A bicycle odometer (whic2 7g anbv *fs;s5v whq/un23bnjza9947luhql h measures distance traveled) is attached near the wheel hub and is d bbslg/4vzn99 3lw q*nn ;fa2js2h57 uqha7vuesigned for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a vuiae:fudh3l w* ; **x 0hi6fnpoint on the rim have radial and/or tangential acceleration? If the disk’s angulavu:wlunf0h h*3ie x;a* i6f*dr 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 angular vhp5jav d*q1qto ,r-p-flkkm )j3z/3 felocity $\omega$ Explain.

Can a small force ever exert a greater ecltq/mp/m jo 6+m( *xb)an hhi0z8)itorque 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 si:swu0te-xgj0-(hy semu */v kt-up with your hands behind your head than when your arms are stretched out in front of you? A diagram may help you to answer thw(0ev e/tyushm*- :-k0 jgxusis.

A 21-speed bicycle has seven sprockets at th/yxtwje4k;(yu /*4, pu a jcyre rear wheel and three at the pedal cranks. In which gear is it harder to pew/, (y u*4xrkuej /;jtcay py4dal, 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? Why?

Mammals that depend on being able to rcsq ow348- gq)3b.:qij)ww(ye r udggun fast have slender lower legs with flesh and muscle concentratedqd 4uqqg c3go3))webj:g(iw-yws r .8 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. 8–35eoh woad,bt5 +15;i-mx71n gue 5 qggz) carry a long, narrow beam?

If the net force on a system is zero, is the nebj5q vauylfk 6/o zc*7c/5 m+et torque also zero? If the net torque o j7fcze*kv/uy5cq5 l6a/ b+mon a system is zero, is the net force zero?

Two inclines have the same height but make different angles with the h y1px.0ool unw,u;.42 z.pvfk mpttwfx32my2orizontal. The same steel ball is rolled down each incline. On which incline will the spefx omyfy 14t m ..n;u,p32p2zpuvlwx 0ot .wk2ed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest)o) x+d b5 byk+fsi5.hq down an incline. One sphere has twice the radius and twice the mass of the other.k)dq.sby+i+b5x5 h of Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. Thk-fzp yb1fx * *dgwfx6w 12ka9ey start from rest at the top of an incline. Which reaches the botto*9gx6afffxw-dkb1k*zw 1y 2 p m 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?

We claim that momentum and angular 5qsr, 2-5zfc9v id*iq yrg3dr momentum are conserved. Yet most moving or rotating objects cr9 rq3 2 ,s 5d-vqf*5riigyzdeventually slow down and stop. Explain.

If there were a great migration of people towa dcsehv fg4-h;uv1w1a4 u7 ck*rd the Earth’s equator, how would this affect the length o uv4c7chu11e gk4 av; df*-swhf the day?

Can the diver of Fig. 8–29 do a somep1t4c:l2+xt g, srhhe rsault without having any initial rotation when she leaves the bxt e1 :2thgrsl+,p 4hcoard?

The moment of inertia of a rotat ;3fytjct m70jaxh8 8cipjp,9ing solid disk about an axis through its center oa hc c8ytjjt,pf;i 9837xmj 0pf 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 rotatingmvtm-mj2 b3*e stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velj- m2meb3* mvtocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is hollow and thyr,k mx4f8z4 k+c1pw*rj+zece other is solid. Describe an experiment to determine which i*+1 jz8 rfr4xwc 4m kzykce+p,s which.

The angular velocity of a wheel rotating on a horizontal agk m,rgi ixumt;)c)z ax,/0+ lxle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decrea mz xc0ug g,),iixt;+/r)lm kasing?

Suppose you are standing on the edge of a large piz auhr- b;ewlxrnco2.8;44 freely rotating turntable. What happenw -c u;8b2.aoerp4rhn i;zl4xs if you walk toward the center?

A shortstop may leap into the air to catch a ball and throw it quickly. As he th 4wwx t46rxs,ly1ds9a rows the ball, the upper part of his body rotates. If you look quickly you1,yt4wa6 rdls49wxx s will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservatizon d4t trv12s e-8e9yon of angular momentum, discuss why a helicopter must have more than one rotor (tno 9rde2 - tv8zes1y4or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radianyv9og/md1 w5-iz e 4vbs: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincidence. +r tsxrx;v z156t9ax28t wm aaCalculate, using the informaax 6+ma8t wzs;t 2x1tv9xar5r 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 directeduvc*syl d(;(t at the Moon, 380,000 km from Earth. The beam diverges a*y;u(dt scv(lt an 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 rotatnh v m4k0qs 1fc7i.nt7e at 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 accelerati.7h7ifsntn1qv0m k4 c on as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.bfn;05gsg 1;jgm /xop5 m to another child. If the ball makes /j p;s0g5m 1xbgnfgo; 15.0 revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revoluqv30-,/ ls + ryhfkcuptions do the whe,+ q 3yvchsrp-0ukf l/els make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate its6 ytsi:b5xu zag(y75bj0dxi*pi +/a k angulxx:ki syy7g p 0*5 a(du5azi+ijb6/bt ar 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 9k mjz*f-tel: one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seat:kl*9f jez-mted 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) in0ufl 4-ed)q vh:i w0te its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed orj jc8zu7 f4b(f 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 oycn8 p 8raii/;;c2vkfrom the axis of rotc8yr pk;vi/;o n8a2ic ation is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformlyf0g.,rt6siug,-rhu * vjh 2gy about its center from 130 rpm to 280 rpm in 4.0 s. Determ 6y,fusg j 0-.,vrit ru2hgh*gine
(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 radiuszi* y3dfci.i35a8tud $R_1$ 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, -sofcl67o l 4bastronauts aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. Ac-4lf6oosb7 l t 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 acceleratevb,+ij)* ougjy kqd* 67 jpm.es uniformly from rest to 15,000 rpm in 220 s. Through how7u*.k) oe*yjq, bg +vjji p6md many revolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 +l 12i0ph0mdlxlceql:;jmdrnoy/36 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 jets9 wglk4: h- +fthh1n7fx ducl) in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 completk:lx4t)g+h f-dcuh 9wnf 1l7h e revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm tmjex+nj80 ug- o 360 rpm in 6.5 s. How far will a po0jmux en-8+ jgint on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned1lcm arheaq5w k -v+:f;pq2o/ off when it is running at 850rev/min It turns 1500 revolutions 1h +;ae rawckfq-polq/2v m:5before it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces itk j-hns66 vp7ls speed uniformly from 95km/h to 45km/h The tires have a diameterjn6hk -67lsvp 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 red t8 vnj3 likb :cl()s1 n)4juiktv: .h:ltoc;yuces its speed uniformly from 95km/h to 45km/h The tires have a diameh 8;ijlvb(:n1 kvcsijonult ) .4lk:y:3t ct)ter 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 weh 8nvqth*080zgtumo .3; hj(w 4toakgight on each pedal when climbing a hill. The pedals rotate in a circle of radius 17 c* 0nt0.q8mgu 3 g4(;ojk aohttz8whvhm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cm wide. What is thqr, )j7zs(ob fe magnitude of the torque if tbj7o(,) fsrqzhe force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wh:zgll* 8x4 jex9sxg3/uzyd3ueel shown in Fig. 8–39. Assume that a frictio xg 3 gy szlxz:ed4x*/9ulj83un torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a massless rog24mg3t9sk c ad which pivots as shown in Fig. 8–40. Initially the rod is helga23g 4sct mk9d in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinb ui6yl9a7r:4v+ y7z7czxu tc der head of an engine require tightening to a torque of 38 4 z7ryu7vtcbl+: ca 967yxiuz $m \cdot N$ 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 mjv, ,kfeu lf):/q+zrg when the axis of rotation is through its gz/ ve u+fj,lfqkr:,)center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. The rimcw)eif8 s8zl9kyy .s;khi g4+ a+k y8zf.i8keg ;)hsc4i wl9ysnd 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 hor43p b ( q9khb,njg yvd6pxz,r+izontal circle of radius 1. g (+dqbpr npzv,94xh,j36 ykb2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl onudd)*i: v 1q3gy ki+b poc*;vb a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and the clay is3q ;*vbu)igb1 iodcpv+ykd: * 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point o( g8t,qkstd k5bjects shown in Fig. 8–43 abo5 dst ,tkqk(8gut
(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 whosee1j w2-mkln:l ;flt+h 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 spinning at the correct9 dtrcg u 1 vql0/)*a;gy;,jv/kjydti 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 q ;ytjdjrak//uv*9vtc iggdly)0 ;,1 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 uniform cylinder with a radius of 8.50 qo e/yanijfp.7:k1apvj -;i27f( sx ucm and a mass of 0.580 xyj(iq o:epk17ai 2 pu;av/.sf7f n j-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 mrcm4vdyf6 :-lc1 ecm /e6of5to 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 hany w+yi)i xu2(0 3uakmlirt 1n1d-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 hy 1txnmykli)u 01ui2 riw a+(3as a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut jcczs555slt3( h:jed 5n,trfoff and is eventually brought unifo:z rtf5j (3cnlcjs5t5deh5,srmly 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 accelerates h. c;5(e 6sm.: g gx3geclesoqa 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 throwpb)p;v q0hp0k.q+ob hn solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle,hb.0boh)qpq ;pk pv+ 0 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 considuzmtlp9jv g376zcpv:;g, y .q ered a long thin rod, as shown in Fig. 8–46v tp9:uz7m.yzgjp v lc;, 6qg3.
(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 consi*c8ig:7tnji+tsq j rn oc/p+b(*2kvbsts of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest within four full turns dpfh k6;,8q ucgbp:7s (revolutions) and releases it at a speed hgf;q7,bcp8d: 6uks pof 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 -r 0qk22c:dy.. mx bqz5sglik$3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine 97 )1kaqsm-1 qb crfetdevelops 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 fbngf( z ei404radius 9.0 cm rolls without slipping down a lane at 3.fn( 0fz4gi4b e3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the lq+2a-o 3m kkxSun as the sum of two termsq2xk3kl ao +-m,
(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/u qy1g h3ld-o kg and a radius of 7.50 m. How much net workd/uqy13 hogl- is required 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 9td.)tw/l;b- p5 nd /oah pykxz2aj- hfrom rest and rolls without slipping down a 3/9.jdzt;opk5axwpd2 y )/ h -bathnl-0.0 $^{\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 t3 g29w/dj5teokxkufgr; .w 3h o fall and its lower end does not slip. What will be the speed of the upper end of the pole just bh; .fj 2rge5d9kk/ gxot w3w3uefore it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on tjgl*qb+ ;y 3zuv4*rcshe end of a thin string in a circle of radiu4vy+* ; qu*clsbjz gr3s 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 cylindrical grindc+4pob;gxo/5 kjbak *ing wheel of radius 18 cm when rotating at 1500;*g4jaob kp/k b5xco+ 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 ki1.k pwt 7w0tk ns:30w9 lahtis rotating at a rate of 1.3rev/s If he .nkkph t wwi:t03stlak w07 91raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can *.k dpus8jd e+q )ppy(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 in the tuck position, what is d)(k yupp.s pj+e*8 qdher angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotatioyt ;rpo(5 ta6hct 1bj1n rate from an initial rate of 1.0 rev every 2.0 s to a final rate of a1hjtr p;1t6bty(5oc3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axi5cmi+1 int3 yhs 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 th3ty+im1c ni h5rows 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 ,e. j t;mdtke0.-urzuat 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 momentu8-p6d7ejwhbx9,sbuz, teqszsk : i0 ,m 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 ine6kwyu:auph88d xeu* (rtia I is dropped onto an identical disk rot8y u8:6x*uheukw(d apating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4 3 v2c6 r6mbjxp$rev/s$ 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 merrytr qz)w;:bg8dp .pk: m (uccq(-go-round platform of radius 3.0 m and moment of dq( u(wpbm8; :pckzt)qc .:grinertia 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 rotatingh: zsj(w+ si9q freely with an angular velocity of 0.8 s+h:jqz si(9w$rad/s$ 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 colla 0/ha2evoy hs )qb1oegz7+ 7kqauc3seu(p0c4 pses 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 carrs3q)cb07 1a7qev2oekugh+e(h /csa4u 0 pzy o ies 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 in excess o5t-i2c3 mvvnaf 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 byxkw09 4:eltm da ex+58tv )d$ 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 a cfgbfl* lv:7jnw;p:n hc/t 2;t rest, that can rotate freely without friction. The moment of inertia of the person plus the plal 7n;;h:l*jf wf2b ncvgtc /:ptform 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 persxp w tp,bk8;-1oj/zjwon stands at the edge of a 6.5-m diameter merry-go-round turnt8tx opj/jw-1z ;wbpk,able 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 groqu*l:,4pl0y,.w x/ )iirf n znzao uq0und with the end of the rope lying on the top edge of the spool. A perao0un., plwr*yuqz n/f:z0lx4i) i, q son 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 the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces t/qar:gtpqf-i )-tx / mhe Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter:q) -qg//m- xfrpt tai case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rateb8/k kgfjz90ckmgf3 t zhf b:48l/w b- 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 radswd*.ntjntp;g 1vpt( nycfggh6 6 )7 /ius 0.20 m acquires 1n t6gnd gt y p7wp/cvjfn;6h(s*tg). a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disk0 vl*4bzg5 caxs, 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. Usv4gzbc* a5l0 xe 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 ise*: hyi) rh)ztr q9hh,y4w7ax/ q*weq 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 9 /ipqw/.qbco4ay2o n, but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarteniyaq .4b o2opc//9wq rs 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 autom do /lz((dn)swobile is for it to use energy stored in a heavy rotat)od(d (sw /znling flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without 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 : h e7 o4kvy5xzrsvt- o2st49dan $H_2O$ 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 (h6cq .g23avqmq2r p,mz 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 ptf:gv5 *: ;qst rcaf4hivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fif 4g;5s : hvt:tq*rfcag. 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 radiu+a +(5pn l0imfyq owg/z)k oi8s R. It is standing vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step kg8naiq/ f(li5 +o m+w)0 pyzoagainst 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 isxow,,ohe6g50 gd,svhi 3e1 qk making a turn with a radius The forces actingi3 dgo swxv0,hk ,5eq,go16he 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 tabh znzha49l 1z20j :of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to 02z z4tl b1nj hzaha:9a 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 a xn5lh 3r 74i-n4*tw0xz p qlsu5,q hy4acp2fbsolid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for3q2h x,a tlh pilsc7bny x0-zf445unq5wp r 4* a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindric;tpl .3xlb ny1s2i ra5al plates, of mpilrt5x l12 .s3b ;ynaass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 lod2t-,-rhvxn qoped rough track of Fig. 8–58. What is the minimum value of the vertical he,rt--qh2v dxn ight 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 qa7fdo/z, fo0*pt hj0xy +/obnot assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uni4)grs 3t rs5tdformly from 90km sg)r3t4r5 sdt/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s