A bicycle odometer (which measures distance traveled) is attached near thj o m9 l:3oh8+rlg6jyie wheel hub and is designed for 27-inch wheels. 3io ho g:+l6jrm8jly9What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. bl 97z9 nth3yxww+qt/Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular+byq7t9wz tlwxn9/ h3 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 single3zt8.gteasto g8;rk 6 value of the angular velocity $\omega$ Explain.

Can a small force evwc:wiax,+ p/uer exert a 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 behindd7n415ykz eu1hro 1jz your head than wh1 157dzuh4 ejyzo 1knren your arms are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle hasjiy)92 qmpw r9 seven sprockets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sp q9jy)29imprwrocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower lt+xm: .e5xyx0 q cneg9(; kqsregs with flesh and muscle concentrated high, ygx:( k 0rm ens.5ec+xxqq;9tclose 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, nd.ldsaq 6/um* eq9k1- fsfi;a arrow beam?

If the net force on a system is zero, is the net torque also zero? I)zg;dc hpb yv ;s)88(nx ydbgc 6e2hn,f the net torque on a system is zero, hvgzb 8ys)d,c)e6cn2p bh x( d;;ng 8yis the net force zero?

Two inclines have the same height but make different angles with the horizontalh k bp;t50sxip8hx :(e. The 8;h:sb5ixke t ph0 x(psame 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 sta(lshq-e 8 9cuwj6q3r lu9m(jvrt rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at8ehv sqj93jl( u(w6c-9qlmur the bottom?

A sphere and a cylinder have the same radius and*676cqlo r)jj .nvdge the same mass. They start 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 ac)jjgd6*vlq7er6o . n t the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum aremps d6o /yu5d: conserved. Yet most moving or rotating objects eventually slow down and stop. Expla/ 5mopu6:s ddyin.

If there were a great migratiowi1s; w-o b4gqn of people toward the Earth’s equator, how would this affect thg-w bw1s4;i qoe length of the day?

Can the diver of Fig. 8–29 do a somersault without havin 3cp6(ikoh3nt1. my iggm*4ph g any initial rotation when she leapgi3*4 .gp6 m(mhontci1k3 hyves the board?

The moment of inertia of a rotating solid dis i*:s;f dbavzbwkjya, .4 95ldk about an axis through its center of masssd.,5v:kwayz 9fb*4a lbid; j 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 tybdk8) knk ,6lhgrwsdrq-.)a)v .+o 2-kg mass in each outstretched hand. If you suddenly drop the masses, wil.r,kdvl6 drgkh )+w y8bnaksq)t .-o )l your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the say/.rq z*; nkbrmxs66 xme mass. However, one is hollow and the other is solid. Describrq*y6sx;x. n6m kbr/ ze an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points h,s.c gx 3spjd-z 3bdz1)s k n(g3pi4mwest. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decz1gphmzs)s 3 3 cdsb(,.d4jkpgin-x3 reasing?

Suppose you are standing on the edge of a large freely rotatin 6wzbx i2 x:ujoo6rs)z.z-f.ug turntable. What happens if you walk toward the cozuxrzx 2ui j 6s:wo.b .f)6z-enter?

A shortstop may leap into the air to catch a ball a5z :vk g6jz:aqpo9)cf nd throw it quickly. As 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 the oppfk6 59oz) cj:g qv:zpaosite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of tpl7 h9lmgbg f7:z8fa /nj+10q iy)9bfq8kq zangular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss 1qk8fz78fab+ b9ifh :9 qp/llgyzntm7 g)q0j one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angl+x(r q3 tmz(tg,fz7phgujz ;1es in radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazi0gn(p pu18ypu ng coincidence. Calculate, using the information inside the Front Cover,gyp1n pu u(80p the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is direp4z lm,2icqbd ; sid2*cted at the Moon, 380,000 km from Earth. The beam diverges at an anzd*4 il2m bqic;d,2spgle $\theta$ (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 moto9k/gwf5 z 5argr is turned off during operation, the blades 95wr k5ggz/fa 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 af9lb) x-2y;6 juy -tcvmuie6c level floor 3.5 m to another child. If the ball makes 15.0 revoly6;fiuue lvm)j 2 y9bc 6cxt--utions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutiou3:e30lk:ndu ot ,b6p exqf4ons do the wheels mtl nopuq : x o:3e3fbk6de0u,4ake?    $rev$

  (a) A grinding wheel 0.35 m igayx7e7 /ul q*ke7lg0j)j oh* n diameter rotates at 2500 rpm. Calculate its angula7/julk )qg*ea7go x7hlyej0 *r 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 rev42(mmlhqw x y5olution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from 2mmhl5yx (4qw 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 eg r 6h/t1ci7o 5pgkd2ww1)dmaround the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed +r9z jn 5tjwc8y -pyl-of a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particxh 9jd)btn(6c )7firohqekd8 +o 4m v*le 7.0 cm from the axis of rotation is to experience an accebivfqx dodhej67 r)9 kh 8m*+on 4)(tcleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel acceler l:6 6 bqwm-ml *8jkk8q+onjbzates uniformly about its center from 130 rpm to 280 rpm i:jq8z-6 wnj m*m blqbk6l8o+ kn 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 radius m,t* knsbj11i $R_1$ 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 putpqo+uui3087 q-7ojb:qkmuk g themselves into a slow rotation to distribute the Sun’s energy qk0bj+kui7g-pmqo 3 q:78u ou 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 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 f.x2 pb2glxzn * 2hfnl+rom rest to 15,000 rpm in 220 s. Through how many revolutions di2pgxl nf2h.nx+lzb*2 d it turn in this time?    $rev$

  An automobile engine slows down from 4500 fxh + npco bg/43/ewk5rpm 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 flyingls ew6+vx1(wj highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolu(sw 61velj +xwtions 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 accelerathl4ne /0c3 dwc e/pfi-es uniformly from 240 rpm to 360 rpm in 6.5 s. Hfcpce-e 0w/i3lnh4 /dow far will a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 1500 rev-mp ua,f) v/uyolutions ,yuvu/ apf)-m before 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 r2kiroz)jbo 9)ghq:r )evolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameteqg hz)o b9oir2): )jrkr 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 frvup vy,2f2+cq, (q xgxom 95km/h to 45km/h The tires have uyx,q,v+g2v(pf cq2 xa 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 climbiy+zpn1ze l2ug 6 zzv2h) +cl:qng a hill. The pedals rotate in a circle of radius 17 cm. h+l 2puc2nz6+lz vq):y z1e gz
(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 o8 r.d 0e7gjmsnf a door 74 cm wide. What is the magnitude of the torque if the force is j0g .r7 e8smdnexerted
(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. Assd r j xrwc44r2pw)f7;i4t/j xkume that a friction torque 2crdr4p xf /tk74rjx;)jw4w iof 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, a:3f () rvcda)/iev*gb yyl(la re 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 thevg) i/ e(ydafacll)*:3vb(yrn released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head hn64cu(gnu l/5:/ lxrr ,1bl7v 7nnj9cis jiwof an engine require tightening to a torque of 38vu(wjnglcb5/1lnx7rlji6c7u ,r n:/ih9 n 4s $m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphe j3y/.fchr( b9xo7otg re of radius 0.648 m when the axis of rotation is through its cent73f r. xgoyh/9ojb (cter.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm ino s)1 uc*pij,f diameter. The rim and tire have a comb1 ijfc p,*)osuined 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 ist8t7c8)(2 zen*x po xgjn -oyx rotated in a horizontal circle of radius 1.2 m. Cal8txg*-tnxoy7c2 8xnozjp ) e (culate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating at constant ang z sd0*r(tqp1eo0 ah:cqs-l g5ular speed (Fig. 8–42). The fr0hdq5o(r cp 0agel1 t*- :sqzsiction 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 inertaiqnum/:f9s0m9tzd 6 u (*+m xmbf8vsia of the array of point objects shown in Fig. 8–43 about *u:usxfd9 t /va+q8mm9 fi6nbm ms(z0
(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 whoseg.jt rxm 1v im/+2vek. 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 spidq -8rt8,r ,r2fh4bsqwr :e qpnning at the correct rate, engineers fire four tangential rockets as bhqr,: rfdq-848w sq r rtp,2eshown 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 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 ;cu,k5a9ke ,rxp w dbx(7j z: nlhvda4j(7x2duniform cylinder with a radius of 8.50 cm and a mass of 0.580 kg. Calc:4z,jhxc, ;k dd u(nb2xavr 75 l7a9xdkwe(pjulate
(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 w,m6o uev6o2j, accelerating 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 tangentiallycddqk 8*s6ouxh ; kk1. on a 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-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 kc; s1 k.xodhq*d68ku produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is eventu0i (6cfijin 7l3vcr4c88 ae dlally brought uniformly to rcvcc7d8 i3jl4nrial0 (86e f iest by a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a 3.6-kg ballxo o jh6rq-9,m 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 kz9yt91/sjqh by the action of the forearm, which rotates about the elbow joint under the ack 19 /htyq9zjstion of the triceps muscle, Fig. 8–45. The ball is accelerated 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 lol dwdfc 2(7r9ung thin rod, as shown in Fig. 8–4fw( l2du7crd96.
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine consists of two ma1d 8*vkgh 3ezwsses, $m_1$ 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 accedgibgg(.9cjt i . :6 n9midxwd* pp,.rlerates the hammer from rest within four full turns (revolutions) and relea(i.96w rg9..nd pgxdpm gidb *:,tcji ses 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 of;u5-pxw z9yqb8 -sscjg w .a6g $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine de/g9zt;vc g18j2mh ch,ib*gh ovelops 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 rolls withaot2cq7ls d6+ j: 9edjout slipping down a lanesc:l9+ao 7dtqj2je 6d at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic e6mwyj g5+2gy dnergy of the Earth with respect to the Sun as the sum of two termsy2 md6g jgy+5w,
(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+1fkmbkt o:p4 p4u5:vlfyh 0k kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolvb14pfk0p:l yfm:u k +t4ohk5 ution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts fro9gk;u nz(y1cim rest and rolls without slipping9(ncyiu; k z1g 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 bhjj 3t u*foo50alanced vertically on its tip. It starts to fall and its lower end d oof5t juh3j0*oes not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg balm,e2c.0es7z7u5j d-ogtah fs l rotating on the end of a thin string in a circle of radiuzsj dae , 2m77o5gf0tshu-e.cs 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylig(-s 9b;kl ffz,e;j4g yn3dyk ndrical grinding wheel of radius 18 cm 3;;nk9-g ylz4gk( eyfd, jbfswhen 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 platfork2dbi4n 82juy -hm8wg m that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed ofm 2u dgk-8yw h4ij2nb8 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 her moment of 8cyp -wu/: luqh3gp5hinertia by a factor of aboul3wyph-cugh q: p/ u85t 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increas)y6lfyoe+gw nhb y 723e her spin rotation rate from an initial rate of 1.0 rev every 2.0 s to yf2wy+6 bngeylo3h7)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 rotatitynb6 rm/4dv uod9p ;)ng around a vertical axis through 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 yod n/9rtpbd)m4u;6 vshaped 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 8min tfl0 d/gu*5vw 2, *zbszjitt9* gspinning 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 momenq wn*x o48nn-ntum 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 onto avsxj 7l r0p( kfuolo/ty1:,d(n identical disk rotating at angular speexk,r ll (d 0:s1/oot( ypvf7jud $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns extk 2+j vcm(3at 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 cene(1q t 8wnr)upter of a rotating merry-go-round platform of radius 3.0 1eu8()tp rq nwm 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 rotating freel*t. 0xkh0udxu y with an angular velocity of 0.800x. *uu xktdh $rad/s$ 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 in2w*7usc 7w4lift*mcz to a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming th7*f u2ci4ls*7 wzmwct e lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds*vdjq.l-7eeoj y* c-s m21bu s 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 d 4)xunenj9 -c$ 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, i. eeff8s7awvz 5o 6s0)hpwc8that can rotate freely without frictioe ioh7 s8pw58zecf)6 avs 0.wfn. 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 at the edge of a 6.5-m diameter merry-go-rl pxv3i((g5 nround turntable thatlvp 35((nigrx 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 edbk c22*b8c /)rlw ., unxgs2o ep1jylying 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 rollw eb)*cjbyr.ed2 ,xp8n olk/2 cg2u1ss behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such thatb6uis i (c:kh; 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 momentum. (In the latter case, treat the Moon as a particle b:ksh c6iui(; orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate oj l +9757 x4bod7f:hy kfmcvtbf 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;n z8aqto)a q0d;qj da9mz1/p radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque deliveredaqz8p) z j/tn ;0add1mq;qao9 by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical uw-09 lfmhf8gi:sv9l u,re0 odisks, each of 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 it reaches the end of its 1.0rge l ,uvi:o -wl9ufs 09m8fh0-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 speed of t453,nkmt in )wvstqx)pq 7s zeag317k he 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, wpan/l/8wgks *ere rotating at a speed ofanl/8*pwgks/ 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 angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a loho0 q*t)m /cnsw-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)th0*n/o smc q 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 illustratesz *kq r6*h;hq +xem7wp 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 (hu)kk(ra5 knz6 oop) is rolling on a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., we mi zmp9bfj9u.w 6*2b v;r: sheignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released9rbhwm9ue;2bfv pzm sij.6: * . 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, and ar ,30 gwa ym7lmt3bv-we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step ha7 agaryml0,-mt 3vb3ws height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road isu5 /bvbpzc28 m making a turn with a radius /b82 mzbp 5cuv 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 and zj yx z.:/1dqmbegins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate oyjx: /zdq1.zmf 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thin rodsd (u(z-v0qap6: sagye, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretsq0 (y6gae :vp(d-auz ched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindricql- 6mpkj)6h1pjwr9xnnee a9w;tx ) ,al plates, of9n- jx nktmaphwje1 xw;,)r6l)9q 6ep mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 alo7sk yukx : jaxw)e-ug.o6)i1ing the looped rough track of Fig. 8–58. What is the minimum va1)) esgx i:6wak-u.yok7 ixjulue of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, bu-nmwprnbe0aq 0oh z 3; ogg i+5d25xc5t do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as th27my- og a:4oqnu bph6e car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m-p7yn:ombqu 4g26oh a. (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