A bicycle odometer (which measures distance traveled) is attached near (9j9go oj+uhj0x02wm r;3gu*odbf xpthe wheel hub and is desj900pjg9xu m;o wx(h+rdb3gf *o2uj oigned 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 tf(.)trtqw+ iqcr50 i a35. zyzzfk b8point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have rab(r)z i53f tw qf.yqk8 ictra+t5zz0 .dial 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 h,esoh:1z()woq)uqq 1r at4 uvelocity $\omega$ Explain.

Can a small force ever exert a gread2 m0yyq-.o,gv+j z svter torque than a larger force? Explain.

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

Why is it more difficult to do a sit-up with your hands behind your hjy2cz ra)c9p 0ead than when your arms are stretched out in front of you? A diagram may help you to answc9 2pay c0j)rzer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three ate;lz9 yp * sk6))o5rrfi .io.tgfwzn+ 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 pele9;i 5k *sry)zf.o)+ wp6 ofri.zgntdal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower legs k*d-t i.,zzbe1f9 :gsp4vpbcwith flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distgcz4-k.b*t,:ipp9bzv f1 d se ribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow bhl+i* +j 33yjtoxp 9rm,8rzseeam?

If the net force on a system is zero, is the net torque also zero? If the netolb vv3t6lcu 2b.z2u+ torque on a svz+l6u vc3l. b22obtuystem is zero, is the net force zero?

Two inclines have the same height but make different angles with the horizonf i + vs:/q-apbu-yjwu/tf a42tal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain./+2-p iyj u4s :/wauqb-ta vff

Two solid spheres simultaneously start rollqk6y(*, bj n8 drg2zjging (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 8*gb6, dznjq( r kyj2ggreater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the sameh mq0da+y/:lu v -or8y radius and 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 at the bottom? Which has the greater rotatdrhuvym 8a0/+l :-yqo ional KE?

We claim that momentum and a9nwvzd t8/0 r4o gg* yu(fop w,8cwj/angular momentum are conserved. Yet most moving or rotating objects eventually slow v/o ,c9fn u0*wtj4az/gyrg( wpdw88o down and stop. Explain.

If there were a great migration of people toward the Earth’s equatortn i 3g9,enx;u, how would this affect the length of,n3 xnt 9iegu; the day?

Can the diver of Fig. 8–29 do 52bmr uo6lol6iril.cbxn q*evl*vl( /- 3 c4na somersault without having any initial rotation when she leaves therl*v6(i4/vl6xnub*q. bmn ll3- ioc ce5 rol2 board?

The moment of inertia of a rotabhw9hj. bx6tie x +v27o.l3dg blkb* :ting solid disk about an axis through its center of mass is 3bbbi.2 *79hx :wd6lj.xgbot v h +kel$\frac{1}{2}WR^2$ (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 sittingn ,aym-cpjy2a k9 ab3- on a rotating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop-2anjc-93 ybm,pkaya the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical ah. gq-oovh63g nd have the same mass. However, one is hollow and the other is solid. Describe an experiment to detehho3 vq g.6og-rmine which is which.

The angular velocity of a wheel rl*8 ta y1k(xzvuogb6:otating on a horizontal axle points west. In what direction is the linea8: z*k (xtvalu6boyg1r 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 decreasing?

Suppose you are standing on the edg2udmwor us3 09;:vqwbe of a large freely rotating turntable. What happens if youq oub;2wvsr9:muw0 d3 walk toward the center?

A shortstop may leap into the air to cal2j /ejuwh.p/tch a ball and 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 llu2/ e/wpj.jhegs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, 8l;:hz9km, rgau-j ps discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicop 9k :ujrapsm ,gl-z8;hter stable.

  Express the following anglespjlr:22wq 6e xjq2h/al16f qnl7f ce5 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 Earth2t 9din wqnu*z-non7mp(6 .ta because of an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (9n(*2 nq-.aun mo dw6tnzt7ip in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beam diverqxtwjgt3a7hg*8r*cm h 6i )-pges at an angleq ighjw*c)mg3rta -87x *6pth $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. When the motor ispi,:qql scq3;ubto,i*:v3(mfqv 4 vk turned off during operation, the blades slow to rest in 3kmbqf:q i,u:3l,(qtvco3 p vi4 sq;v*.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the l,r8r i+mp0k8dxna*m ball makes 15.0 revolutionskni* prdr+m 088a,l xm, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many r1rwetj a j;0*t 3qmpa8evolutions do the wheels make?1 rpm*te 30a8;jjtwaq    $rev$

  (a) A grinding wheel 0.35 m in diame(egln/bgll w:+ f)d u*ter rotates at 2500 rpm. Calculate its angular velocity indg(ln lweu bl+ g/f*:) $rad/s$ $\omega$ =    $rad/sec$
(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). ( wp-l gc+os)mrw;n(s*b g2o 3oa) What is the linear speed of a child seated 1.r g*)s-b2 w ; +o3(gwnolmspoc2 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 aroundyp e))uc)i7izhl z 2(l the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear spe )hz0hr 8 +ilyx,c 4ybeyv;t.wed 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 particle 7.0 cm f+ uxr7usr4a0 9mmsg 7urom the axis of rotation is to experience an acceleration m4s m7srrg +ua79x0uuof 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly aboux81dnjje4k qh peb.hn 1r:; 2ot its center from 130 rpm to 280 rpm in o8dejk:hnqne4j;b21. xhr1 p 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 radiu (o)pij byjq0+6n: hh8.irlszs $R_1$ 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 put thi1p4 xn(gdp36jn dez 9emselves into a slow rg61edpixdznn9 4p3j(otation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest s-7 j/r 88kvca zchwspo/y3f4to 15,000 rpm in 220 s. Through how many revolutions did irs/p y /84zj8-f 3 awhv7scckot turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5t lv6gmzde29) 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 highspgf 9g os2gk xwh8i /pyj89rg+8eed jets in a whirling “human centrifuge,” which takes 1.0 min to turn throu88hg9 gi sg8+wkrjpyxgo 92f/gh 20 complete 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 whwgkx w,73q2 u8nlo) 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have trnw, h3l7uxkgo ww2)8qaveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/yppzp ak(:zud0am 2kb2ddyy ,s 887t 0mla 68amin It turns 1500 revolutions beforp d8ls0y u8daapmtz m07ay8adk2z( k,6 y: b2pe 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 its speed uniform3ael *ufi+sb2w p:7-udsfsm,mek 65t ly from 95km/h to 45km/h The3be e+iu*5f k:dm2msl6,7 pa-us st wf tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car mak9 aeeq o8 nv5tq.r:)pre 65 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.80 mrvt8: e e) qnr.59apqo.
(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 6 .swo: 4odyeha bike puts all her weight on each pedal when climbing a hill. The pedals rotate in a circle of radius 17 o4 : oh6wsdye.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 on the end of a door 74 cmo6;o bkbjz .h: wide. What is the magnitude of the torque if the forc:kob ;jz.boh6 e 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 9osmr9s*q8u wb(e2 xxthe wheel shown in Fig. 8–39. Assume that a frictioous2xmq(wexr b* s899n torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of m wci7m /s;u)fo62 ijvhass m, are attached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal posishwofc6u 2i)i ;/v7mj tion and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylitkh:lp l*(z4 4kngj/pnder head of an engine require tightening to a torque of 38/ p glk4zl*njth: p(k4 $m \cdot N$ 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 spheiqta3 zw.18a5pf*bok , dend3 re of radius 0.648 m when the ax3 a1, .ewab onqpdd5*t8zifk3is of rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter.bwtx12 pry 1s4rf05 erj+ill7*- cctx The rim and tire have a combined mass of 1.25 kg. The mass of thesxebl-2ftxl4+r0 ct 5yr*w1p 1i jc7r hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light roh9x/.hzn)iqjg :*dbnd is rotated in a horizontal cirz xbh j n./*hn9idg:q)cle of radius 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 bowl on a potter’s wheel rotating at constant anr/7 h+o3hu-a q t6 dhgebl2ch8gular speed (Fig. 8–42). The friction force between her hands and the clay6tb2or3d +heu-/ h l h87gchqa is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point objects shown ik+ olv7nx wjonhe(23-n Fig. 8–43 about k( 3v7xo+2ohel wn-n j
(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 tw y0j-jbmc4, 8/itk3htoiwn9 qo oxygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satelli8sqx,d+ u 3uhk 9all-l6 bx1cbte 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 lku6xuas9 q8x,h- b1+ l3dl bcthe 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 cm and a mass o) sj t dnm cbeymj+p98),;r/prf 0.580 kg. Calcu8pc,j dbrjp ;/ nst)+ymm9)erlate
(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, accel+n9 j6jo g8gg6d:xc qx.j+.rn1ky7 dw cbzz/serating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a smalllkk0/9 wvrn 6o0v q2tum,ym o6 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 merm o 69v2okny, q 0tvw6kmu/l0rry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and idx :kk9 a8dmc6hc8y3vmh,dq )s eventually brought uniformly to rest v3qkya x8,hcd9 hd8cd) 6 mmk: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. 8–4 p:j;syrc2hv r75 -obk8.vu(dfe y(am5 accelerates a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by th0l 9)k fn/( jh4fkmk4cn.j lple action of the forearm, which rotates about the elbow joint u. kl(hf n l0 kf/kp9jjmn)4lc4nder the 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 can6h4qqk jl02 ok be considered a long thin rod, as shown in Fig. 8–46. 06ko4 lhjk2 qq
(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 masse *pbj3 bgjffk-;n5xr 5s, $m_1$ 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 tuxsrun9 *4 -3y 7pglhiurns (revolutions) and releases it at a spl 3u9-h* u7x4yinpg rseed 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 inerti.jwq-qj+7 i r3(emtp ox,in u1a 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 9c2(3j-ri ee:,kcg zj dm*xwrdevelops 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 withbu +j:aqi05kf 3uuk 2rout slipping down a lan 2 uqujkrb+f5:a uki30e at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth withf,p-w:n4fd j ou)iz nnqge /:/ respect to the Sun as the sum of two ,:n/ pqn oz/4fui)g:d -fn jewterms,
(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 7.50mrtj, dyq f/-8 m. How much net work is required to accelerate it from rest to a rotatit8 /mrq fyj-,don rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and massxjm( yq+gq q;1mrfi5r8 ag/e3 1.80 kg starts from rest and rolls without slippin rq/q(gyqrjemi+g x5ma 8;13f g 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 itjxqjk*s9( 8 f*zhal1t s tip. It starts to fall and its lower end does not slip. What will jq la8f*shk(x *jz9t1 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. qh9a u dy 1:*ai2jvn,(dhawu 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an angulavnw i a. dy9da1jq2h*u,:u(ha r speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.+plxlm8e ,16e;m+eapzu bn5 v8-kg uniform cylindrical grinding wheel of radius 18 cm when rotate+mzexlb8a;p +e6 lpmn15uv,ing 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 i : cu t/fr8/sk+dfb7bbs rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the//b f +dkb :tcufr8sb7 speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce her moment of inertia vma afi ndrua, - (f2(/u+c9 mqv7n+tvby a factor of about 3.5 when changing from the straight position to the t(vaf t/v++mnr(cnq 2ifa m9duau v-, 7uck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate f nul1 nwzi 7lx7d34.rnrom an initial rate of 1.0 rev every n 73wlr uz4ld.n i1nx72.0 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 rotating around a vertical ax5bt brocv5; ;jis 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 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 rb;b5t;5vco jsticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinninkpgj n:u7z9 w w0qc.0xg 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 momnx yqfs6 n-vq-t)4*v h xv3vjr t;im/5entum 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 inerg,prp : :6 nvz43qzmvgtia I is dropped onto an identicag3n6r4v: zg vp,zq pm:l disk rotating 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 uev5f 417a szo$rev/s$ 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 rota: nzl.momj,0(wzk/u5r/ pe ddting merry-go-round platform of radius 3.0 m and moment of nldz/mu ,rkpe/j5: 0w z(md.o 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 rotativ ):uo 4 vh(v1x, d+lzbfov,mkng freely with an angular velocity of 0(mv:u xkvvhfb 4zv)o+ 1,do l,.8 $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into oc,((iuwtc3g a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would thecwt cug 3i,(o( 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 t.s+ k4bp7odk 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 )a tw9m0l0cim6z* jab$ 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, initil/a viu;e,ih tcx. f4z ;y/ah2ally at rest, that can rotate freely without friction. The moment of inertia;fe x/ty2uv/c i.az 4hl hia,; 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 diametgt er-it8 hi0:q5zr3x er merry-go-round turntable that is mou:trt -0zr8x eq3g ii5hnted 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 groundc+uby;ju.eyeu ;0,ar, u0gh u with the end of the rope lying 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 behind the person without slipping. What length of rope unwinds ,r a yuc.ue;0h,ju0+bueyu; gfrom the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same sidev; me3p7w ek+i always faces the Earth. Determine the rak me+pv i7w3;etio 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 orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at ap+m u jm:8t*dlrvxmgghq6 ,6 : rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of rad 17m-pd;gr qwdius 0.20 m acquires a rotational rate of from rd p1q7m-rdw g;est 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 discklo (5t m*nvwb, .z*4:q 1pqf hw4hcvks, 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 reacflk 4t*c4* 5ozh1:qcnwvvb .mw,h p (qhes 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 a4rr uq7/q9/ee3t pb iungular speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times as great, w/ plak/,3v5ukgfz 8v7 dihk .dere rotating at a speed of 1.0 revolution e/kg/iz7adl,. vdu3 fpkk 8v5hvery 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-pollutuvou-wymfolc 1g + 9/;ion 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 fly/w ufcv-u ;yo1ml9+g owheel 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 an c5m8xi0v y:4oqs v4pzejd(a r cfb(.8 $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal surd1od97jb jq;a5bbst8 face 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 1swei.-gf *pr bn;am2 M and length 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 tp 2*;snre-.b wfamg1ihe 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 venvfd g v75wb80rtically on the floor, and we want to exert a horizontal force F at70 gvfd5vwn8 b 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 mdsu : f0vr-. lpkxjl07fjbd 807pdux +aking a turn with a radius The forces acting on the cycl0uxljj70r : vk .fuxpb 8sdlp-+0d 7fdist 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 ap,5sw-bhcs 6o) i5(u7ps rjb n 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 rate of u5srwsh-op, ) bp75 6ns bi(cj120 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 7b7ch3 uo w/m-rn,itbarms as thin rods, making reasonable estimates for the dimensions. Thentmbh wo77ni/b 3rc,u- calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clumv7c7,wdf9r ;ez2 yhe tch assembly which consists of two cylindrical plates, of mass ve z7f7;r2h e, 9dcywm$M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 t ng ,wg8* 4thl:(tk+erfz:tbkhe looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble mttlfn:kb,zt:g 8(k4 r*g+he wust drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assumex.n5h.:+yifwne3kyl 5-)kk pdpa ( rj $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car j0jw l g,k)g+f(cow4r reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration 4(0l,wj fgow )kcrg+j 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