A bicycle odometer (w2b+ 6l;haycls hich measures distance traveled) is attached near the wheel hub and is designed a2sblh;+y6c l for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocr)6s 7 wf:8 ;ggzmiuinity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration8inzgsr6w igf ;)u 7m: change?

Could a nonrigid body be described by 0t:vx w+:pirt a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? f:e0ry-7v e9u rchp8b7go .g)j0ncgi Explain.

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

Why is it more difficult to do a sit-up with your hands behind k l9 vy:hy-t5myr(/hwyour head than when your arms are stretched out in f(9ht mhl5r/k yyw v:y-ront of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at 9 botlt*wr+yx(a;q+n the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small reaoa*9nt +t ;wl q+xrb(yr sprocket or a large rear sprocket? Why? In 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 fast have slendfj+ejuwvy1db gcd330xf 8h.5er lower legs with flesh and muscle cfdy+13jxcg8 50bv w.jdfu 3ehoncentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow beam)cmw3 c v61bcm?

If the net force on a system is zero, is the net torque also zero s+b*f k2vhk*v? If the net torque on a system is zerkb*v+kf vs *h2o, is the net force zero?

Two inclines have the same height but make different angles w1bs3y dwhqz6ib0 qi; 0ith the horizontal. The same steel ball is rolled down each incline. On which incline wii10isd yw;36qbq0zhb ll the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rozftg g;0r n4r+lling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater tzg4g+ nf0;rtr otal kinetic energy at the bottom?

A sphere and a cylinder hafz)vcc g, vo)y+j un71ve 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 v,7+1 zugcjno)cvf )y bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most movin+trng6u09d gjop1 gi/g or rotating objects eventually slow down90tg /du g1rn6go p+ji and stop. Explain.

If there were a great migration of people tkyc.b-p 9 wq5toward the Earth’s equator, how would this affect the lengt5wqck9.ptb y- h of the day?

Can the diver of Fig. 8–29 do a somersault without havigexv) 4/h9 pqyj p.j(tng any initial rotation when she leaves the board?xpvh9e4/.( jyjqt)gp

The moment of inertia of a rotating solid disk about an axis through its cen7 yq. g.7 b k o1*zb2.bidam8 .8hj)wqxmngcanter of mass.. hn qb1zxw.k)bm7c8 dq* 7n.2myagjgbo i8a 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 rotat;n;,b/ f rekbd3wplu4-4ne2 wei0lyxing stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decreas,-;x0leub pf34l2 iwrkn;dwye e nb4 /e, or stay the same? Explain.

Two spheres look identical and have the same mass. Howevez1f a.e;tp:n c/ gq8lfr, one is hollow and the other a :q8.ncfeftz1;/p glis solid. Describe an experiment to determine which is which.

The angular velocity of a wheel roi mwh 3g.y.j(,5kbpgytating on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speehmwg(j yp yg,5i. b.3kd increasing or decreasing?

Suppose you are standing on the edgel)ht7b l9jt9 1c7rerr of a large freely rotating turntable. What happ9 ct jrrrtll 7e7hb)19ens if you walk toward the center?

A shortstop may leap into the air to catch a ball and throw it qu 0xk -e+s7xhflickly. As he throws the ball, the upper part of his body rota+lf7sk 0-xh xetes. 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, discuss sp-pd79ag ( mywhy a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeypp9smgd7-a (ller can operate to keep the helicopter stable.

  Express the following c8lvcaisn f yah zv:)5e.b03(angles 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 yj91(*c ulhv(o, mahz amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in l*(huzoymh9v a1j (c, 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 f,4 0 fa6nadyu/kqf+ lhrom Earth. The beam diverges at an al 04hfqf/dk y+6ua na,ngle $\theta$ (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.i)1 k8tn vdu9s When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angula)8dvu9 ski n1tr acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child0 ql+ sa lgr/i4;b7ivv. If the ball makes 15.0 revolutions, wharvibs q /llgv47+;0 iat is its diameter?    m

  A bicycle with tires 68 cm ihbhga e2 6hz eeuec9d61h /3j3n diameter travels 8.0 km. How many revolutions do th e62g/ab1ejzeh93uhde 6hhc 3e wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2q/ocb- 8 (gtkb mxtliu/w6+3 o500 rpm. Calculate its angular velocity8gq /-3omtk ioct(+u6bxl wb/ 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 o hguiww9rnyx/2ln/63 ne complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear spee/wyhu9lnrgw/6 2inx 3d 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) in its orbit around 2,y owh)lyul a, 5sp9fb1m:eqthe Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a p*/owlete .9-s-)wjsmcc .r mvoint
(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 paoiteog3-nl 24 rticle 7.0 cm from the axis of rot g tnie-o23o4lation is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly k/m;ng - -nf7 t2 1jtij q3(t,ben*neibxwu.nabout its center from 130 rpm to 280 rpm in gk17tnt jn/,j b eqb*-nmuefxwn-(i2;. 3itn4.0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radiuzga 2akv bn+)g5z.;ochp .e;ps $R_1$ 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 spacecrafd8boq 5 i ihbgv0+-ca0t put themselves into a slow roto ic-ab5 0 +bidv0qg8hation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates unifzeaor+:p:a9fkx x0 +sormly from rest to 15,000 rpm in 220 s. Through how makxo z+ x ::raae9+fsp0ny revolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. ki0jf*)jl j0)leyf4 sCalculate
(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 ati0 j3zp2h ,8h jn4ckusgy.cmi+/wq 8 whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final shw8t0zyq3mn+/gicks,2u jj c h 84ip. peed.
(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 diamet+i cqoi0ne)q(5l*ziige:w 1 ier accelerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled i5iq0ewignie)+ ( *oic:qzli1 n this time?    m

  A cooling fan is turned off when it is rj3g :it 65fdatunning at 850rev/min It turns 1500 revolutions before it comes to a sto:i5 afdgtjt6 3p.
(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 revolutmel;l56ntq mk l/*1vpions as the car reduces its speed uniformly from 95km/h to e *lmtmvl;p 561 /qknl45km/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

  The tires of a car make 65 revolutions as the car reduces its speedr dz*jidupk0-3p/t5c uniformly from 953dip d/-*u50pckt zjrkm/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 aj z tt; 2bxk0ovg(d0g4 bike puts all her weight on each pedal when climbing a hill. The pedals rotate in a circle ofj 2z0t4;d 0vb xg(kgot 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 w)rg+dry -e6zyn b07ed ide. What is the magnitude of the torque if the force i0 e bzryey6n-+) gdr7ds 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 in Fig. 8–39. Ai,nzc)vlg i0goo:- r :ssume that a friction torqg,i0:r glo:c -) iznovue 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 th3t9ipl5. t dd0 dq9mp i.q3tjqe 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 tor3d jd9t.pd3pt i9q.i5t qlm0qque on this system.

  The bolts on the cyliz 3 ybdgxm*wnf,k)7q8d3i/te +u) t1nbp0is mnder head of an engine require tightening to a torque of 38 m)mwd3 fii e3), tsgzbp/8ndbxq1k+7 *uynt0$m \cdot N$ 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 o nikq ymagien-co6al8 06y6;0 f a 10.8-kg sphere of radius 0.648 m when the axis of rotation is through its cen -gyym 0inael6k; 6i8c6o0nq ater.    $kg \cdot m^2$

  Calculate the moment of inertia of wnh,b;z h.uvvb*8cv ;a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. w; bh.u8 vv,; cnzhbv*The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin,l :ge,(r9ko 4spsi)fy light rod is rotated in a horizontal cfy,ope(glsri : )9k4s ircle of radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowlc63+ vx pcho,n on a potter’s wheel rotating at constant angular speed (Fig. 8–42). The frv, x6h3oc+pcniction 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 1pa 6ybvb s7u3the array of point objects shown in Fig. 8–43 abo1ubvy7s6p ba 3ut
(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 ox1v6)dh rt3s jpc--g w0vz:fsdygen 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 spinni(:bhegw6 a o/p t7x)aeng 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 p 7exh)ta( e:o/6bagwthe 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 8qr8r4l bq:-ob .50 cm and a mass of 0.580 kg. Calculatblbr oq-48:rq e
(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 fr yhsl(4cl)x*ebr:)gdom 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-drcf1hx90 q8tnm ;yqob7iven merry-go-round and is able to accelerate it frq9 ;yhox70f18qt cnbm om rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotatinnnwqctwq9 (, 1g at 10,300 rpm is shut off and is eventually brought uniformly to rest bt(qw9w 1n,nqc y 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 acceler+ ( mb (s ja1zj.svze7er0end5ates a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the actiov6way2 d*3ntan of the forearm, which rotates about the elbow joint und3wyn*d2a6 tvaer 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 rod, as shown ,6(r8 (w+qfzn7fboaa6kjcvk in Fig. 8–46. abnc 8(fv,kr( +qfz7j66oaw k
(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 j/*01 i a8qmjkn2us qktwo 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 from restij fm)1wv:5dm5zsi gm:e ,+/odv9tp o within four full turns (revolutions) and :wgeimz/i 51fsjo v9,om: mt)v+ 5pdd 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 of iui03rp uj-j: q uk6z l,.,nt 1an2sjpsnertia 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 tor8hxp gd(7 r6shl4djkjtou 991que 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 without slipping dolv*rza8(d2 ydf8eg, 6qv r6 ftwn a lane at 3.g8f6d(z a*dy 6qrvfet,2rlv8 3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with reszl;0mq6x *g4 eo7 rrohpect to the Sun as the sum of two terms,4lxe0 r;*mrgqo z7 h6o
(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 r hcb /*sr.0da 4vpc2eand a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revoluthpr4 rv*d s.2ac ebc0/ion per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls widd+m0k2mwodf o ol9h(9/qy:u thout slippinoldf wh0o/mo9 :qdy2kd+( u9m 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 verti 3e hvinj2:qsn e6-i2vgjfy53cally on its tip. It starts to fall and its lower end does not slip. What will be the speed 3:35ishjv qg6nv2 fi- 2ejyne 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 balltoi/fs9 wittzjcb s4)w0*s** rotating on the end of a thin string b sc) 0s wf*wszt9/o4*j *ttiiin a circle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg unifo yk5m7 f7dc2wlzdk88hrm cylindrical grinding wheel of radius 18 cm when rotating at kl5mw2y dkf887h7 czd1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, handsvq,ag pbpu2fw9ui,k(f 2p . a. at his side, on a platform that is rotating at a rate of 1.3rev/s If he raises ha i,.pwup p a(2bguv9 2fkqf.,is 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 one shown in Fig. 8–29) can reduce her moment ofs.-wj1 l9u:com6ba mu inertia by a factor of about 3.5 when chajwa:l-6.c 9ub1mos um nging from the straight 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 rotation rate fr jf,gy 6ihpiam 3.+/egom an initial rate of 1.0 re,m6 +/.p g yjehiai3gfv every 2.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis through its center at a freq;f 6ul,pef jde.c9:s 0je8 onxuency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency jedse 6 9nuj.l; 0fcep,f:o x8of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum o69/cq vwixng4r-l 2p kf a figure 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 momentum of cis np7fxp.0xj.rb-4 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 onto anin). yu 0mkk:p/c2vi c identical di/k: y2cpk.i nm 0v)iucsk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns +ip(m : 6ytbo 7z4d ok1 grw2el6bjh)dat 2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotating merry-go-round platfoxtf7n 6qs 559x mqd4owrm of radius 3xqd 5 snfmt4qxw956 o7.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely wit8/- igx t-nqi2mjvm )nh an angular velocity of 0.8 mg--i t8/v)2 minjxnq$rad/s$ 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 collapsx:f ue,eu33w kes into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the f,e 3uuex:3 wknearest 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 (a facyr0h3fnxzl *y*eg 9+*b 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 k6 v/)y b0upnadp)gz* $ 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,(ia, mg:x w- .hw -2y:6 r;raueeygzpj that can rotate freely without friction. The moment of., rxpwaa e ihmr zwe: u(2-;jyy6-g:g inertia 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 merry-go/8tx xcd8)r6s suwy,c -round turntable that is mounted on frictionless bearings and has a momentwu 8ty/ xsr6,cdcx8)s 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 lying on tn o 7y3eg v:dyxgz;l*fjmmx9/y,;w9 uhe top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. Whatzo; l:yj efmn*mwgx;7d v3gxuy99 , /y length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side alwa-a;o.nvmd kxk8fbq8 3ys faces the Earth. Determine the ratio of the Moon’s spin angula fv8 mb o.38;dqxk-nkar momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from restsu pq w1:3;hjd 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 g6 i z;x/no-c01uf tdxthe shape of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the totcdxx 0 u1-fn6gi/; ozrque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical d ig(ng4bg;8+qhe-pt x r1viv) isks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid-(qi+8pgb grin)e1t4x vgh ; v cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular*6r:/( lw ;b:ely h/l uukb74lkdogoh speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.ugonmvp(.) /o1upqv ggx)u;4 0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitat moogx;gqu p)v pvu(4./)1gnu ional 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 auto f1g y, wk+apxrk 5pk (slfe-u(/rl2.rmobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a to(+ar/(u,l fx.pf kkl- gr5 wsey21prk tal mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrat6 7 o*sl-bq ljaazxf346. ;vtb+rujhges an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal surface at speed vb5t fpg4hza/qt .(g y3/t.qet =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 e( ,dh.fcr: hzfriction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rodhfec (.,hzr:d , 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 verticalcu i2x2 xyry2wkk +g r1b5(qrt);d f5m9 vitl0ly on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step againm2yc w)5 d+129i (qr0rxuxtr k;iyk2v 5lfgtbst which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a fl.wbj7 2 adgr0 rw(etjkx;h z;9un+2fl at road is making a turn with a radius The forces acting on the cyclist and cycle are the normal fowr+uz9 w j0 7;xnd2; alr2bgkhfj(et.rce $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ 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-/1v ojc3ws*u:)t g jwtkg 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 theojws3tucgj / 1*vt):w torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thin rodxvzqc 66 couezepz c1 5g7;8,qs, making reasonable estimates forgzu;eq5 6coe6,8v zc1qzp c 7x the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindrical lw/ard- km60a.mm yk6plates, of mass a0r wdkm yk/l6.m6- am$M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 6a7d2j c+ioskrolls 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 of the loop wit o7cdais+6 jk2hout leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assum 6dn1 wk qwf1+wd 08zaox8wd mqc+)d3de $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reducw(7rt zi7++c 0wv9a:qhhf /ve gxrvo8 / gkv2yes its speed uniformly from 90km/h to 60km/h w/vi8gk 7:rx0+rz+t 7ha2 ec hvo9fqvvyw/g(The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s