A bicycle odometer (which measures distance traveled) is attached near the wh-ta --df k:wnheel hub and is designed for 27-inch wheels. W--d tn-w f:akhhat happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular ve s16ahi 2fery9locity. Does a point on the rim have radial and/or tangential acceleration? If theiyfe9 1a hsr62 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 b,l09 ummb:g zby a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than argdzh y / p9jf5l-p7qj5a 68kd 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 your ha 9 ./om*gdomqvnds behind your head than when your arms are stretched out in front of you? A diagram may help you to answer this 9oogmv./dmq *.

A 21-speed bicycle has seven sprockets at the rear wheel and three at thhx7*/giz ffm2 qo 4j+ie pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In w 2fjz ifgx+4q*moi /h7hich 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 slender lower legs w: 42rqgf3*onyinkjg 6 ith flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explair j34n kfgyo6gn2i*q :n why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narr- oim(1kr3 jnyow beam?

If the net force on a sy82n 9oomyd6etstem is zero, is the net torque also zero? If the net torque on a system is zero,e2dno6 9t moy8 is the net force zero?

Two inclines have the sam(:9filbvk fp,g 1s xzon- ,9zse height but make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explz,z-fg 1s,:pn l( xkisb9fo9 vain.

Two solid spheres simultaneously start rolling (from rest) down an inclr vv,ao:2w gx1ine. 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 energy at,w:v v2og1xa r the bottom?

A sphere and a cylinder have the same radius and g xnggzvy: ;x,pq(erd 79t4l4the same mass. They start from rest at t l(qx9v4yz , 4 :grdgxg7t;nephe top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum ar e)zbgc 8g 3pzvyua8/ se2hz wd.q6g.)e5*b jue conserved. Yet most moving or rotatin*b56wu g8/ husg.e)v )3 zd b2ejaqzey8c zpg.g objects eventually slow down and stop. Explain.

If there were a great migration of people edi; (6+yd9ac8nimpc toward the Earth’s equator, how would this affect the length of th iyi8cpd (edc 6+9m;ane day?

Can the diver of Fig. 8–29 do a somersault without stc4d mw7si6x y. .mj/having any initial rotation when she leaves the b4s6i s 7dmmy .wx/tcj.oard?

The moment of inertia of a rotating solid disk about gi ,qe6,qq zyr-ksx5k0er,+fan axis through its center of mass,frr, yq ,qkq+gx6s0e -zei5k 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 sittixrw( :0lfr v2e f6)ucdng on a rotating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Expl() e2:fcwu60 lfxr vrdain.

Two spheres look identical and have the same mass. Howe- 1muh1sn y:jfwuii6- ver, one is hollow and the other is sol11niy 6u:s h-m -fjiwuid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating m m;f-nvr4;fn)b7g fjon 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 pointnf4 7j mrf;fnbg;v-) m at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating turntable. Wf(t,jnw:- itd hat happens if you walk toward the centdi: -f(w jt,nter?

A shortstop may leap into the air to catch a ball and throw it quickly. w)fc s8h2ph2y qi5;usAs he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and leg ywpchu )825ssfi2;hqs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discuss wh6ge5 (f dq7gnvy a helicopter must have d7v5(enfg6qg more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (a) uht cj*ov7fsfh44 m)t qr3-k9 s+fh+k30 $^{\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 olqy *ld)u6r e45f;hlr f an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diamefl4 e r5);*ldur q6hlyters (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 Ea. (9(wwc jwwj 3rigyw 2sa-ut8rth. The beam diverges at an angle j(wi c- u9yrwag ws(.w328wjt$\theta$ (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 r76;qoh ( y)p )xbyfzcli9kh+oate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slowf o;+b9 y)7y(xoi6 zkl)hcqph down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the ball make(kjkvi nz.r09 s 15.0 rev k9rvz kj.(in0olutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travelt vw/9)kex/-0ac7q m-wp ck ucs 8.0 km. How many revolutions do the whev0 p uck)cq/--ce/t9xmwk7wa els make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500pkeds,t:npdg. -yweq: t h ,fm(1eq( 3 rpm. Calculate its angular velocity ineht w,3 k -e1 ,pqyfqnm dt:(e(d.gp:s $rad/s$ $\omega$ =    $rad/sec$
(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 79ye*as b5gab revolution in 4.0 s (Fig. 8–38). (a) What is the linear spee9*bs7 5 eybagad 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 arof.1 2o h4a3qsrjrfzjiy;hfo 7q( l6j4und the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear sy28jofwb21 y mpeed of a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotat:tip4*hsw-c cfy(mk5ve d/r, e if a particle 7.0 cm from the axis of rotationrswmccet4( /* fh-i,vdpy 5 k: is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about itsjfu )tq 7 2o(:x7hgvrg center from 130 rpm to 280 rpm in 4.0 s. Determinef2quj )7ht7ggx( :orv
(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 ;h- qt p:j0sbs$R_1$ 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 themselve7warb c-r 8dgm4 5+bpo db9ci)9vnaa(0 qpz c6s into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated froc8pc9cr9ind6v b 7p)0 5 zr+-4aad(ombgbwqa m 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 uniformkih9044l 8enwf;k :jp0h bs dply from rest to 15,000 rpm in 220 s. Through how9 4jikpws;8de hpb :kf0hln4 0 many revolutions did it turn in this time?    $rev$

  An automobile engine slows down from 450lid(t zmmrqa* ,0d9u 60 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 strel( af ekz,4x5v)dj 0jksses of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final speed.e)40kjdvzfkj x ,(la5
(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 rpv1- w0-wb rvhnm in 6.5 s. How far will a point on the edge of the whb-vvw-1nrhw 0 eel have traveled in this time?    m

  A cooling fan is turned off when it is running azekez + 9(zeq(t 850rev/min It turns 1500 revolutions befoz(9eqzze e k(+re 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 reduces its spk.(g .ml,iae )ipw:a qr-ir u1eed uniformly from 95km/h to 45km/h The tires have a diamei:a-pur.r .kei(wg 1l mq)a,iter 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 speed unifo8/l:j 5-t4 y dvipt+wyl,ek iwx.x 6zlrmly from 95km/h to 45km/h The tires have a dia wlpd+ l.eykxj 5i t68/4y-vxlzt,i :wmeter 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 ;m cnhrc:u0f)puts all her weight on each pedal when climbing a hill. The pedals rotate in a circle of radiurc;) 0m :fchnus 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 wide. What is the ma g4bhs pzqb/s0qrwha8tj2.2 , gnitudes.absq h,pbh2z 84gw2r qtj/0 of the torque 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 axl0i- g+bfddm t.e of the wheel shown in Fig. 8–39. Assume that a friction torque of 0.gt d m0d+b.f-i4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attac5v xru60jk 3+ vywrh(uhed to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is h3vw+ k h6uryr0u v5j(xeld 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 enginee8 l ty4vgp-2-m. lfjq require tightening to a torque of 38 4g2pl- lqvye m.8-tj f$m \cdot N$ 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;xv3jpy9flt7*.z nea 10.8-kg sphere of radius 0.648 m when the axis of rotation is through its tvj.9 lpz3*7 xnfey;acenter.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66yp wogcn;jd1+26 y mu5.7 cm in diameter. The rim and tire have a combined mass of 1.25yy6c d1 unj5;g +mo2pw kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in a horizonta/z 9zqoyaa j q)frl+. xgj1y59l circley )/9q f.oxz+r1 yj5aqajg l9z 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 bowl on a potter’s wheel rotating at constant angular scjg6::*hioyn9 2trlyt.zq7 d peed (Fig. 8–42). The friction force between her hands and the clay is ly.zt j gtqrnh: 6ic*o9y2d7 :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 ofxfb5b 12urd;w point objects shown in Fig. 8–43 abou bb1;w 5xfrud2t
(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 total mass is 2gyyaiz4 r* zv b88qy)$5.3 \times10^{ -26}$ 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 the correct rate, engin dm/dj/-rfsgfq5p h;7 eers fire four tangential rockets as shown in Fig. 8–44. If the satellite ha7p5h dgfr-/ /fdsm jq;s 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 cylinder with3 a2g()jqitu n a radius of 8.50 cm and a mass of 0.580 kg. Calcula t2ij3n)(a qugte
(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, acceleratin)w9dy bkrnbrnb/o 4; ;g it 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 ippr63+rcnq,hu5 a:q ms able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglectincrp6mph: r3 qn +5aqu,g 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 bpjc)a.z;qb j(fl r *h.ol y,*brought uniformly to rest by a frictional aoj zql);*.*brh(fyj.c, blp 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-kgtvixjl5a q o;gx +95vey)ng+3 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 sol3u hpdc m9f98nely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 898hu3f9 dcmpn –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 sh + y/ *cjqjop8l1bud*nown in Fig. 8–4/8plujo dyc1j *qb+*n6.
(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 consif es,dr+.yh 67*y43o-wf-dk5 ocwhasdpu s-ksts of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer 8i q6c tqzbdyp+u6s,) from rest within four full turns (revolutions) and releases it at a speed oqstp8+ idu6zqybc) ,6 f 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 momac - pz.lw2t3l*h9 keyent of inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a tor,e9hjn-v7+t*wo gx iuque 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 rafmbz4qxd 641pu+o v+tdius 9.0 cm rolls without slipping down a lane at 3v+6qom b 14dp+x4zt fu.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect t dq5mo- z /jx5v:47g+ud sxekzo the Sun as the sum of two terms+ :7mog ed qs5vz k5xxzuj/-d4,
(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 1pc5c,s, ffnw, 640 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 rev5ffp,, nw scc,olution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts frnm liizv c-04.u 7chl8om rest and rolls without slipping l4-lc ciz 0ihnm.v78udown a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

  Two masses, $m_1$ = 18 kg and $m_2$ = 26.5 kg are connected by a rope that hangs over a pulley (as in Fig. 8–47). The pulley is a uniform cylinder of radius 0.260 m and mass 7.50 kg. Initially, is on the ground and $m_2$ rests 3.00 m above the ground. If the system is now released, use conservation of energy to determine the speed of $m_2$ just before it strikes the ground. Assume the pulley is frictionless.    $m/s$

  A 2.30-m-long pole is balanced vertically on its tip. It startw;(m(ush*m, 4 t0jwwslp + yyss 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 conservu 4w, smp((wsjw*ylt; ym0h+sation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg bv e:h6y1twf+yc l-( ozall rotating on the end of a thin string in a w+voe 6c- h:(lyyt1fz 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 uniform cylindrical grindingwjjj 4:) *rkxm wheel of radius 18 cm:jmrjw)x jk*4 when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on ail2w/dc fzj7ami w.f 605to9u platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases to 0/dz.w 6a 7c95oltfj i20u mwfi.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the onnv-0 pzz0bil yhu.-*-/n fkhv. j6aaoe shown in Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the st .n*ukhab a.-o-fi/z-j 60 hylzv0npvraight 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 from an initial rate of fdxv4 08)zi4r2zdjgx 1.0 rev every 2.0 s to a 8fg24rd j4xzvz d 0x)ifinal 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 ,-gic re qce 0p.4pd.sa vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approxgc.-sdr ei.pp4 eq0c, imately 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 spinf,lv kcl0 (7tfning 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 thdrrdgjm -78pymy 6ux0p b5 -w6e 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,up/29pocxn1x gyxm 8 r q1-zo identical disk rotating at angular speed 1qzuc2pxg rmo9xo8p 1,ny/x -$\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.1ap 68l xr- mb5cqajpp0azp714 $rev/s$ 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 olo cx z80t*wvz(v 7d8of a rotating merry-go-round platform of radius 3.0 m and moment of inert78wt (zc* 8vlozdx 0voia 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 merryy1wq sx+vgux)c+uwgx+ ;y8. u-go-round is rotating freely with an angular velocity of 0c+uqx+.x) v ywuy;u+x1wg8gs .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 dwa3haenl8xa6 rdu- r**arf, 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 e8a-h al *63*drnxuar?(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*l 92og8fc io4sv pu-g* iqbn- 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 wb jl/a6i+j;d$ 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, init,ylb6v1u;q c pv.pausu 3 z3/cially at rest, that can rotate freely without friction. The moment lp1uq;acyup 3 v6 .3bzsc,u/vof 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-round iizma)h /7v2y turntable that is mounzam7yhv2/i i)ted on frictionless 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 tsn1o9btajc3 c/on( m (he ground with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slippin(a n o9sm1bntco/(cj3g. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such 74k4x7reue mlne,cf4q ak0m 1that 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 momkre,l 7mf44 meceq4n ka0u71xentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest ip+ 4d7.q ntlnwau;n/at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of radius 0.20 mjwth qg 9v27voapf7( /rzmy) / 3:jacm acquires a rotjra m mqyh/77t: zawjv9f2ovpcg3/()ational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical diske49m o(e;ttd 5cz/wkl s, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, ifk ezoed94w/; 5(m lttc 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 wheel a gyq/ momt qc49h11yezlx:qq, i+88j($\omega_R$) 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, were /t1d 6d3sv z 8ep9zgfsrotating 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 ws s9tv z6ddzf 8g3/1peould 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 automobil(mj8k7j8b *a,v7ve fcjzll :ee is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to tr b,8v7cefl*ma( vl8jz j7:ejkavel 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 illustratezcupp dds75c6c(7 , oos 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 rollimqi. t +)3z 9 znutjlxz)3xpk2ng on a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot yp+lbzo5pijn*b 0z 71 freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: Se7 jb5 z*ippb0y1nlzo +e Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on ts+ c:ct(l h/dlhe floor, and we want to exert a horizontal force F at its axle so that it will ( s +c/llhtd:cclimb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m.* bn5 k*ouf /qy-bxbe/s on a flat road is making a turn with a radius The forces acting on the cyclist and cycle-.5*xb bb/u yqn ok*fe 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 l naq s46youo3sbpnml6a 0+t, +ength 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rsuq 6oltayp+ma n+0s6, 4n ob3est 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 w/) at dez;jrr 7-nc;ias a solid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of th;w-d ri/j)rn c7zae ;te 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 plates,ynp, 1x8,jxa d/qy/*qp lo(mu of oym/pq,dx/ jn *8lyup,qx1 a(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 roz73jew o52 xljlls 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 thee jx3j2zw7 o5l loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do6 ffzewt.mq oo:7o9,h not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the ca sh3 pd3g 4twltk)x)j)r reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What wjt hx)kgd3) )3p 4tlswas 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