A bicycle odometer (which measures distance traveled) is attached near thec2nyhf6., 05a, oivxe)t qabz wheel hub and is designed for 27-inch wheels. What happens if you use it on a biczht2),no,c f6 .a ebixq0y5vaycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on th/6 fw.s;kfh n0oeojwycq, +w/e rim have rasf, ky/w fohwc+/o0j.w q e;n6dial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by*xuno 2/ b1.kwejmk 4rv1i rq/ a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Explai,ht(hn4 l (.exrl bk9an.

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 hx -/f;zd7,9v sobqgdgy4)xtfbox30a ead than when your a)f4dxvx x;o, q9- boz/bg y f3tasd7g0rms are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprocketbhf- oilmjeg4*.6 :fgm+7qv ts 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 6 qiv+h :fm7gm lb. teo*gj4-fit harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to uu0l.h/ml8-- - rqawj,u1tihr(qt z thw( y0 hrun fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of a h0tu-j1.y i8m/0r(thlu -whl ,zu t-wq hq(rmass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry y3ru 8 s1up:fga long, narrow beam?

If the net force on a6i3* eiz1pntq/ (2 mcr pihcz( system is zero, is the net torque also zero? If the net torque on a system is z3mi ( ( e1 rhpzpnqit/czc6i2*ero, is the net force zero?

Two inclines have the same height but make diffeia +wa,5rcgm* dx) qo8rent angles with the horizontal. The same steel ball is rolled dow8q,wa)cmgx dr aio+*5n each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from xlx 5yx6fh d1/rest) down an incline. One sphere has twice the radius and y5d1x l6xxf/ htwice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have thejfx4qq:*41r. u btb nf same radius and the same mass. They start from rest at the top of an incline. Wbt qu41nfxqb. 4*rfj:hich reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum ar3st 8sutg5 cf9e conserved. Yet most moving or rotating objects eventually slow c gs39sf8 utt5down and stop. Explain.

If there were a great ;k/m 9uixm;ru8ft4 o)z,brfsmigration of people toward the Earth’s equator, how would this affect the lengtu/9k)fito,r 8zm r4 bf;usx ;mh of the day?

Can the diver of Fig. 8–29 do a somersault without having any initial rotation w y*;t-jdvdm2whe-;*jmvdw d2tyn she leaves the board?

The moment of inertia of a rotating solid disk about an axisn8i7z) he8f: qvgjp8mfxaw g; y3e6 y: through its center of mass iyyx8ne8i z;:mj3e w67 qgh f8:) vgafps $\frac{1}{2}WR^2$ (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 stoo*7 ms3qpbprw 8l holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, wm rpq3s w8*b7pill your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identice 0l+ -n0 .vpxgpkz*ejal and have the same mass. However, one is hollow and the othern*p0zel p.v0 kex-+j g is solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points west. In1;oko/1b g huwfbt;*a what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, bfow 1ogk/;uh 1 *;btadescribe 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 l9z/y-3c8t sa :d, tubsldv 59a dbaqf(arge freely rotating turntable. What happens if you walk tow bdctsdz/fv (y8ta:a,q59a9-ubdl3 s ard the center?

A shortstop may leap intom q5)a/ xwxe8/ut -lbdqv.+sr the air to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly rx.bq/xa-+)vwt 8/u5ed lmqs 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 -72ckf, o,j2ead,h tnj bwwj)oh+3kp of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss oe +2cbd ,o72hfkj ,pjwtjh-ow3 ,nka) ne or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (fyccqefsg n6q28 ,5g8a) 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 u)vgxb3sp,g5h-6(fm a i8psf amazing coincidence. Calculate, using the information inside bigf)8xssp -6gv3fpma h,(u 5 the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beamdvkr/z0ue63kyov (o pul o4 ,3: nn9jg diverges at zuoeru:v3 j k/kvo(d 06p43ynl9,o gnan angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. When the motor isyyp tvc1h *l6 gik)f,5 turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as) 16y c,gv 5yilphk*tf the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another chilszzg6 x jmje,x5n36j)d. If the ball makes 15.s5g 3x6, jjn)jem 6zxz0 revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8/,xlw hm-8sl6th e0yu.0 km. How many revolutions do the wl 6,myu xse0w-8lt/ hhheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculatnztb,3 4vmh -, q w-ngxgtwt*;e its angulargt-wxhnw nmq, bgt 4,-*t;v3z 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 ,fn:clxqlz* rs;7 uj(.0 s (Fig. 8–38). (a) What ;,uc lznlr7xfq:*js( is the linear speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) inpl2z93v.o d,evaw3cyc ,j / sw its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed -+ip up)x0 ,r r qoqmsab1+/st tce:6dof a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.0 cm from n ,7m bx,c np2mgj(6vmf/dqm 7tuya,5the axis of rotation is to emmuf n7,p/jam7xmgyt(b6v d5 c,2n ,q xperience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about itsxe d2dyq kqsf.n9km 8 (9ja);s center from 130 rpm to 280 rpm in 4.0 s. (s; a 8d)qnm92.kfsjdqx ky9eDetermine
(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 radiuy2x0hi7dm +ed 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 Apollo spacecraft put the:dh s8ixu4ad .mselves 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 d:ds8 uahx4i.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 to 15,0l6rm ;f b9rrn9c8:rvdr5/ l0f;.u edli ku1 ln00 rpm in 220 s. Through how many revolurc;nr; r9 f8kdbr.ie9 l6:v r5luul l 0f1dnm/tions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 g+i3txs) ls3g ay(g 1rrpm 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 f6 ;+en/ vzegrq 79cdzzlying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn throu vzr6zdq9 g+;e cezn7/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 24 jdolx:aop6p,w28px 0f)hb .j0 rpm to 360 rpm in 6.5 s. How far will a point on the edd) j2,6 a8wpx opoh xbp.0jlf:ge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 1500 reaocvo9no wn6;3(qnt yd97t5yh,/ed qvolutions beforeqatw73ho / ,5n dc d(to9yvnney6;o q9 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 reduces3u8md dnbnv ,j.or0o. its speed uniformly from 95km/h to 45km/h The tires hav.njr,0 3u.n8oo v dmdbe 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 its si: :y :tzgq;/lhdcr0+o p0etkklf- n 4peed uniformly from 95km/h to 45km/h The tires have a diameter o -yi: o:d/pq:t+kf rch0glz ekt4n;l0f 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 putsh.vp*vrb dc41f)2t jn hqy 67s all her weight on each pedal when climbing a hill. The pedals rotate in a circle of radius 17 c t7 phr6 ) jd*nhvf1.y24cbsvqm.
(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 7rp. 2nrx-a6hrt e ,iv)4 cm wide. What is the magnitude of the torque if the force is exerted t2) a hvrr6-r,nx.ip e
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel sho+9 :qwl4 vze*wa v5hl;ccn2ccwn in Fig. 8–39. Assume that a friction torque of 0.4 +: w;9ahl2wc c v4lc5cn zvqe*$m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of i jgq*9lu0f fwkp j:5a -sc);xa massless rod which pivots as shown in qa fksgw:fp0-x9cuij)5; jl * Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engcpyih(- bn fuz;45g rir,tt1;ine require tightening to a torque of 38httb 5ifgun(y rpi;c z4r;1 -, $m \cdot N$ 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 o0k *fabhd9d1 gf inertia of a 10.8-kg sphere of radius 0.648 m when the axis of rotation is throu0 h fk1ad*bgd9gh its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diamln 4j4d7syd4 8odt*: uw6 7uza x4bhkueter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignored (whyzyu76dau8w74 ks4*b 4:djdu lt4hon x?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thh8pb2u9 o fi7yin, light rod is rotated in a horizontal circle of radius 1.2 m.u of9yi p7h2b8 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 p6 atj e;; d6xj7oaa 1blsv1zd4otter’s wheel rotating at constant angular speed (Fig. 8–42). The friction fo64 abz7 ej1 1ax js;d;tl6aodvrce 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 point pls)omds5,8i b xl93mobjects shown in Fig. 8–43 8 3is,mbxsmp)5ll do 9about
(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 oxygedfvaki xes0 j;1+*rvr+k2abtq3 / t.vn 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 spinning at th3 s;eo/pjelo; fbiie*0*c )xu e correct rate, engineers fire four tangentiu ;/biesf ;oj cx*e0e*3 )olipal rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cyli* .9hagc+wr*g(oej gqnder with a radius of 8.50 cm and a mass of 0.580 kg. Calc qg ch.(oejw9gar+ g**ulate
(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 ito f- ug/*ssk:b 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-driven merry-go-round and is c( 8d;ec1h hwqm np:h)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) cphq(nw )1hech;8 md: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 is eventually brolf :9 dme6of,ubp4* ;y4vcqvhq fp: 7lught uniformly 7q ;u*fe ol pfy4v:m9f :pvqhcl b4d,6to rest by a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a 3.6 o y6aw:8cmgu ggm7e64-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 balgnw40/yk80:o xytd h9g gs+jr l is thrown solely by the action of the forearm, which rotates about the k0y 0 hn8r+gx: dgyg9t oj4/wselbow joint under 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 can be considered a long thin r-3gyshkmbt+-uj8 1 tr od, as shown in Fig. 8–4hg1uby m+t rt-8j3 ks-6.
(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 masses,7dmnl hor3b as0j;g.7 $m_1$ 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 accelera0ja zi*f8,yertes the hammer from rest within four full turns (revolutions) and releases it at y* i,ze 0afrj8a 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 has a moment of i coozo3 -0;fgmm cs7j0nertia 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 developsejz.0l k3teapy3 ry)7 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 withouf rh(1)f nlkq-t slipping down a lane at 3r-(f)l1 qkfhn .3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to thh30t58h5pf a sb,w(q) ummn mme Sun as the sum of two )b w ,mf pmsmq8a0hu t5hn3(5mterms,
(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 mas+nc9 koop *yn9(wxrdgy 2mo.3)5wbaqs of 1640 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 reo*w 9)ok pqo3ny( 5 +2dm w9xbc.yganrvolution 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 amx5 -kxulz :cti5vye7+q+8ua nd rolls without slippinq-v a8myli e5z+x5+uxt kc:7u 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 :h:kfy 3x1f:1dz2hs r ( elidaon its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservatios2 3kdf:i: zh(a rhy11:def lxn of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg;bl*ly xro gd+3uh0fkm z302u ball rotating on the end of a thin string in a circle of rax3go+ kd fyzr;0 3hlu0m*2buldius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 4bq02ke9ve a i, itlun;7wj)e2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotating at ew)ekta0i b 79e;u2, ijqlvn4 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotating at1/ ubr6irzm f* a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decrear 1b f6/zmiru*ses 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 re:uren40:;n,qa mo*hsft4dp-n .w lkxduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tucno dn*4wl:qtafxp .h4,- ek um0n;s:rk 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 rotqpnc+ 5ug:qi6 x; so0sation rate from an initial rate of 1.0 rev every 2.0 s to a fiu; i: coqxg+s60s 5nqpnal 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 vertica nto2e ixm34aw(gv8:al 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, approximat8ite:2(a3w v nxo4 gamely shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spin.93 5 y (homcmwmhya9gning 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 otbsa.(zcoddh( 07tfx-,o4 cvf 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 dr5p:oo/p ) v8s uzbtzy9opped onto an identical disk rotating at ang9oopup/v5)yt b s zz8:ular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns9kbtbk:0 ow(b)zy6 iq 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 stan0k6uj3 gfx:m;-ag;dbu z bk:tds at the center of a rotating merry-go-round platform of radius ;06jx3uab-g g;:tud k:zmfk b3.0 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 fre ; ejnvgxpcaszy7l6t0,. (7icely with an angular velocity of 0 e(n,iygsz76;j0c7 cvta.pxl.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 a white dwarf, losing ab5w64v rbmu/( v8qwq lap(*ceaout half its mass in the process, and winding up with a radius 1.0% of its exia 4w5p(ml6r a v*/q8ce(vquwbsting 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 win5-6ht:vd.fx oh3dpm uds 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 *qdpz zxgnrm- 8*rw:($ 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 rotate r7c mtx72gs 8m3i1gezfreely without friction. The moment of inertia of the pers7gitxger28 z1 mc3m7s on 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 ofnphx) 9z0; 9zqahs yh8*w. yua a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia of 17008z0y)*;spna uhqyh h.9wx 9az $kg \cdot m^2$ 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 o f++ er76bqvbs: f.iemn the ground with the end of the rope lying on the top edge ofe e:67. smqff+vb bi+r 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 from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such qt1)xq5whwv)b k7 ) xqthat the same side always faces the Earth. Determine the ratio 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 particlkw5wt ) xq)7bxq v1q)he orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from restlt f-z5i3 :yro 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 unifordj*fmfy /m 2uda2 *t7fm cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s2fm*j*m /7dyua f f2td 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 disks, eacwx,): fjz2ngpa+1ibx xy6i/ m h 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 c6,ij )y /x+amz 1:fbpxx wgn2ionservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of the rear wheer)sh td ;4l.o6pnh 8.l5ce lpfl ($\omega_R$) 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,9qp gxyddp;o/jc, ,4 v were rotatiq ;/po, gj,dxv4p9cdyng at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to us l00dzn;m 0xf)1.lxr 5eu.pz+toc gaee energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 140ge xl0 5+)ztc1a dm.n ;l00 eoxfupr.z0 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates hluz8+2-s rfvfg w87man $H_2O$ 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 f0mm: dwc6v7ra 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 pk4nr;hzq, uitzav 6 4v*u d7fpyv66) 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 acd qik*u,6 p zrvtzpvf76v446u;nahy )celeration 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. It is standing vertically on the floor, and w.so m.;3osrg8vyp.p edp/xi ) e want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height/8iygp mserpx)..;.v3op o sd h, where h

  A bicyclist traveling with speed v=6. 9(zjogx zfd4.2m/s on a flat road is making a turn with a radius The forces aojzzgf.(d 9 x6cting on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.5 m and becds;ck* 8ms d8gins whirling the rock in*m d8kc;cds8s 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 the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms ab,or;j*hrsof,+ g2kb o gck 40s thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of,oo rk2,0rsg f+h g4cbkjo*; b the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly w4sr-: vqi+tok hich consists of two cylindrical plates, oftvqo- r:i4k +s 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 the looped rough track of Fig. 8–5.h.2 xjihu yi t+0+istn/kn9 s8. What is the minimum value of the 2i+hkntn+s9 .x.yi ti/j0 uhs 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 de )yr0-s pdljm 95(r nh/rdcnk6u)v)zo not assume $r\ll R$ h =    (R-r)

  The tires of a car make 8 8 cw+5i+endxjmvdgm ok,;:*+qqb/pr 5 revolutions 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 acceleration of each q erj+5n++b d/wox8;c:, mmp*g kivdq 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