A bicycle odometer (whi0 7gy voev1(-pq s4ih-,fy6zndp /me ach measures distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicy/gd yi f em( -4nqevozsy-,71vhpap6 0cle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point oxr8h ib,x+ ,yrn the rim have radial and/or tangentia,i 8x r,rhyx+bl 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 val5j.yj6fy -+qqp m-jjtue of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Ex6b*7 u6-3hjukf w ei,qzkm ti(f+x-pbplain.

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 behksa /emz5h; qrv4+n c1ind your head than when your arms are stretched out in front 4e/ 1sv;rkh+n5zqcma of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear whana; ad(lhl 0.eel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedana.0l (;ldhaal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being ablqw6/yy1uqfyg 2wvt6ot .jy1; e to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this dis /uqy1y ty1yw.q fw ;gj6ov2t6tribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carg+ z.wbi: u akk .,yt7kc 40fdzuc-t/mry a long, narrow beam?

If the net force on a system iv-e3 bs x iz;u/ //)epupa)dnis zero, is the net torque also zero? If the net torque on a system is zero, is the net fa iez3nud/v- sux;e p/ip)) b/orce zero?

Two inclines have the same height but make different angles with the horizown /28(el ;1mfa nm5ckj1bj+ ,qroj guntal. T(amjg, /1ojurk wljqm2b8e+5 c ;nf1 nhe same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling .feh 7 nfw/it*(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 total kinenf */htefi. w7tic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. Th l+w6csv drox4zf8, .vey 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 enrv 46zw+vf.l d8sc, xoergy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet mom-*vs/)ju aut szmcuv*1h(0q st moving or rotating objects eventually slow down *vucjt s/uha(z0 -1sm*)vumqand stop. Explain.

If there were a great migration of peov e(epuv+h8( l .m0tbqple toward the Earth’s equator, how would this affect the length of the dale(8ve(.ut q+bmhv0 py?

Can the diver of Fig. 8–29 do pze(-u6v 2xu ga somersault without having any initial rotation when she leaves the boa6 (upx- zguv2erd?

The moment of inertia of a rotating solid disk about an axis thro-v6elw4 ox 8lh,c+klv rgw 7h*ugh its center of mass isw84+lgwl xoevhr7*k6 vhl , -c $\frac{1}{2}WR^2$ (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 eachwrtq: yh0)y g6 outstretched hand. If you suddenly drop t6ry )h 0:ytgqwhe masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is hollow a t s7,b*+ poy3 okvhkrukis5 1;ny+2xrnd the other is solid. Describy1x ,s*;vkktisor u pr5+y n k72boh+3e an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle p;mr+ t1z.bjgv oints 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;tb zmj. r1gv+ 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 freep76) ,r tyaznfly rotating turntable. What happens if you walk townfy 67,)rt pazard the center?

A shortstop may leap into the air to catch a ball and4tta84mfhgn/ tx9a jc 2ngi38 throw it quickly. As he 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 opp8c38xm4tgnh4 n agi9tf t/2ajosite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discuss why a helic( 1x(yh2bc0d )/wkzskrguhu+ bf7 kl *opter must have more than one rotor (or propeller). Discuss one or more ways the second propeller caukhs h+7gx(zf)1kk*(/dc u 0ry lwb2bn operate to keep the helicopter stable.

  Express the following angle u/*;cp*yy,ac( kbngg s 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 amhi(j3dd4go 0t/ v 8r+x(icdpme r :)orazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (i )porrrjgho+4((im ci3:x dtdd/0e v8n 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 (9)fv6gnyyauavtz/3w6q d 3hfrom Earth. The beam diverges atwy)tgn3ya93fdv6 v( /zhqau6 an angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. When v2u*3 tv mkt5athe motor is turned off during operation, 2* kavt5u3mtvthe 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 leve1qyvwforrlzl;: :*p- jp/g z slb4e07 l floor 3.5 m to another child. If the ball makes 15.0 revolutions, what is its digjplol pyzf:1/rl: wrb0; z74qs-v* eameter?    m

  A bicycle with tires 68 cm in diameter s ciqx6u+zxtjt 6c.p-9n 6ek7travels 8.0 km. How many revolutions do t6p j6c6z.xiek - t q+sxnc97uthe wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 250 prb9;yq4ub(i0 rpm. Calculate its angular ve9;br uyb i4q(plocity 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 4.0 s (Fig. 8–378lgxhp7 zj .f8). (a) W8g jzhpl .7fx7hat is the linear speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a:um uzwcfl ih;kn-h/b1(:9 gr) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear spel- u cvn/8yo-pk *qa8qt-x .mbed 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 centrif/*(esglpbp b. uge rotate if a particle 7.0 cm from the axis of rotation is to experience an acceleration of 100pe lbbs /.(*gp,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accenlvfww6h20 ; glerates uniformly about its center from 130 rpm to 280 rpm in 4.0 s. Determih6 gn0;lwfwv 2ne
(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 radiu, ml0tt1kb b4ns $R_1$ 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, astr- wzwev ;c;)zronauts aboard the Apollo spacecraft put themselves into a slow)zcww- ;zevr; rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpm in 220 s. Throu.vftpbjn2cx , 0 hu9u*gh how many revolutions did it turn20 . xpfhvju t,9nbcu* in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calcxxsj,3 ;fj xj6)zm0km ulate
(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 re4 krjnnti( 2 a*q3wa*u3n5nacb(+uin a whirling “human centri3b542 qntr(( iakaue3na n*cjn ru+*wfuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diaz h,c)/pnr nv1meter accelerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel havnhvcn/r) 1z, pe traveled in this time?    m

  A cooling fan is turned off when it iso2g za-o 1b( vexcj q3z*pb4)g running at 850rev/min It turns 1500 revolutions before it comes to o3 b4ez2g* bgp v)cjoxaq- z(1a 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 speed unifo.tz g qi6q9yta/op 6b*oj2htw1x9k3;tb rms5rmly from tq3 9m*rx6bk bi2691zot/jwpghq. ;tst 5aoy95km/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

  The tires of a car make 65 revolutions as the car reduces its speed uniformly al9b : f ce xisdfy 58*s8u3r.blzkn*)from 95km/h to 45km/h The tires have a uysdf ) lbl8zr f*8knx 3s.*ibe59:ca 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 ,d,rv2 nr c3tdher weight on each pedal when climbing a hill. The pedals rotate in a circle of radiutc3,rrv n2,d ds 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 thegzn/h 2gq6 sm7zc0d 6a end of a door 74 cm wide. What is the magnzgza60 sc7nhmd6/g2 q itude 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 axle of the wheel shown in Fig. 8–39. Ab(q5i n jsybb b64(o3yssume that a frictioby j (5(6sb y34oqbibnn 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 massless r qazq- 7(ima +y:xqybrd; zv).od which pivots as shown in Fig. 8–40. Initially the rod isi -:+z.(x)ya7raqv z d;q qbmy held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tig0xta2thop7y (wj )i 2ghtening to a torque of 38 2 j 7 xthiwgtop2(0)ya$m \cdot N$ 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 spherjx*4eyc)c7wg e of radius 0.648 m when the axis of rotation is through ixjc ewgy)*4 7cts center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm wn(v 30aw866 4oblqha:ni ,7phq pf wcin diameter. The rim and tire have a combined mass of 1.25 kg. The mass of thna, 4:b 8wq3q 0(nwph6cp7ovfl6wiahe hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on th33 p*(z+xtj77ge8 jejak7i ca wmyhd/ e end of a thin, light rod is rotated in a horizontal cir yjz8+ x ph j37dcg ai(k/ew7t3me*aj7cle 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 jg648r e4eu vykl*i.hwheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and the6jhv8 .4k*gy e lur4ie clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the iymk6) tcicwqt(79:jms *d r- array of point objects shown in Fig. 8–43 absdj*)-mi ctm(c kq w:i67rt9yout
(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 conf ijz3p-1 ehgxi6inef*y6cw8/, rcf5 sists of two oxygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning 6(zt7 dth 1dpt2t)sqk 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, wt)hqztd p6(k d1 ts27that 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 with a radius of 8nzj xt9+tv --rh*)d mj.50 cm and a mass of 0.580 kg.x-d+t9hjztjr mv* - n) Calculate
(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, acce*y.sr 9jz j.1x76u)mrdq5q7 nv sldsdlerating 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 smas5t;ugtvn01ld )gsb6w4 mpd7n xq87ro 1ym)s ll hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kq 1)x8ttm7 d6;pbsng)orvn4750 m uyg sl1sdw g, 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 rotating at 10,300 rpm is shut off and is e (1(shuc5e/yazddf(dac8j( oventually brought uniforczs 5ao(ehd/ fy1u c( d(jda8(mly to rest by a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a 3.6-kg ball :1hx m4hywj hj8s) vrhh5fc;-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 k9*s ohreg 6f/en)fr 0ball is thrown solely by the action of the forearm, which / r9r ) esgf*ofehk0n6rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rodfq :vg3s +byv5, as shown in Fig. 8–46. vs+qg3fy5b:v
(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 3fm2lw:x. qdl 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 accela2h ceka)8r5keq-sc*e0j je/erates the hammer from rest within four full turns (revolutions) and jqser/2a8 kjhc)0c 5eke-e *areleases 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 iner)omm1 gm sqvq)(*( a mjp1y)u i2dahd5tia 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 torq30y y .c)q/;t3opp ww9kcpk si(w pcnr+6fc z1ue 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 slip;( )rljn5c7yf) adw qaping down a lan7a( )ldanfw cy)q r5j;e at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Eart 1hd1x gz kncc+g1)gs1h with respect to the Sun as the sum of two ter nc)kx1 ggsgc+1hz11dms,
(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 and a radius o4,o6my/a zz+jpxn 9eq f 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8p+q, 9/y4n6xzjmao ze.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm h.,t -ayeu vmo,k:q; hand mass 1.80 kg starts from rest and rolls without slipping down h ohk;.v t-u,eaqy:m,a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

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

  A 2.30-m-long pole is balanced vertically on its tip. It starts to fall andr5y xc8well mum*.)3 x i+1wex its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation ocm3y58 wxe 1wx)rl.eil*um+x f energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end of a thin -;r twwjb6c2y strincby j;rwwt- 26g in 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 mom h-:5vl jj1(onafx55xutx5i(glr b e.entum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 gxe5 (b1:5 .hnajvtx5jxf o-rl5 (l uicm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotating at a rateh u l++gagfn /m-hu c4pw i0:zn.a;f8a of 1.3rev/s If he raises his arms to a horizon/;4piu8c gmzhg+-+ 0wa:fa .nhf unlatal 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 our:mcb-o5/ ws 4fki4x ne shown in Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.f -rmb/ciwk:xsuo4 545 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 initiacl wb5;;0+dnjenf1dmf )kizek*n0 2hl rate of 1.0 renjl d im1*2cenebnhk+ k00fz;;) 5 dfwv 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 th idqx,rin)1v2 va+lhny; ng:-rough 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, approximately shaped as a flat disk of radius 8.0 cm, onto the center ofrlh; dn:2inv,ag-xqn1 iyv+) the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentumx0i 6wdhcg,z-lv :)i 9tjf2u n of 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 o 8jst:ce,bj8/ vsjy1of 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 cylindri7i kvduq9 *vd2ml8hm l 0qf,e.cal disk of moment of inertia I is dropped onto an identical disk rotating at angula l lm.i,*hkfmv9du qv07 q28edr speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns atp,+9gr c yhgp;4 1yf0z v7ykkv 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 /1eah(zde,.m;n /v /r iixmcd stands at the center of a rotating merry-go-round platform of radius 3.0 m and m;z ha ex m.ie//vm (,icd1nr/doment 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 with an angular velocity ofb*9 oe pze/eh: 0.8oep : zb9e*h/e $rad/s$ 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 eventualfe 56dg 7vv6xes8m6ut ly collapses into a white dwarf, losing about half its mass in the process, and winding up wiemu5 vvg6 txds6ef678 th 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 win2 73pv.kevx0uqu w99bpv*tb nds in excess of 120 $km/h$ at the outer edge. Make a crude estimate of
(a) the energy,    $ \times10^{16 }$ J
(b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 $kg \cdot m^2$) of radius 100 km and height 4.0 km.    $ \times10^{20 }$ $kg \cdot m^2$

  An asteroid of mass 7ig2y0 jzvqw: $ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, that can rotate freely wp +u)wi8+9dtzne0may ithout friction. The mome+ pzey9d utm) a+0ni8wnt of 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-47(gfu*0 c,m.v i-zjfmss qo lgo-round turntable that is moun* mm , io q7ujz-fsv.(0fsc4glted 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 the ground with the endez0,s k4qo 67v,pzzjtsf5j8f of the rope lying on the top edge of the spool. A person grabs the end of the rope anos,jkpz f5ft sj4 ze0q87vz6,d walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always jmldcqt ((a9: faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about itt9ql(djmac (: s 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 fro kw*t b cd91zcst8 t:w:1ml)olm rest at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of radius 0..-5rif xt r7qu20 m acquires a rotational rate of from restu7 .ti5r rfx-q 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 cyli1 mt1;u84jc(;gmsgr ww pfgs2ndrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of t8 cgp(jusmg;t21sgm;f1w4 rw he 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 a9bnqp: -mgx/c2*vxm 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 wi ,/gsz 1i7o :eosfybr/th mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a 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 carrieso/ifrg, zy:s o71s eb/ off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use energ3qt30c2 amzn(8 5qkmjl)fd g ty 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 shouc2dm5) am8q 03 zgjfntltq3(kld 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+ba5 fsqj+ov9g,hs d-czs0 jr+z9a 5x 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 horiz/s8 ldwscj:8e4mp o0oontal 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 lengtb:6dyr: nxjv,fd62 zth L can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular accelerationyx6b dn,d:fvj:z6 t2r 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 standingl5ccifovz 0th,8rdw+f9f u;udwgyf.l;p;1/ vertically on the floor, and we want to exert a horiz; w+ o;0fg iwf.uvdl8f y1 z/9plr5cd,hctu; fontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with spehrupijs n jj-8 j9;4z wei5v)yw;2-qded v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist and cycle are the normal fori5dp); sjhn8wr4yi2 -juvz j- qe; wj9ce $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ 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 lengd -e/ uijty4+ 9ue2pkdth 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, acceleratin9+iteeupd4 ukd y2j/-g it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thin rods 6ux.vo th8 6m1-jrzoucv b-+j -su5nh, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds vjovs 6zur+-j18h tuc.6mon5-ub-hx for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assem dx(t;uskl8 jceom/a(7o-ng5bly which consists of two cylindrical plates, of massanc-d(oe;mjts(l/k7u 5 g o x8 $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the looped rougfg;o2af4,w8rbu j 3rwg h-v2 hgr80zr h 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 porbfg 4;hgvg2 88uaf0j2wo-zh,r 3wr rint of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, ba2 xv9*, ppag18dm. heu(svnrep9pu,x3qmf/ ut do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uniformlyi+(tbm*:m2 ufdswnm ( from 90km/h to 60km/h The tires have a diameter of w*(mb( tn+mf si2: mdu0.90 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