moving or rotating objects eventually slow down and stop. Explainw3o/js2 pj(c i.

If there were a great migration of peopqpxag627j6ygd q o57 sle toward the Earth’s equator, how would this affect the length of theo 6j5 p2gqdsg6xq77ay day?

Can the diver of Fig. 8–29 do a somersault without having any initial rr 8v)-n7g5wdimkw.c l otation when she leaves thew .g7vi8l-dw knc5m) r board?

The moment of inertia of a rotating solid disk about an axis through its cenxz,5 l:-s twsater ,x-wls z:5tsaof 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 hol1zy- c6qi.dn5i ah(iso4 qj 0eding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decre6q dse5h iyo i0i.za q1-4jnc(ase, or stay the same? Explain.

Two spheres look identical and have the same mas1bf((pz*3weo v*4cc zyz jox6il . pc6s. However, one is hollow and the other is solizp wfo 3jzvopze4(16* xc y*.blcc6(id. Describe an experiment to determine which is which.

The angular velocity ogaeptu0 c(;tmw4j1 (hf a wheel rotating on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If h(wtauejm0c pg1(4t ; 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 a large fruicbt5ke1t,w -c t o8 0d9u)dheely rotating turntable. What happens if you walk tott0h udu c1 w),89cto 5k-iedbward the center?

A shortstop may leap into the air to catch s;zdy)e8m.zmyg 1 o3v a ball and throw it quickly. As he throws3yzdozv 1).gsm;y m8 e the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conqpq 9 eckn5njo g(4,y,servation of angular momentum, discuss why a helicopter must have more than o j,4( g,ep9qq5nokyn cne 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) yjh -gp5r;o/z0tvg-7qmee9q ixlh* 0 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 coincidenc s+b8yl,ayk q-e. Calculate, using the informayl baqyk-+ s8,tion inside the 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, 34y010e / jdczvy mj;wg80,000 km from Earth. The beam diverges at ;d0j0mg/zjc y vy1ew 4an 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 atp .r2s-q .z*cs4swbdk35 bzzr a rate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the bladbp3 z.2rksdb scq *4rz5- w.szes slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to an;b 7.ek;lndkd f(c s.nother child. If the ball makes 15.0 revolk; .ld(7 ;dbnek ncf.sutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many rez2tz g1h0iocucd y:ds i:2(4*9 jxnohvolutions do the whe 4d(t* :hcyu2ohncdsz9:z0ig ixj 12oels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates4u xif. meixtp1. 2tcv*py- 9w at 2500 rpm. Calculate its angular velocity in yt1ecx vxt* 9f.mu-p wpi42.i $rad/s$ $\omega$ =    $rad/sec$
(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 4pqae)fq )/d+lw f c25z.0 s (Fig. 8–38). (a) What is the linear speed of a chilf )q25ca/wz f)ledq +pd 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 the Sun z mfc;2((89vrjpjo+ igbsa-s    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a f (plz/j.eu(u c)p :mvpoint
(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 tr 4/z.xrp+y ei6uc/fx 4-v gothe axis of rotation is to experience6rex4/fz-ti.r 4 cy ug+xp ov/ an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformlyfua.95 ggmkv //l5j so about its center from 130 rpm to 280 rpm in 4.0 s. Determ vu9g ./a/ fok5j5mglsine
(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 radiushut3fk.zi/qc87ux 3 t $R_1$ 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 4nv3v (l xv7gtspacecraft 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 123vv4x vg(lt7 n-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 r*rwpju:l9/z-x8e) nn mve h r0pm in 220 s. Through how man/8)lrp*-vm h:nee9j n 0x uzrwy revolutions did it turn in this time?    $rev$

  An automobile engine slows down fromn3 zz7m3m33cbdaw mo 7 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 flying highspeed jets in a , wmel.pj:84fmq 6 rl) 8mtoplnm w98owhirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reachin8 l4:omm q8pwel np)f8lm6. 9jmo,wtr g 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 rp 1efp ml* j3hf(emip5w /xzh6j*vq) e,m to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled in this time? / pmeihve*1z)hf5,pf6w*l j3 (mxqje    m

  A cooling fan is turned off when it is running at 850rev/min It tha-li9 i(xy-ifd o w5-urns 1500 revolutions before it comeyfw-9d -l(oi5i xai h-s to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed unifo 2(y/( w bzwbul:r.mnbrmly from 95km/h to 45km/h The tires have a diameter of 0.80 m.(bl: 2nwy(br.bz wum/
(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 ss h g6gqww80hhr/a+ 7rpeed uniformly from 95km/h to 45kh0hg w/+ar 867grw qshm/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 putsdd 34r1pkxf n. all her weight on each pedal when climbing a hill. The pedals rotate in a circle of radius 17 cm. .rfkn d4xpd13
(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 z:sw8,do ri7ja door 74 cm wide. What is the magnitude of the torque if the for s:iz8,jdo7w rce 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 abovm c67p6 dxx.aut the axle of the wheel shown in Fig. 8–39. Assume that a friction torque of 066 dxmp7 xv.ac.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are att)v90h ggq7+8i8 xsl nbrk -otvached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is helg h +78gix to)qvnk08blv-sr9d in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder hene/vn c2c 24nbnca(w(ad of an engine require tightening to a torque of 38ancn /(bcec(v n n2w24 $m \cdot N$ 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 8i qab.mz3ic52wvj s27i pjy9 10.8-kg sphere of radius 0.648 m when the a zjv2i52csi73pqjba8y.m i w9xis of rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a )zbow yl0z493 lyvmw. bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The masvy0zwz9mw4 o .ll by)3s 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 horizontalq8nac0hi pv-: l2k7g u circle of radius 1.2 m. Calc 0lh-ak 7p2vq iucg8n:ulate
(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 vkb 7.gxmuj6 yw2 jff+4i20(yv,xcjy a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and the clay is 1.5 N fx7yv,4vg j2kmc+ bf.xju y (i2y6j0 wtotal.
(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 arx0 im45g*m gtqray of point objects shown in Fig. 8–43 abo5mx0q4gg t *imut
(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 0ecew fout3.:i6r u5smass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the correct rat 9cvgjd -/,5n ,h7fxb najogu1e, 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 tgu1d/ c7a -hv nx59g,jf,onjbhe 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 a6n1,imbgd ;80enmhp4rc o au2 uniform cylinder with a radius of 8.50 cm and a mass of 0.580 kg. Calculatm nc4p 1 roeihu26nm ;gda,0b8e
(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, accelerating7l)yb1s ol2 oj it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-drivenyn )yqa) 9h9uum+d)ws merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the 9mua) usy+hqn9d))ywmerry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at mbbs 4-niyec29 *kq0 i10,300 rpm is shut off and is eventually brought uniformly to rest by a frictional toyb cnim0s*ie2-q 4k9brque 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 acceetsrmvj4 g0x5 7s) .nylerates 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*,we-bs4c, k7:ub l giplh )wkpq(us5, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accek),7*s(we w,uc5gp4ksb lqh bpui :-llerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, as shown inr i(msy.at6+7b42qf * fboj au Fig. 8–46 ob27+a sm .bufjr*t4i(af6qy.
(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 3tx x(0t7 cu 0ffwf/jr2gksl *consists 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 acceleratennbbf3;686 nh; u mk8wyaj b6;jeu)jss the hammer from rest within four full turns (revolutions) and releases it at a swjb6 mjkeb;8uf;6 sb3h)y ;a6nnju 8n peed 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 h s5(k5 qv qz1xif3dh)fb2m ;vlas a moment of inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops sirl,)(akd o( 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 andh5b/4g cnag,a-zuv t 1 radius 9.0 cm rolls without slipping down a lane atabg ,un 5h a/-4zvt1cg 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with re (ojdy14y -i3uv7lkihspect to the Sun as the sum of two terms,ivy-(h1 diyul 7 3k4oj
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius of 7.50 m. How much net wo8 gbj rm .6(qxcz28 *dayci:york is required to accelerate it from rest to a a d6(*c qxy8 bi8j .2crymg:ozrotation 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 1.80 kg starts from rest and rollswt ku0 9ef4 vz)1vshe/ without slipping down atvke1 f49u0v h/ewz)s 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 t:1 h rf q+ouwk8t77.c x6pa fogfp9r4s*:l doko 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 conservation of energsfhpxqr77p6g 9dof:r14cwut. o8a+fk* :lo k y.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end ofd 3 d(wqdd*b:1w8qt(j s6seio a thin string in a dbd i(w(jo s 1:qdwedtq*386scircle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular moeke b1n1nqq1x 8z .qc*cgx( i.mentum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm wh *z81cb1n.qc1x(qein xq.ekgen rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotating fr ;1ls:n8f n:,(cz7plah zn*fbkhp5 at a rate of 1.3rev/s If he raises his arms to a p*kbhr ::z8;7fnf (haf1,nzlscnl p5horizontal 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 theni0;q3u jdz 6;;eq dka one shown in Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when inqq;zd360idune ;kj ;a the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her eh6u 33 b;vrjpspin rotation rate from an initial rate of 1.0 rev evu6 br3 vhe3;jpery 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 rotatingeozox849)hoc j atdmzinmmc: 33u(+8 around a vertical axis through its center at a frequency of 1.5cxd4 z +jmm:an)3u9(i3o 88coz temhorev/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 spinning ep /0k2 i zdf(p,i;gqqat 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 momb1pljnv/ h11u,ee sn-entum 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 cylindrica */yksfi1 ozjw+),j jtl disk of moment of inertia I is dropped onto an identical disk rot,zw1/*jt y)s jfj o+ikating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns6ktk6l4 2h*clletz1e8m vq *u at 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-rop kw9hq ywzk74w4/1lpund platform of rawwk414hl 9qy/kpw 7zpdius 3.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is j83fv ;s yamh ,*w7xwlrotating freely with an angular velocity of 0.8 xvlys 7,;whw*3fm a8j$rad/s$ 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 ha4,meg(eq1zuc,w3n+5wp kkir (i 5dd flf 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,qc1z id emg(54rwkine5(k w f+3,upd 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 wg8rat x;v4-oiinds 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 em0dq56b 0 odtp k;.jj)hp,oz$ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, that-ahqj yi5 x2w(ctb /5r can rotate freely without friiy( 2tqac5r-wh bx 5j/ction. The moment of inertia of 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 a n d7c1ody9fs*ub:l4tam-:oft the edge of a 6.5-m diameter merry-go-round turntabl4n7 f :*:m 9-bcfoultd1dasoy e that is mounted on frictionless bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of the rope lying on the q2 ,-sa8jckd1qpo+fi 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 unwinds from the spool? How far does ,kiq21j-+f8cas qod pthe spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces th t9dfc+ex ,lj(rt:+wne Earth. Determine the ratio of the Moon’s spin angular momentuldwnc: fjr9e ++(tt,xm (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 restp j28.u/dlqnm at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of radius 0.20 mzlj, bn z,d/8qt,h-jk acquires a rotational rate of from rest over a 6.0-s interval at constant ang8zh zj,, tqjl nd/,b-kular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical diskpe 0gq-o8e3 6xwv a:als, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin qepao8al- 30w: vg6xesolid 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, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is- zh-,:hv tgct the angular 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, but with mass 8.0 times as great, wbt+g4,4wwf;5j i+cpms,rc0i * ym puiere 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 mass carries off no angjfi5 c *bt4ig 4+ upmr;,wy+s wpcm0i,ular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollu *,9eulpx maq0tion automobile is for it to use energy stored in a heavy rotating flywheel. Suppose9,* pqxu0elma 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 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 andcknad-;,a2t e ; njlc. ww5n* $H_2O$ 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 m .m4tlvw) 4rctcni du 7)sbnx6;+ y(bsurface 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 pivzn lfo pz366-pmh:,ch ot freely (i.e., we ignore friction) p-zpl366czm :ohn h ,fabout a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. 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 stan9:4o drq+q) irxl7h yskfj+j;ding vertically on the floor, and we want to exert a hk)sifo r j;hjxr794l q++yd:qorizontal 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 h

  A bicyclist traveling with speed v=4.2m/s on a flat road ig0jg(qacd h3p w mr.ti8lg 9ds ))h7(as making a turn with a radius The forces acting on the cyclist and cycle ardw0t 3)c)8ia. 9gg (aqg(lh dhjmprs7e 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 aleg.c vfdn*z 1v+k *(lnd 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 t ck1nvlf vde.g*+(zl*he torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid2oplai4/ z r vuy:*8c,cb wj+t cylinder 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 agac+y o*l/z:tb4 rv8pi,wju2 cainst her body.   

  You are designing a clutch assembly which consists of two cylindrical plate/t -iw 6s6tc7b qxil7hs, of mas7i6 6ts t7qwb/i-c xlhs $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the loo* w* 7zc7s dtnceofwiz0h()r. ped rough track of Fig. 8–58. What is the minimum value of the vertical sze7chwc7wdft) rz0i *(*no. 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 assume72m pvd- ysx10iqc e/w6)enli $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car mgnx,* wtqc-3 z-b dk(reduces its speed uniformly from 90km/h to 60km/h The tirg*(zw- c dmnt-xqbk ,3es 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