A bicycle odometer (which measures distance traveled) is attached near the wheel3p5cftw aw/ ro,(qvgs jr/2e-o ,x5el hub and is designed for 27-inch wheels. What happens if you use it on a bi2xp a ( -wroc o5,lrteqv //ef5wg,js3cycle with 24-inch wheels?

Suppose a disk rotates at co2toi-x.z 2 z z2x2 :wr2ysacoenstant angular velocity. Does a point on the rim have radirt 2a:-z2s .2 cezwxixyz 2oo2al 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 a single value of the angular velocita79/*pzz)i d z (pfry:fbcb8ty $\omega$ Explain.

Can a small force ever exert a greater h7uq2xd0gjb .;0s .d smml h7pb*9uextorque than a larger force? Explain.

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

Why is it more difficult to do a sit-up with csacp ;,/fvb: your hands behind your head than when your arms are stretc/ cbfavp ,sc:;hed out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seab-g:15 ytk 1h.rpoq rven sprockets 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? 5py .aktr-:gq1r1 ohbIn which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run -rqag, 6mx l,p8eat( efast 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 distribga6pat(e- rqe8lx, m ,ution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a llxppmmpx(3 )m.x,rz+ ong, narrow beam?

If the net force on a system is zero, is the net torqm;*y dmo6 :bsbs3 c4xxi6;j nn*j3gufue also zero? If the net torque on a system is zero, is;mg3uxj ds*nnjob3b*m cx fi:4 ys6 6; the net force zero?

Two inclines have the same height but make different aw j6g;wc,9*hsnhbv 4 /lct:f cngles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at thb6t:wccshfw g4c , hn/*;j9 vle bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest)15a6u/0 zjqjwajj q s. down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic eneruazw1.j0 6qs qj /j5jagy at the bottom?

A sphere and a cylinder have theh 8p0,iwzk(j0+ m1hamhj pve0 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 bottom? Which has the pzi01h(hv whmm,pka j0e0j8 + greater rotational KE?

We claim that momentum and angular -tzka e;7 b9w2kjrt2amomentum are conserved. Yet most moving or rotating objects eventually slow down t r2 9 wt2-akezjka7;band stop. Explain.

If there were a great migration of peopl8 i:cxnms7aek -dzx dfo+a 7r7 ku+4rp5 z2pl-e toward the Earth’s equator, how would sex-l7c z+k: -m82 prkaif7upx5+ 7ddz4 ra onthis affect the length of the day?

Can the diver of Fig. 8–29 do a somersault without having any ini- zc/:av am;oza,q) oyk-ub2jtial rotation when she leavk -o;jy 2av c,/ao):u z-zabqmes the board?

The moment of inertia of a rotating solid disk abo0d9hj * 1t- lyumvmnkrmm- li0q*** tgut an axis through its center of mass im-l nlr*gykjmvimd**0-m0ht* 9 1 qtus $\frac{1}{2}WR^2$ (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 oun .;b(-vppwpt6c5v7 / pz m+pg tro8vxtstretched hand. If you suddenly drop the masses,n gppvvt6vp p;7zm+cx -5(w t/o8p.rb will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. Honsgd dnb( e)-clot 23,wever, one is hollow and the other is solid. Describe an experiment to determine which is whi-e ddtl), ocn(2s3bgn ch.

The angular velocity jqev l)a2dm m ad.1i2wc.6c7kof a wheel rotating 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 acceleratioiaa 7 we djmcdl.kv26)qc.1 m2n 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 large freely rotatini en// m1lw*v/tvalv g2uk51y g turntable. What hv*e1mw/1v a/v ytigkuln2l5/appens if you walk toward the center?

A shortstop may leap into the air to catch a ball and throw it quickly. As ha wsopgn h8 ;f00bib*r-xt40z vmng, *e throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explab gtnw*- *xbi o0fm,p;n8g0zas r4v0hin.

On the basis of the law of conservation of angular momeno6n9i sgwv6 0itum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can69 i w0no6visg operate to keep the helicopter stable.

  Express the following anglesdftz +/truio0;;ow*6ct( e z t in radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincidence. Calculat96 gg +dgz7ahxe, using the infor zggxg69 ha+7dmation inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The b4j*c(g + l:fwjg,bkwg eam diverges at anf:g, w gbwkgjl4*+(j c 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.zfa wm;,k xs9f03k7xjscva 62 When the motor is turned off during02x ;7v sxa3mf9k akjc6wzs, f operation, the blades slow to rest 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 toay8su anwk4gd6z x,t) 4 bs+d9 another child. If the ball makes 15.0 revolutions, what is its diameter? d46+wz kga yxt d,as8b 4s)9nu   m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many rev,jkk15dolfn:*go b,:v sf 0kbolutions do the w,d51:lfb kkjo bn ,:0*gk fsvoheels make?    $rev$

  (a) A grinding wheel 0.35 m in diam(p y(jvkoms/ j*53mlk n)r1tq eter rotates at 2500 rpm. Calculate its angular veloq( mst/ jrp5oyj1 v(kkl3nm*) city 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.:8q9s(kgl 2p;ubw / //ij ru:fh)zogomrob4y0 s (Fig. 8–38). (a) What is the linear speed yb ur:oub8hw/(lmz k /of:g9/gip; o)srj2q 4 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 od5yck;o cslt8 3/x dzj(v;0tn f 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 speed of 0gzrm m hybm7l4::mz x+1ootggn1z )1a 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) mus ;-b+zdvbb1el6+pz q yt a centrifuge rotate if a particle 7.0 cm from the axis of rotation is to experience an acceleration of 100,q6y z-lze+bb vb+d1 ;p000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its cenk 88 gpjnw)wi/ter from 130 rpm to 280 rpm in 4.0inp)/8jwwk 8g 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 radius 9ki: n(i8cez z$R_1$ 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 Moonruk 39)pvc jrx0b/r:k, astronauts aboard the Apollo spacecraft put 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 /r:ruk pr0k9)xcjvb3 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 unifor /nz.b)0in6 ij4o5ce9fdn3 mc 2cbagzmly from rest to 15,000 rpm in 220 s. Through how many revolutions did it turadz2 9i0c n/zn4 ).ej bf m35icg6bnocn in this time?    $rev$

  An automobile engine slows dnt,m;z58 ol0 s/aaam down from 4500 rpm to 1200 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 jets in a whirz 1(gbpsn,lz +ling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutionsbsg( zln1zp, + before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 360 rpm in 6.zdq8putwg, b- xc9e+(5 s. How far will a point on the edge of the wheel ce qbwxt9-p+d ,g(8zuhave traveled in this time?    m

  A cooling fan is turned off when it is running atjm)dba yw,(s,x ,z+csbt -zb1 850rev/min It turns 1500 revolutions before ibdms-) z1tj,,cs, yz(aw xbb+t 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 o2f :tmcx8:peiln7u 8the car reduces its speed uniformly from 95km/h to 45km/h The tiolie t7xu8cf8:nm: 2pres 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 car ep(4w-g +szwb reduces its speed uniformly from wsw-eb g+z(p4 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 bike puts all her weig s6;kf 5pyjkwf02 g3nght on each pedal when climbing a hill. The pedals rota n23swgkj g; fykp50f6te 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 ofu87e)(+g kt cdl a9dbh a door 74 cm wide. What is the magnitude of the taedc +hu97 td()8kbgl orque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown zjzg:7q3d,gq 3bj b3w in Fig. 8–39. Assume that a friz j3,bq3 7wdqz:b gjg3ction 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 masslyx:xum5 - ntqx0r jcqjln56u tj/64;ness rod which pivots as shown in Fig. 8–40. Initially the rod is heldu 05 ;u:4nny6 x/r lcqnt5j6m-txqxj j 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 any;rj,zer(33-* h5rf rtth s fh engine require tightening to a torque of 38 zts 5trh3j3efh (r*-yr,;h rf$m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of radius :q*6 v oeki f3mpo;yk;0.648 m when the axis of ro*oke; 6op;mki3q:yvf tation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm ii.0u8/+8nmz vo r diven diameter. The rim and tire have ai8niv8e /o0z.+um rdv combined mass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end p4y3jtzc qr6iqtv0 o(e*q2f6 of a thin, light rod is rotated in a horizontal circle of radius 1.2 m. Calcula4re 06ovtjqqi3f* (qz2 y6cptte
(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 bow)czh 7. t a,8,g3eao7f cddjmsl on a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force bhc ad7 ga)7tz emd8,sjo.3c,fetween 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 p:t,3 txj )p c2m2ve mplie1q 33bw8thof the array of point objects shown in Fig. 8–43 abo3 htc3xv p:)qt2 ep1il8mj p2e3wb ,mtut
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen atoms whose top y6jt/il 0mvqe2 hx70tal 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 nqn.oa;4kyj6 jaq0 f9 6jdn2k satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what iq0oyk k9 ajqnf nd.nj2 j;46a6s 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 8.50 cm and a mass dg4g1 +qz70r; xdpkw gof 0.580 kg. Calculat0wg 4x;+gg 1qrp7dz kde
(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 p((aaqf 0m spr v a*89gd+3px6tae7lh 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-dvcwux-4 he:z1 jqv* 9eriven 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 9-v41h ecx*qz vwuj:ewith 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 an0r+dtc 2, a rq(+hujjvv38 prod is eventually brought uniformly to rest by a frictional torque of 1.2cjop(rvhu2tq03rr +d+vj8 ,a $m \cdot N$ 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 a9p;:jsd o-5f ltdfbq +ccelerates 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-kgyz2c9rzb 9u/ng rnb(k bu3y )9 ball is thrown solely by the action of the forearm, which rotates about the elbow jor/u)n k9(3bzbyz99cnyu 2gbr int 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 thin2p m/ac41xcfsh;n7 ; na gkorqx9 a6x( rod, as shown in Fig. 8–4haaqp cn9;4x2gc/ x71 6f;xsmk(onr a6.
(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 .9 dzyxpw8)fk 0 :jumdmasses, $m_1$ 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 q xlshn4,:4ali1c, vqwz0 yix.a4rr 4j w6bj, the hammer from rest within four full turns (revolutions) a l4nr1 x,:.vswhqaq ,64biz4wa l,cjirx4 y0jnd 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 has a moment o * z*w/d*m9ct -m py9y cy y9zx.mhebx+ryu*z7f 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 280m0wisoo7u7-y lt5fpfg 48z n - $m \cdot N$ 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 without slipping down a lv/j5*5 za y,walxjq, banejx5avljq 5w,*/zy, b a at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with 6 ioxh2102x (x2g d thzy5glgzrespect to the Sun as the sum of two termgx 1yhho z 065(tlxdigxz22g 2s,
(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 anf/;/ tyrh1jg g.d(ixbd 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 8.00 s? Assume it is a solid c;i jxt dy/ r.f(hb1/ggylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and ro7tuxtj -m 8e5clls without slip5xutjt8c-m 7 eping 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 is9b :mcqhus- .vd3c0cv balanced vertically on its tip. It starts to fall and its lower end does not slip. What sm 3 c 9ccvbhqv:-0u.dwill be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ballb;krrvfz r,h.6 n z hj z gur73uc-;a.m(-7fof rotating on the end of a thin string in a circle of radius 1.10 m at an angular spemjn.uzz 7 h hbf36rcrv.go( -uzr,f;f;a -7kred of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniforef/ 8m.4horner8n kn 5cjj-s0 m cylindrical grinding wheel of radius 18 cm when4orn khc8sjren5e 8m / .n0j-f 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 platform that is rotating at a rate of(h8)uczpss02m*,h v t4mwdjb 1.3rev/s If he raiszpjh2bdh vw,tcm840 m) us(* ses his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the onej/k3nnwz cufb- 2g87 *.mbwe d-hg z/a shown in Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the strw*eg kdn 2jfu/3 a7. mb/zngbhz c--8waight position to the tuck 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 ro.o wjrf+z/r f dg5*tm9iju/x -tation rate from an initial rate of 1.0 rev evu .g/* wr/xj5 ftrjdmi+oz 9f-ery 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 around a vertical a :r9xdm h6(c lt..dwguxis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diahg m9dct(x6. r:.wd ulmeter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skatensshdm6 (kz7 .yrzj0. r 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 momentum of th y9dzy7h*w-vce 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 onto an ih:3g ega,l 3ladentical disk rotating3ela3h la: g,g at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns ats0m+y fsac* dd2d8vw + 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 atmo/ dzt e q),zs*b8*hv the center of a rotating merry-go-round platform of radius 3.0 m and moment of inertia 9th *d,/ze8z)m qs *obv20 $kg \cdot m^2$ 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 an angular velocxq2 beum66eh(til j*h*,t 2vsity of 0e 6l2ht ej,tv*xuqbs*(i2m6 h.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 coj/o-gvm (d0joiib: i7llapses into a white dwarf, losing about half its mj:/iviibm oo- 0d(jg 7ass 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?(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 of+0c5qqwgo-vl g 3-8-mljn mkn 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 d 5n4zy:j ;jpnwo -g,m$ 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 platfc)0,m 5i.qwqgv kxqrsh2 r,d *orm, initially at rest, that can rotate freely without friction. The moment of inertia of the persorm,c .h*0xiq)5w2ksq ,vrdqg n 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 merry+/p tn;jgv2mj-go-round turntable that is mounted on frictionl/;v2mn gjp+jtess bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of the rope lyinws)tg1z4)lb3c(vn jkg 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 spoo43gwvjk)t ( snl c)1bzl 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 thatohbjem.. , 7vx 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 particle orbiting the Earth.) o hjveb7.,mx.   $\times10^{ -6}$

  A cyclist accelerates from rest at a q ysf. dj6l.48/ kxpcrrate 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 ibro 2+srcn ii,z3r / qi9xe10ln the shape of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s in2 b9+rxczl oi/3,riisne1 q0 rterval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks4.g 1vjdnrv ep51 ggg4, 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 e.gvg1r g1g5djnv p44 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 qxe zfri83nodt)*2+t)fg f+ r1/kzru angular speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 86+-tpc/ip3q p/wqa:x3jsjkd)d lp-u .0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse todapc-) i s /3q3q: jutkp+x-6jpdwp l/ 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 i:vht5 yxe ((rgt to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass:( xe(yrvth g5 of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates cm1lh +wn4x .lan $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal surface at spee 3pdq:lqkbhoeur/w9na b77 *)d 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.,o,j+f;y i ck6p x:n-if 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 releax,f 6 f;npi-yoij:kc+ se, 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. It is standing vertically on the floor, and w : :a*mf5ijlqoe want to exert a horizontal force F at its axle so that it will climb a step against which itiao:5jm* l :qf rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.28nvy 5:jn 0gqt/ z3bbrm/s on a flat road is making a turn with a radius The forces acting on the b5 0 q:8nvzntgj/b3r ycyclist 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 be*z 9ecf.ho8if gins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm af 8fz heo*.9cfiter 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 as thin rods, uvlj:rq t9u7+)f3kjx zeyd 9(u k (g/imaking reasonable estimates for the dimensions. Then cd3ku (k v iu:texgjr fq+j(l )u7y9z9/alculate 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 clut *p1hxg4no1mnv +*fw4u 8pz oxch assembly which consists of two cylindrical plates, of mass n4xmzw1* ux4of+hgvp81 n*p o $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 20hap lt7e l*u:mq+bh along the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point lb+m:2el *h 0uq ahtp7of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do 7 f1im+a3zl)pps kd*pnot assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed unhiw -ywho l5bh 4a*1*xiformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each tire? h*ih *- w5h1oa4l xwyb$\approx$    $rad/s^2$(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