A bicycle odometer (which measureszwg9.p 0/q kl7fb /udj distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bi9zu lwjd.g7 /bkfpq/ 0cycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on the rim havnk +gj/.fyx.ih7 jre( /zmdm3e radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial andn. jhye3+ mgrf/ x( d/jim.7zk/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 o,6fw3xb w + e/r.o/pbzviut4if the angular velocity $\omega$ Explain.

Can a small force ever exert a greate*lv3l5fx 858jq rj k2byb )wqnr 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 -q 4cij*pw 6qgzkyjn6tm 09)z7ee) xu hands behind your head than when your arms are stretched out in frontj9) xcpq0e6-)4 ie76wtzynjq* kzum g of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel) +zpqb.: v3v 2pbpaxt and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Whv )+z:qbp t2a 3px.vpby?

Mammals that depend on being able to runua(7 1xb p5 84prqffkihubs+5)b/k ie fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is i iu)47b85x p e(+uqsbf/a1r hpk5fbkadvantageous.

Why do tightrope walkers (Fig. 8–35) 3v5hx qi qn lgxyz5th(f4pimj*j 22(6carry a long, narrow beam?

If the net force on a s4 r0p5t3oc(ve2 hiouwystem is zero, is the net torque also zero? If the net torque on a system is zero, is thpu3vi w 2h5(roco4te0e net force zero?

Two inclines have the same height but make different a 0zg6+o( ,xr e,fhqheo.rs 7awngles with the horizontal. The same steel ball is rolled down each incline. On which incline will the ,7hxwz e,org.ho e(60fqra+s speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (frj/vh;b nv+8ug om 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? Whic/buv + jgh;vn8h has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. They start f bfy1pc:2sg,q rom 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? Which has the greater rotational KE?1 sypfbc:,g q2

We claim that momentum and angular momentum are conserved. Yet 1ltnmdn .dvwq3u/8 ynhd3m 84most moving or rotating objects eventually slow down and wmd/y u3n41n3 d8mnvhlt8dq.stop. Explain.

If there were a great migration of people toward th9i h af.: k)9tmspunu4e Earth’s equator, how would this affect the len ): pf9kuuhi.m49nstagth of the day?

Can the diver of Fig. 8–29 do a somers6e .ut*v rrwi;ault without having any initial rotation when she re ti*. ;wvur6leaves the board?

The moment of inertia of 73pqeg 0ziyy8tfygt)w :,s x(a rotating solid disk about an axis through its center of mass isxp(tw),8 et iq703sfyyzg:gy $\frac{1}{2}WR^2$ (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-i4+k+( hvwcb0 q 0jc58xg 8aeaob4j tqkg mass in each outstretched hand. If you suddenly drop the masses, will4k8b0w+b8 votgih5qcjjx+c4q0(a ae your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and haveyu(9sv+o-inw ;p8lc aynus 88d (-rth the same mass. However, one is hollow and the other is solid. Describe an exper-(y 8o -l+tvr duwna cyh(98 issu;pn8iment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points wesl4+yn; a f+xvgt. 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 whel+gx4yna +v;f el. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a 7.t gtwlhyp23 large freely rotating turntable. What happens if you wa gth7 y.2pt3lwlk toward the center?

A shortstop may leap into the air to catch a ball and throw it quickcddq(qr/wt4d7 (9 ye uum5,vzly. As he throws the ball, the upper part of his body rotates. If you look quickly youmv d7u /dqt,uy4zer cq(9d w(5 will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentas9dte3p3un ny43zi (2kqim5 um, discuss why a helicopter must have more than one rotor (o4( kzny32p93asim 53tneiqd ur propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angx5v(lqm 9d(gysjs joh83 5n( mles 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 9om7en bbzq. 56*jfu q((iekjamazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Ea(.ojqez em 7qf(* k5 bj9iubn6rth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 3hq.j 7vgb-d183ag:8rk uji3r 3t w gml80,000 km from Earth. The beam diverges at an angle i3wdjr1ajuhg8t3 mlbvr kq:g8 g . -73$\theta$ (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 thel 3qa*ormi .m++8kfrx motor is turned off during operation, the blades slow to rest in 3.0 s. What is the *makfor8 . l+rx+im q3angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level flhw a3ijef vga7 c)d:48cgx 24eoor 3.5 m to another child. If the ball makes 15.0 x)aev4j fg7ecd 4i8g3ac h w:2revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diat5,rnyxww7j 3q( mq3:xo sky-meter travels 8.0 km. How many revolutions do the wheels-3,xywwt( 57o mkj s:x3y rqnq make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculd d4wnkk;so5 i+np3n/ate its angular velocnw o5n/4 k pi3kdd;sn+ity 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. 8–38). (r.n5jxf6v5h3 n6zw/geen j m4i4yf+v1 lfa.o a) What is the linear snn 135 mf56lo fyif6nz+h.wa re /4xv vjgj.e4peed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orbit arou4due d66owslfb e*0qf) h+:mqnd the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed o w+u e)dgd0rr su q2aexf,/7ao199c jpf a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.0 cm from the axis o9 7*4twnfbb5fbt5w16e q jwldf rotation is to experience an acceleet t 54f* bdnw 7l5b9fqw61wjbration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel acc hy h 1 +q,zzfzt0b3yl2(jvif1elerates uniformly about its center from 130 rpm to 280 rpm in 4.0 sz vh,yh t(zij1q+20yz lfb f13. 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 ) 4.bhvgf/ ia d9wft7t$R_1$ 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 cu89tcm 9o9z6fu. y lt.* wea 1ssk,8ll+bbjmspacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, etjlt*l1f.om uwlc,a6.su8b+9 8cy zb9k9smthey 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 from5gmto1hg,t j+ rest to 15,000 rpm in 220 s. Through how many revolutions did it turn in this to5gjt t,1+ mhgime?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Ca/79 l 9ts5rybwmc . fqs.cigb4lculate
(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 jets5ht57d nu:kne in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions bed5 7t:k5hnn uefore reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates unl8 5v m ojk5(4fr6tz5bvoqnn*iformly from 240 rpm to 360 rpm in 6.5 s. t vnoj*m lrvb5qfz48( o5 kn56How far will a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned o 2qbl0 eeev *wm3 .;;mwb7zt-s7 yflbxff when it is running at 850rev/min It turns 1500 revolumm7*3es.l w bqf lv7bwt;e zby0e 2;x-tions 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 mak;tkoyl:0z bye3/83dx, e h6mifcll 4p 5 h*goce 65 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0. 5hygbct63:hi l*f8xld lc;4e0epyzo/k m3o, 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 uniax zokd(3s; 2cformly from 95km/h to 45km/h The tires have a diamet (cx3koz2da;ser 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 pe+hgsj1pdt w(z63)o bbz4v7 5 goilcz5dal when climbing a hilv37( h1dsl54zt)g5cgz +z bboji6po wl. The pedals rotate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N d,;entt/v9a k on the end of a door 74 cm wide. What is the magnitude of the ; dte,van/k9ttorque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle ofv d2a t2p-9 nx1q2hsox the wheel shown in Fig. 8–39. Assume that a friction torque 1t2-hpq on2xv a29d xsof 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to t:2 nqt znmm0wr6i.zs:-lzyf f 206sfthe ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then r q fn snzzw26ty06 i:fs:z-.02mt lfmreleased. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening to a torque of 38y yh ijlp v)l(:;xonh 6vfa*-- lpo;)6-vhjaiv* -ylxn:f(hy $m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of radius 0.648 m wf7dev8 ieu ,-khen the axis of rotation is -7i fu,8vee dkthrough its center.    $kg \cdot m^2$

  Calculate the moment of i6w69tqgjwr k.1a n8 abce3uv: nertia of a bicycle wheel 66.7 cm in diameter. The rim aen qa3 8kuv.6c1ab69wg: jwrtnd tire have a combined mass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in a horizrmo ee7d6k:vhqb ib/p: er.4 ( m8uxq3ontal circle of dx::pbr 3ib/ 8eqoq( e v rmu.4e6mkh7radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a b+uz qtsm6 0zhlnmo :*0owl on a potter’s wheel rotating at constant angular speed (mzz0qu6hn tsm*l:0+ o Fig. 8–42). The friction 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 ofchdl(2sq* n0/ywqu 9h inertia of the array of point objects shown in Fig. 8–43 abouyh sq*hc dl9(/ nuwq20t
(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 atom4df teg0+0ehfk9 ra1b m4 y;pds 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 cy2vuyoz;lvz sy:*zy31 jfr2 .o lindrical satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 zsoy2 vly2zv.13uyz:f; r* oj rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 cm and a mxjhs77 v/bsu ep+ j2(rass of 0.5802br + 7pjshsjux/e7v ( kg. Calculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerating it from rest to 3 04,m rm 0r/ 5h+d7kxsp,vcyyi jwlf:m $rev/s$ 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 haj2: b/z)cak1ge 5 mk.ruf0qnv nd-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 r kfe)a1bzn2mqvk0uj./:cg5 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 w yw-6krrj :fl0*a:ng at 10,300 rpm is shut off and is eventually brought uniformfg:6y:0-j* r nawrwklly to rest by a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accel*- m)x0z l a*khf3bao)4deo omerates 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-k7 .kuhb4d isa:t8,lw ;iwd2br g ball is thrown solely by the action of the forearm, which rotates about the elbow joint under the4wr, ki dbiw8s7u:d .ahl;bt2 action 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 considere fght n:)k sp s.c6z2..ncu/jvpxj.9dd a long thin rod, as shown in Fig. 8–4cjd.p sh:..6stcnp)gk9 n.ujv/2zxf6.
(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 x5m6 ra2jy q4nmasses, $m_1$ 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 acceler*g e5pax x4mlc6-p7y sates the hammer from rest within four full turns (revolutions) and releases it at a speed of x elp4gax *c7s5-myp628 $m/s$ 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 oq akr ksriw-l np .1/k/n87jftud85/d f 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 a torque d.6wlkyq s0; pe; ju6uof 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls without slipping down a layssrz-9xv9,h c/ kve 8nxyv vr c8hsk,z -s9/e9e at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the sum j51urql ,n8k iof twoli n85u,kr1 qj terms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius of 7.5vv0m- 4s myr53ney y2m0 m. How much net work is r-r0 3y ys4mvmy mv52neequired to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts fro 1xr2vyc9y(xv8 blv .;cb sqc9m rest and rolls without sl91x;sv lccq(y82bx9 yvr vcb.ipping down a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

  Two masses, $m_1$ = 18 kg and $m_2$ = 26.5 kg are connected by a rope that hangs over a pulley (as in Fig. 8–47). The pulley is a uniform cylinder of radius 0.260 m and mass 7.50 kg. Initially, is on the ground and $m_2$ rests 3.00 m above the ground. If the system is now released, use conservation of energy to determine the speed of $m_2$ just before it strikes the ground. Assume the pulley is frictionless.    $m/s$

  A 2.30-m-long pole is balanced vertically on its tip. It starts :uzj sp)6sffabv )ag0ysd ( */to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Usfzgjd)6a (s py0v /:u*)afbsse conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end of a x 078wyt7jj71qkq a umthin string in a circle of radius 1.10 m at07j8ujm7t xaqk q7 wy1 an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum lnos04 o-ow8oof a 2.8-kg uniform cylindrical grinding wheel of radi4ls 08nooo- wous 18 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotating at a rate5gr jh: gxvw36 of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48,wxrj 5h36: vgg 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 reyd qgh8 f1of 6lb6spyp 9x1s.9duce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0fx6.99s8fdy ly opbs 6 qp1h1g 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 - q9n eqei0ap*rate from an initial rate of 1.0 req*a p i9qe-en0v every 2.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotat,f nq+oszfv 2q1i b90v(o k*uaing around a vertical axis through its center at a frequency of 1.5rev/s The wheel can be connf oq(uvzo 0v*q,9kb+2 s fai1sidered 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 x8bw fqr;w0 )tffa3w*qpst+ - 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 the55be5hquv9 vyb2cft; 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 dropp6fjms c;z;wlh, ;nz2eed onto an identical disk rotating at angular speed szhje n;; 2w ,fc6;zml$\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2q )1 kv(x7il4q jr+pzh.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 cen,zp:1rqinp .u+ouwiw 4 s3g da6f5i*mter of a rotating merry-go-round platform of rad45:u spn,wwuqg p+ oaf1d.6miir3z* iius 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 rotating freely with an anx16 8zp*ijzvr qd 7na,gular velocity of 0.8pr d,znj8x761q *izav $rad/s$ 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 eventuall6: fwbjmvfs1 3a.kn:iu4n pg.y 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 angularakf1bfn:g.sp:u.jm 4 n36w iv 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 in ewix b f3lm fu,z+/v/37ns-gnk:t-kiu xcess 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 6ydobskr v:; l); 5kg*zbf0qi $ 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, tha4;np n j.ynh22 lb6kn0p1uyipt can rotate freely without friction. The moment of iin.nyh0 up 1j2p2n6 4kbpl;y nnertia 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 diica/bu) i fz0.ameter merry-go-round turntable that is .b)fc/ uiaiz0mounted 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 lyrx,u4s))qy a ting on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls)qxyt r4u,a)s 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 sug fb ghm(37m5+qhibk,cw .b8 mch that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular mk5,fmg +8 cgb( mb7q.3whmihbomentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at aqjz q8jv.29* 9mj wheu 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 unifgk/oj((x*qe l v(cn k9orm cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constan *k(o e(9jn vcgkqlx(/t angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of.uiw. 30b4 qnosc5q om 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 nb.wo5o 3s uqqc04mi. of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the ang (9ybfo(4v2tv i dq 9omn1q9qj12bbpn;vw 0cdular 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 ofy5a7z5m j* zfk our Sun, but with mass 8.0 times as great, were rotatz*amj5 7 kzf5ying 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 angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use energy stored9m/ ctfa,wq9 b in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a unmqw t9/fbc9a,iform 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 q:q0i8zhyvj-*mu r3 b an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontalk9r:2t jsbt 2b 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 levv 9j;x ,,nukkngth L can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then 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 sh; 9x,,nku vjkvown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vq9k0it9l5wmay* z ynp b8tf 08ertically on the floor, and we want to exert a horizontal force F 8qwtyz *598pa09nt milyb0k fat 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 is making a turn witszu 0:z;bgw o*h a radius The0:szo w*; bzgu 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 roca 7,gxxl:w ),iapdax9k into a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a ) x dalxi,7aa:x9g,pw rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as ls94;6vtc uwhh8nau( a solid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skuhwl(s 6atn9u4hv c ;8ater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assemb)r4ch is4m 1dtly which consists of two cylindrical plates, of 1m)drchi 4ts 4mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 looped rougvaiaw3 nj 2-vgk d 7 zm7+t;l,1(i yamc3)kigah track of Fig. 8–58. What is the minimum value of the vertical heiz3 ki)a7tw+mya3g i vni ; cvd2mk(a7-la,1gj ght 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 notp0i e:ki-*wmcy q (bquoz 0:/v assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uniforv6 fz8hy 2fvn(g7m1 fk tkkr:-mly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular ac(v-76y vgm nf:8fh1tk2z k kfrceleration 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