A bicycle odometer (which measures distance *v : uk(whxto5traveled) is attached near the wheel hub and is dot:v x5(*wuk hesigned 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 point on the rim hazl 2 +f7yc.fzyve radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, docfy27+l .zyzf es 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 sipo z1d 6s 4byed*vaxr+2u3,g ungle value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger forces2upwzw m.9( u x;t;,ad1vzuf ? 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 au4ss3hvrx1.key 9y b7 sit-up with your hands behind your head than when your arms are stretched outs3 s 7rxybu14h9e.vky in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprocketi:r.oyx9a cr ;s at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprr.9;:xicaor yocket? Why?

Mammals that depend on being able to run fast have slende,1p2ewt1 c*fqpan 0uhe (vts 4r lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is t , ( cs1qna epuetphf0*vw412advantageous.

Why do tightrope walkers (Fig. 8–35) carry a lo*eu.q83dlk c dng, narrow beam?

If the net force on a system is zero, is the net torque also zez0nyy3nn o0*z4 srb e1ro? If the net torque on a syst n s*y bo4z3n0en0yz1rem is zero, is the net force zero?

Two inclines have the same height but make different angles with the horizo 0,u-b1sax l k+ggltp,ntal. The same steel ball is rolled down each incline. On +,,t xk p1l-aul gsgb0which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously+pzhv8fw5 g6ktgc* 7fxm3p*bhc u61q start 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 hk1+zf pu tm*g3hpgfb hxq86*c6v c57was the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder havei i7uk s9g5m( bvzl*u8e+ kzy2 the same 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 g 7vs u2u*ki8z(9imyel +z5b kbottom? Which has the greater rotational KE?

We claim that momentum and angular m6;yb e.wjfr1(hubx y0 ez,moh+1uj 8nomentum are conserved. Yet most moving or rotatinbh.wumuye0+jx f1 o(zry,bje n6 ;1h8 g objects eventually slow down and stop. Explain.

If there were a great migration of people too4ty:s; rbpyj0kg1 ow4n37cs xig;,f ward the Earth’s equator, how would this affect the length of the day?jkg;:g 03,4n;yi tf wr ob4cxs7p1soy

Can the diver of Fig. 8–29 do a somersault without hhb,*8oln( vmvmvxap +y8 m*x1 aving any initial rotation when sm x8,h op+myvv8 1(m*a*xnbvlhe leaves the board?

The moment of inertia of a rotating solid disk about an axis thr1pb3: ku 3i2ltp7n b54jpn*cum:gv xiough its center of mau 7i4bjv *i2u 5:cpl1 :t3pngxb nkp3mss is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2-kg mass in each outstretr k*ly;w.o7 vvched hand. If you suddenly drop the masses, will your angular velocityv vo7yl.wk ;*r increase, decrease, or stay the same? Explain.

Two spheres look identical and hak-ymwyn9x-qbh)x+,i u3v f7 ive the same mass. However, one is hollow and the other is solid. Describe an experiment to determinh3)vbwu+7 i-m-n ,x9fk qi xyye which is which.

The angular velocity of a whee lw.mqxy6rwoeh,5ta8my .l, .1 c7i1t bkog8ul rotating on a horizontal axle points west. In what direction is the linear velocity of a point on the top of theqtt a8 r,omli lh7oug11y.6bc x8 m.5,eywwk. 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 edge of a lbqy df 110p: 8mnxehye(-mnl3arge freely rotating turntable. What happens if you walk toward th 8-y e0be nn 1dmm3(hpyxq:fl1e center?

A shortstop may leap into the air to catch a ball and thrown-ez jg 6o uhm)4fgw;1 it quickly. As he throws the ball, the upper part of his body rotatew- ;6)14 emugfjognzhs. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum,lu oj -d/fe8s) discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate e)d/8jufl- so to keep the helicopter stable.

  Express the following angles in radians: n6-hhw y 3.sud+dk.cm(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 dlg-/: pf5w6rsni ,a ep 9okzq,/trj,because of an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radiazr,:n,a6dqog/9k lr/-wfj 5tpi,se pns) 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 dqo y4*efpb6 h7iverges at an angle 4obhy *q6 fp7e$\theta$ (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 of1 bjbw h0s4ky) 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as4h kb10s)jb yw the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the xe+-72ph30gpoa30 f, lfr8 coan etgpball makes 15.0 revolutions, what is its 8 h pean3o70, efc rgglf03x2-oa+t ppdiameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 kmv k62si/n1ok b. How many revolutions do the w 1v6s2bnok/ki heels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Cal xl*.+vi4b fodculate its angular +l 4x*vodfb i.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 in 4 p5mdk5f*e2 mp.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2m2* def kp5pm5 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 thgyt h 2.*fqg(ze Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of y:o wm ff;:/o +k;wy(bm5kefha 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 cml/or1 tts3gvi c-tafcdh +uy7a:97q7 from the axis of rotation is to experience an arv/ct:+- 31 y7uoti7cqh7daas f 9gtl cceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniforml9,ytgvoh5t k 1kg (lj;y about its center from 130 rpm to 280 rpm in 4.0 s. Determ(h,yg 5kl vt;tk9gjo1ine
(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 radiusc ksg1x eg.o*g;kwe 79 $R_1$ 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 3*d9fw78y pjokmt6w rac6 3q(:sx sfr themselves into a slow rotation 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 space qrd(twayf 6o8s x6:9csk fj*pw7r3m3craft 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 to 15,000 rpm in 220 spu/2r;lf+m:hn i,l tprw:hf gb ex5:8 . Through how many revolutions did it tupfnhw:r mt e,58:ghr;fiul+/b:px2 l rn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s.tgr/s uiyw+)+s s42* phr. wek 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 jets in a whirling “1arbk :* v7*cdnov 56 mpd ci-3pblv-rhuman centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaci5 a7 :d3v**lbr-c kd6noppr -bv1mvching 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 mx8+ /6b 3ciphzmyu u3uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have tra6m 3 p8hxby/3+zmcuiuveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 1m .8agpegn+u()a4ll q500 revolutlq+e)an(l4pag 8gm. u ions before it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car fcdi k5f +88hwreduces its speed uniformly from 95km/h to 45km/h The tires havc df8 i85k+hfwe 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 car reduces x,prv mqj xr( baz,(y 5 9y*r7o1wud(zits speed uniformly from qoy u9(*r1,pjzv7dr x,(awb(r5 myzx 95km/h to 45km/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 c.9tu0/5eph (d gbqo fbike puts all her weight on each pedal when climbing a hill. The pedals rotate 5c9bf( tg/q.duohep 0 in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cm widliu,i -4v,j( s4j zf bk5e,czue. What is the magnitude of the torque if the z u,4f ,izb- ejku vcjs,54(ilforce 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 aboun0. zbz1* v)jp 4z p8th. vocyyo+d89:znuvjst the axle of the wheel shown in Fig. 8–39. Assume that adp*zb.tjyo ou h +z8.4zvnyv9z)10js:v8 c pn 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 ub 6gthhodtk:o g) ey2m(2;8wattached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque od6t ge(h 2;t o:h8byug 2kw)omn this system.

  The bolts on the cylinder head of an engine require tighte .c )5(-usmvdrzq;jg;im3 dxtning to a torque of 38 imjrd)sgmt- ;.;c3vq dx5(uz $m \cdot N$ 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 1p1 lhj bh28.56-se :cleqj-vxs:e dya0.8-kg sphere of radius 0.648 m when the axis of rotation q s sc6 dvb. ehy5:pa1hl-:xlj-e2e8j is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia yjj,y*t r6usg5 /m)kd/qm ( h ;qlpug*of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass*t ,qu(mgjmuy/ pq)krj6s; lh 5y d*/g of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball lk.ib(x: )b;f hnj r(uon the end of a thin, light rod is rotated in a horizontal circle of radius 1.2 m.j: (x)bbf .hun k(ilr; 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 angular spa1l 4.1 tvzwf-+ kpwp qdpykw 2441segeed (Fig. 8 ekk p 2zda111vs+wp-g.q 44fwl4typ w–42). The friction force between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of *y1e 2; qa7*z8f/qces1ujg8ocfcy+, samy w rpoint objects shown in Fig. 8–43 about ue7cy1m; g/1*ass+fe2 r yy a,q8w8qjo cfzc*
(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 oxdc) edy5p qk)7ygen 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 satellite s;gv8ted6g w 9k:ub26 gjnj x0aan e6j w-4ko5dpinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is tv9oe0 -n6gdb 6gwu4anj kkj8gxd ja:te2w;5 6 o 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 9-t9az;mlas omass of 0.580 kg. Caza9ao ts-;l9 mlculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerating it from rest to 6s6w g5 oz ms;hssysp8uqf::6 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 tangentiall(o1x wl y.md1wa- ja8tyna4/jy on a small hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torquelw.aw( 8 jdx ama 1toj-y1yn4/. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rhcn*y,3iv 8tx otating at 10,300 rpm is shut off and is eventually brought uniformly to rest by a frictional 8hixcn* t3,v ytorque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a 3.6-kg ball kn(;s t thw.* ilizmh0js8(s: 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 of the foracbbl4 y jz;1m+qb8i 1earm, which rotates about the elbow joint under th4l y 1c+bjamb1;8zq ibe action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thdbtho8 w3x 2w8in rod, as shown in Fig. 8–46. bd w82tw3xoh8
(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 cohee goh6cpz3 s-+dm6: nsists 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 frovg,qa2)k2d 70u saq,c24/umrs itvovm rest within four full turns (revolutions) 2 mv,k7c su42v/oga2avtqsu 0r)d qi,and releases it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has8j(q 5 qvts9aj a 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 torquebx6/0 * hmievq:wzq3 / s8udca 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 rollf4r- a0e7r dpws without slipping down a lane at 3.3d-04rf aer 7wp $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the sum o4sm 5pn+ 8r /-ckbkticf two t tsn+ /crmi-p5bck 84kerms,
(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 ( zfr0 ,uz7syh5kpc/ wig fo:fa).or2and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 807w/a,zou i(.fc kyf :pg ohs z52f)rr.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 rolls wit t j9t4qfa:wpm.2 ,qglhout slipping downq j,a4fmtt.lp:2g9wq 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 fallkm.y a :g1slowi ; w4pg;myuux,p-:a7 and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.] 4ggkw7ya; :p1 pyuom:ws-,;m ax.liu   $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the 8 uxm:7sz 0yloqt, ;evend of a thin string in a circle of radius 1.10 m at an angular speed o0myevu x o:tz;s7l8q ,f 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular mht4 4ps+dv+hpa+jfo1 omentum of a 2.8-kg uniform cylindrical grinding wheel of r p4 jv+1a+s hpfht+d4oadius 18 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platf1y 5x.dn bay+ dkcg/vgtzz-: i +64stnorm 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 rotatio6ga/ dgk.ztd-by c1y5+itn z: v+4snx n decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shvhm e te0m.4*sown in Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If mve0te. 4sm* hshe 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 rott1gno/tfeov -m*:qu 7 szk:t +ation rate from an initial rate of 1.0 z:uq:t1g mt*+-onfko/ 7v tesrev 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 rota *bmkh)12xx+ m70+zeg,4bm xsctpxd jting around a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, apsd)+pekm4gz*7 b hmc xtmx1,b20+xjxproximately 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 fig msu(vw9xr5 v:ure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular brh.,-xh2 sbm momentum of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$

(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is dropped ontb ypl(b d.a/d9l0)t ycupqb./ lj 7a.jo an identical disk rotating at angular l .pt ba7c ldb9 (.y/jqpay 0/lbjd).uspeed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.u8qsyd 8kn7w 64 $rev/s$ 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 rotatqa idp:8hbo)*ipo yfs,+n 1;ring merry-go-round platform of radius 3.0 mio8*)d, rnp+:;s1pyfh qabo i 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 i5 xui2 -,tyamws rotating freely with an angular velocity of 0.8 auimx2t-w,5 y $rad/s$ 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 r4/umpkh m*ccvi x88*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 the Sun’s new rotation rate be?(rounh8 *cv*r4u ic km8/mpxd 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 infhh *qw;/..-re uvuiyda7v0q ;2 g cim 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 -nb kg3 7;2w) ezt0;nuqfmhus $ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, that can rott)+*tmdm4: 9k v*v uh:jss qakate freely without friction. The moment of inertq *jtdsts m:*)v+ a4:kvm9khuia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diameter c( qaca((i lou6f; j6mn-3mqu merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia of 1700 6nl(mq (uqo-ic6f ujm;ac a3( $kg \cdot m^2$ 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 grounu n)q8c12tp8t(r fu rjy a-4xmd 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 beh 8x ut qnjf4t28p1)-(mycrrauind 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 thrbf4ufp. 8y vf,:a77rnwoc3nat the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its o7:f, nra.ropw vbycuf7n f3 84rbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates frnr 86-hz8w96u o nm9fpy-v+no dh0sgw om 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 in7.nc wk+jjwcor f a xy(t 8 x9-udv,,zpy,x-p+ the shape of a uniform cylinder of radius 0.20 m acquires a rotational ra-nxkjywp,aft 9wdo r,-(+xpv+c 87zcx,uj. yte 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 smvcx-,pi 3s0vcn7yz;uzq)j) tib 8. disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-lonimtpy73i bvj;u0,-c) )sqz. z8c sxvng 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 anoy5mwz, a;c6tt *eip9iie2;cgular 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 or3i sy3jj44bpew -d*s 8m lx5cf our Sun, 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 neutrop i3les54y84xcwjr 3-sj m*bdn star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use energy a(*0 r ;zccfcbzk6el +stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and a zcf0 +rzkbc6l*e;c(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 illustrateq p1qnhzn5bkqgz d-k71mq-65 u p0jg) s 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) z5z*8xgag wnql*9zo1 is rolling on a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., we ignore frictiobggurboa3-1 lw48h l;n) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. A ru1b3-la8 hwggo;l4b t the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. nj*,4py nc0*h t jor*pIt 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 against which it rests (Fig. 8–55ny r0tj,n4phc*op** j). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on+:;( vwmsi-zy a7fylqnv tfmq385 w 5n a flat road is making a turn with a radiym3-i5 f almn :f8;qn75 yswtvzw(qv +us 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 oas h(cmc7dxlr*; wf-a 60 kgak9p0t)0.50-kg rock into a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does (k9- 6h0g 0xtam)skdpl;cao*r7 facw the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thi* lpi uv35gxlxez-,e1n rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds foieu3* lp,1lzeg -xxv 5r a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two czwlh6dzv p0(z; rr ;1oylindrical plates, ofpw0v; (;zz6lhrd1zo r mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along thex2 c14g-n iers looped rough track of Fig. 8–58. What exc -g4n1ris2is the 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 not6s0ui 8 yqpbk- assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions (/l va.roxqn9d2mfq 3 as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular accnxqlado 2 m/. 3rf(qv9eleration 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