A bicycle odometer (which measures distance ktk;t: xxs;uu s,+o e7traveled) is attached near the wheel hs+xo:7 sk uu; ;x,ktetub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant an7rs 8 9dt3 *lk squvnrgy4(+zvgular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangentvy z(rsrut8g*nqd 34l97+ sk vial 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 ang ;)vdupk:y 8njular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Explavs2gnf924d8l hen rht *c:8qxin.

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 sitsj, ,vxi/ v 0ss9t0qmrp2) xax-up with your hands behind your head than when your arms are stretched out in front of you? A diagram ix2 a vq sj0rmsxp,t 0,sx9/v)may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the y9+2)): ebnwfasc zrb pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which g9)2:y)e f+nba cwrszbear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower legs with flese fx2*)md. d.n .npbtxh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why thxm 2 ).fn.ne*xpdbtd. is distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–wb)k imn :vd d6la79k535) carry a long, narrow beam?

If the net force on a vshntky((5t*system is zero, is the net torque also zero? If the net torqs h(y5*ktnt v(ue on a system is zero, is the net force zero?

Two inclines have the same height but make differ.(.ls4 yfjlbwj (z7jtro-q8lent angles with the horizontal. The same steel ball is rolled down each incline. Ont.4oylfl7 w-r( jls8qz(.jbj which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously str82 7 shcmh7pl) hx(lrart rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which ha2hmhxsr7 hl)c7(lp r8 s the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. They start ; azv-5whsj*n 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 rotational w*5n hzjva s-;KE?

We claim that momentum and angular momea1x1* a ei3+gsl*8y zdxl qokw6zjp-2s7 2mtpntum are conserved. Yet most moving or rotating objects eventually slowlzew2 +yltjsp*x3-kma1*a8i d62og sqx 1zp 7 down and stop. Explain.

If there were a great migration of people toward thee8 :*.yr qo5 pkq e.ks9qjyel: Earth’s equator, how would this affect the leyep.5l.s* qjkoyq:eeq8: 9r k ngth of the day?

Can the diver of Fig. 8–29 do a somersault without having any ix u4tvg/whj.+grr-:w i 6lq :unitial rotation when she leavr4xl 6vqi /:t u.hg-jug+w :rwes the board?

The moment of inerti p3d5 4viw9g6 10hpe;lscp wpla of a rotating solid disk about an axis through its center of mass lcd;pw9pw1 06l ih psp4 egv35is $\frac{1}{2}WR^2$ (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 holding9k9r z nap(w: 5mgvdn/ a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocinnz9/gd5vk9(:ar pwmty increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is hollow -eso kf2o0gr7 and the other is solid. Describe an experiment to detk0 2 g-seofr7oermine which is which.

The angular velocity of a wheel rotating8oih8k-bww d axs7 tt 4-h+zn9 on a horizontal axle 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 acceleratib9thzh4 -k8dinsw+7ot-8a xw on of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating turntable. What p-f)t3w8c+ bv f.lz6wmi7q ckhappens if you walk toward .bt 8 6v-q7wkmc+fcwpl)f3zi the center?

A shortstop may leap into the air to zh6+s:rp( 1oszz ty jis-f-;gtd, *t vcatch 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 tvp z6tzots siysh+1;t j- rd:z,*f(g- hat his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular mome h ii*r7gzqxwa e)nk;k99(3 fjntum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can oh*w(7k fgaqe)z 9xikji9r 3;nperate to keep the helicopter stable.

  Express the following angles in radians: 9 * lakh:yosr:(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.b jgxdbfr1u nmarcx1s(m27 7jtufgh.58p5:6 Calculate, using the information inside the Front Co ufxgp785u71( 1.m5tbf mhs6x jbrc j:dagn r2ver, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,00:l) d(. s,p8a lhpwkev y4s4jc0 km from Earth. The beam diverges at an angle s dk.8) yepj:v44 sca(p w,lhl$\theta$ (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. W; 05ewycc0c(p y8 j00yudun flhen the motor is turned off during operation, the blades slow to restyunpc yjufw5;c0( 00 cyle 08d in 3.0 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. a0qfu7nv8c5yjavg5 * pjpq7( If the ball makes 15.0 up qj7a *(q8 n5fgj70yca vpv5revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. y sucl 8 +ktwf6f/(m44) oxoq;u:uwqtHow many revolutions do the wheelswf(x;u:l4w uq)t mf6u qoy8c/ otk4s+ make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate its angqex20 q vxp0 asp/:+hkula0 /qhx:q+xve 2 ppak0sr velocity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution i6 u/pyii(mt ,i)u ,htwn 4.0 s (Fig. 8–38). (a) What is the linear speed of a i/imthp yt) (w6iuu ,,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 velba a /phe9ke oj xzo;z *vaz-08,3yk9yocity of the Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speedw4dw z/cz 3+mf 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 bv z9f-c. u9vkeb cyf7ia-9 )pif a particle 7.0 cm from the axis of rotation is to experience an- feuyipvfazcc-)v. 9 b79k9 b acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpm to(: d1x8jt emsul(fm0l zkog. rxj+ s63 280 rpm in 4.0 xmlsf+g8(.zul erj6 smo1kdtj0x:(3 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 radiuw6rm xaz)6 yi,s $R_1$ 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 Agu v;f*g31zy ys1 r pj3w8wb1gu+m gd3pollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenmjfzw * 1ydu gb3;gyv3psr1 +1u8g 3wgly. 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 acceleratesfqi* bo)9fi u- ub5b: ti-mae0 uniformly from rest to 15,000 rpm in 220 s. Through how many revo:u 0-ab-5 fe m bt*uii)9ioqbflutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200; wbcvj/ry-e 8 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 jet*itg ** lmdaj;wyf1h;a8a c y(m z2;jos in a whirling “human centrifuge,” which takes ;fm8*j;wl 1jim(g*; o yha2ayztcda*1.0 min to turn through 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 240 rm2lj2l7dj s 0wpm to 360 rpm in 6.5 s.w7mj2s0d2j ll How far will a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is run55d89atplvqp o;st4t ning at 850rev/min It turns 1500 revolutions bef;tsol45tv8t 59q dap pore 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 revolutiowa8fk :fcs u)gz :(:zwns as the car reduces its speed uniformly from 95km/h to 45km/h:czk gs:fwfu8w)z: (a The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the cax+7cy:aa y0bc r reduces its speed uniformly from 95km/h to 45kc x: 70yybaa+cm/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each pedal when climbihc+l/bo0r 9 qspvc:,eng a hill. Thqvbl:/crp 9+ho sc,0ee 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 vd3*jtl0,zt 83r*k4 bo w+zyp-z vzvdof 55 N on the end of a door 74 cm wide. What is the magnitude of the torque if th3tdzwj*l p8,yr b*v-3+4 v0zv zz ktode 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 touz61-uh;9njw( wewry i4pa7k rque about the axle of the wheel shown in Fig. 8–39. Assume tu w ;z(4auey1 wjnp-76r wih9khat a friction 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 rod which pi iwekv+sv, 5cwj, h-70npge0a vots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitudev5 -i 7hj0sc 0wgw+,e,kv pane and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening to a torque of 38d vw ;v1-zwyzp/gwn;8 wzd-1 v;znw/;g8 pv yw $m \cdot N$ 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 shl1zk6yomn : gkz*;p4phere of radius 0.648 m when the axis of rotation is through its centerz1nopg l4k ykm;h:*z6.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. Tmjpakth+d75vl j89zx 12 l j-xhe rim and tire have a combined mass of 1.25 kg. The mass of the hubx7pd8 vmh+jjjz-at52 llk91 x can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, liprdnq+ gx93w/ ght rod is rotated in a horizontal circle of radius 1.2 3qp/+x w9nd rgm. 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 onvtl:3s 3flfuk:ewi +; a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and the clay iek: w lf 33ils+u;tvf:s 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 tho5;ppm .qnl6u okpb2 5)s iow2e array of point objects shown in Fig. 8–43 abooq l .ob5mp p)n wo;u52ps26kiut
(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 atox/ ) cl/euxt ,mg-uas-ms 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 satellite spigtgfir3b:73 c b 0ch)3d 8xvffnning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a g30x:dgb vrch733 ftfc 8bi f)mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.5zc/ g/uxh*grqbg7/6 6swu nb40 cm and a mass of 0.580 kg. Cal4hc7gr /6*wb gs u6b/g/qun zxculate
(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 itmojta (5ttq5xh4;mm t3j1 2j yt8fmp/ from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-drizpuzhdra4m: 87gm-.4 kadx b45pl +wqven merry-go-round and is able to accelerate iqpuz 4:kpaxr-d4gha+5ld7wmz. 84 mb t from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to 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 mwog5o 9q46wvnb*7unum.*j pis eventually brought unifor4mpm. u5jw*qn9vub6wg o *no7mly 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. 8–45wl /f; gi2r p9eoh1cp +5x*8vdpcrrg, jo cr7; 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 the action gi3qus (dzyyyny798pxar)o.).u ,dn of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to q y ,a9u.dg)y8xrpz)odu (y3.snni y710 $m/s$ 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 mssil r1fsch28-*c r cre:n)(considered a long thin rod, as shown in Fig. 8–4e) m:ris2sc1 -(nl8h*c rsfcr6.
(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 tw; tw3,9roabe o ivy:1*gc 0vjro 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 4uenthuimd t 1n/jklhm0:+(1 from rest within four full turns (revolutions) and releases it at a spe+1nl1/ udheik4(n:u t mh0tmjed 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 au9tnjm( 9ovid 8xi z66 moment of inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque of +b/ hx drv( tldqui;a;h. h4+n280 $m \cdot N$ 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 witu3fravj:; ei+ui-e*mhout slipping down a lane at 3.3jrfiv i;a* mu:ee3-+u $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the sum of 0sn( h i*x+ocitwo term+0hs(n ix co*is,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius of 7.50 m. How much net wo q.cy 2wtf)qe6dvih( + lqjh56rk is required to acceleratedl whc.t6+)e (fh2 5iqvy qjq6 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 from rest and h +u n(.ax1a4h3oly horolls without slippinhlh. uaoyah 31+nx 4o(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 its tip. It starts to fal6fe a n:1-(;ntvnylhpk6a nm -l 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? [Hi n naevf(aktlh yp-66-nn1: m;nt: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating al ng1..agu 8non the end of a thin string in a circle of radius 1.10 m at an angular .nn1 gla8u.gaspeed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2c (95;hrbbscn .8-kg uniform cylindrical grinding wheel of radius 18 cm w;r5c(hb snc9 bhen 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, handshf7 oc;(jp 3cff5pc6q at his side, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotatio c f5jp6o( hf7fcq;3cpn 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 owm. 1pyw0,4u hhgxs 9of inertia by a factor of about 3.5 when changing from the straight position to the tuck posit.9xogp4uy 1,hwm 0hsw ion. 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 rh3)*z bwb-r yxotation rate from an initial rate of 1.0 re*hz 3rbx-b yw)v 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 rotating z) sm2efqg,c2m3 o;fraround a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the freque2r gs),3mc 2o;feqzfm ncy of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinninwo 1;vk, r0mino+j 0 ;7ihema mbvh15bg 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 the Ear-7 t a(nrv+h0:sgzmkwmf 5 o,zth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is dropped o- 6 jc1 4j7br(dc stfv7h:1vunm9yckhnto an identical disk rotating at anguv7crv7hc s61-mh1 dn4kubcf9tj y:j (lar speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns 2r(qjd0*v cl98wohy-l e q6lz at 2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the centem1l v3f;.w3p 8:lupd jaoosj1 r of a rotating merry-go-round platform of radius v fo18m.d3 :l1wpsujpjal 3 o;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 a29ek z yxc:/len angular velocity of 0.8 ez cle/:92ykx$rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, losing about half itsr7,hq ;bp6nr z -jo4nk mass in the process, and wbjqn64knh; o -pr7,rzinding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in excess k;buedw 5v7:s 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 w7 qoj-5 s,qlf$ 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 a vy9im2 g8bkn(b/hj-, initially at rest, that can rotate freely without friction. The moment of inertia of the peby-8gb ivn/a ( 9hmkj2rson 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 stan/i0qhj sg3.d yds at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertij.3yhsq0 /d gia 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 en(7ny42oy* + d lbkw ac,m3)hj nmzki)fd of the rope lying on the top edge of the spool. A person gr(, 3yomil7zjc hn n my awb2fk)+)k4*dabs 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 zpqt.0(rkzk0i ,eu m 4Earth such that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its op uz,kqit0me(0 k4z.r rbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates(qgr :frtad81bk2q e) from rest at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of * 9canzy )dy0nfmrn6-radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant anguny -)m0dnyz*rc6f 9 anlar acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of maztkfepy xs-i4l /8(i-ss 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it iztxkf/ -84sipi e-l( ys released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of the rear whe bf dk9n0,-oukel ($\omega_R$) 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 rzhn2vm+v g97 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitati+ vg vm92z7rhnonal 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 ene/tixah32qw k. ueslcf -pk4 i8zx ./(vrgy stored in a heavy rotating flywheel. Suppose such a car has a xwefxsit l./c2-.vk 48 iaqk /hpuz(3total 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 illustratesp-l1tmhut cr++ e 4)iu 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-zhq j (vj8hdrn :40k +pcn4gs 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 /kugi7/akf8c 3 iymb( 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 cu/m f(i8yak7b/3 gcki enter of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing9 xx4y.0 d dgvw2fzz9i vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb zi g90yv w9x4fz .2ddxa step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=sa 6(fg1hn x i(6e,i81 irrsdz4.2m/s on a flat road is making a turn with a radius rrg, efsn is8(ii66a hd(11zx The forces acting on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock itc6+w xe v-bd 3cdg:4nnto 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 rate of 120 rpm after 5.0 s. What is the torque required to achieve thi+nvtdcb34- :6eg x cdws feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body asfz6 )bh0kdn 2t/ :pejy a solid cylinder and her arms as thin rods, making reasonable estimates for0d6 yejz2tk/n:ph)fb the dimensions. Then 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 clutch assembly which consists of two cylindri,q* fb.xlia-: e18qpm o hvc5xcal plates, of mass fc,x .x8ib* 5voaql mqhp:1-e $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 roughe6m5-5r; (xux0zeg8 b okrmw nxe ko;6 track of Fig. 8–58. What is th6xo;5 ;k5( mw8 6 kxxeur-r0ognbemeze minimum value of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assumelt;1phyfc96 k $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutionmj x ()q57zmwjs as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diamezw5) qj m7j(xmter 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