A bicycle odometer (which measures distance traveled) is attached near the w voq/y xfq ,2m-bj8kb,heel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-incym8ko-,vqf2/j, x bqb h wheels?

Suppose a disk rotates at constant angular velocity. Does a point oh7.5lwjhw;o,xjbwo 9l. s: opn 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 eithhol:jb;j5sl hxpwo w.o. 7 9w,er component of linear acceleration change?

Could a nonrigid body be described by a single value ofp*ok,w;,mq f nmv1qr. m.px4h the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a laru9.t).l pxiqsm4b nk )ger 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 toe nl rzi265:d6mbplo / do a sit-up with your hands behind your head than when your arms are stretched out in /pn5eb:oz l r66i l2dmfront of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel ae )lssp.8f q. lx:x/hfw.i2 qand three at the pedal cranks. In whpxf).eq.w 8il f.2qa :hl sxs/ich gear is it harder to pedal, a small rear 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 slender lower legs with fleux7z03s zewp ;x;yy(pa yp )xm(j1m( rsh and muscle con3 pzax m xy(()r; sxyeuw07(;mzpj1 ypcentrated 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? ggucaf3+cuh7; 4;ug i

If the net force on a system is zero, is the nloik;l )zzhysxi ( q9q) 3y(,(lf/kgtet torque also zero? If the net torque on a systemo/y3;g)h i lq(,kf)9xqk szty(zi(ll is zero, is the net force zero?

Two inclines have the same height but make diffe5ee x6g+ cwib;rent angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at thxiew;eg6+c b5e bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an inclineiwp -7zjp5 9ap. 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 g-j7 5pp9zwi apreater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and-+4a:auj l pov the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater to:au4jpl +o- vatal kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentus (rrz6 t42xm37a3wwo 6ftguy m are conserved. Yet most moving or rotating objects eventually slow down an6 (t7twy2xa r wo4sgum6zfr33 d stop. Explain.

If there were a great migration oqft6n6xsl;3 b *:rzvbf people toward the Earth’s equator, how would this affect the lengr*sn:b qzfbtx 63v ;6lth of the day?

Can the diver of Fig. 8–291j1vl 3; oxedh do a somersault without having any initial rotation when she leaves the board? v;1 hl31exdoj

The moment of inertia of6.zg p *j/n4b2urtvno a rotating solid disk about an axis through its center of mass is t4/6pv2ug* nr nzj.ob$\frac{1}{2}WR^2$ (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 holdiqve+mmml8no/a3p ix8q xe.) q8y; jc,ng a 2-kg mass in each outstretched hand. If you suddenly drop the mas qi8 en)x;mlpq3me/+, a 88omyqcv x.jses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, onxi c,a*e h;;ub ,hwmy86ui0awe is hollow and the other is solid. Describe an expu i08,ywab;;ehmchaxiu,*6w eriment to determine which is which.

The angular velocity of a wheel rowls ei53b0wop20*fz:o 3tdm gtating on a horizontal axle points west. In what direction is the linear velocity of a point on the top of tzofbw030 :d3o*5wg ip2mtle she wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large fjz(erfdq5 0m1se 42uureely rotating turntable. What hape(zue qmudf4 025 s1jrpens if you walk toward the center?

A shortstop may leap into the air to catch2:c p vun2h:x)njm6o x a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hnv:pxhc n)26j 2:xom uips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of a *xb*i 9lxpyo.ngular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one xlo*.* xpb9i yor more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in ra(x0;a j rbf+lf7ifb8q dians: (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 na nlco:51vz7 because of an amazing coincidence. Calculate, using the information inside the Fronoanz1:7vc5 nlt Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380s-5q2-m( r02t.mcx*vax rx ryfyvy p(,000 km from Earth. The beam diverges at an anglv2xcr5svy-2rxa(m yry *p ( .mxqt-f 0e $\theta$ (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 ratel ; y :yyc87vo-zb2fxw)p0gcv 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 boy 8 l0zf-27c gyv;pc: vwy)xas the blades slow down?    $rad/s^2$

  A child rolls a ball oa:v -seu j+0ov;k( elsn a level floor 3.5 m to another child. If the ball makes 15.0 revolutions, what is its diajl ukva0ov+e: e;-s(smeter?    m

  A bicycle with tires 68 cm in diameter travels 8.pmgr ubv ;p:1-+fs/ vdv)fwnm0:,eiy0 km. How many revolutions do the wheels s::0eb /pd nvvf-ig,) yfvw um+r;p1m make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate ic 2lhtls (1+ijtltz)oy 4a* r-ts angular veloc(2h rct +y-*a)t1lszjilt4 l oity 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 make .((fwpt)j) wdqx.kui s one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1u(q(jiw)kf d)x .tw.p.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 ( m, yo/5jzvgs dgj0w:0a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear su4vyxj*aojc(w,8 ky n p4 *g(npeed 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 rotate if a y51judi+ bshxhk (48yparticle 7.0 cm from the axis ofkh5yy(xb i4h 1sju8d+ rotation is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its centh:*0 bqnisjkm/. u+n2m*(q vlb k *djxbd.7gver from 130 rpm to 280 rpm in 4.0 s. Determin. ljxb0b* n.vk/ v *+:bmkud7iqgsqjn2m*d(he
(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 radiue0ikju56 j 0gvx.i0fqamu t:.s $R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, afc -2ywz fvdn hb3o**+:,yo9aspk 7j lstronauts aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of th avchsky7l 9fz, p+*jw3 o*d: 2o-fybneir 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. Throvtf7 ni1) ipq,a*7 lecugh how many revolutinp1 ev7l *7)a,iqfictons did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpeo9vbrzyw -,1yf :v- ro1 ci1jm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highsptg3jt-( ; vtx*f)agwq eed jets in a whirling “human centrifuge,” which takes 1.0 min to turn thxjtt fgw;q 3g-(a)vt* rough 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformlyltg:1i nj*tu0 from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the tt0gn1 ji ul:*wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 1e*yy/oc 7d9x w 3hq:pqc8l *5uw turhf1be11n500 revolutions before it comes 1q1 whq uor:/3cw7 89ulbef5ye ht**nydc x1pto 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 speed uniformly pgpbii+v). q0from 95km/h to 45km/h The tires have a dii)gp0q vbp.+i ameter 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 6:uwd9(+ ia8 + wcefsea5 revolutions as the car reduces its speed uniformly from eu (+widaf ws9cea+8: 95km/h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight o( r28ncx ./ i3)0m(paynjy qx) yhaxmb)w(uex n each pedal when climbing a hill. The pedals rotate in a circle of ra3 .(rxxni0p)m8/ye 2njqh b(y)) u(xxcma ywa dius 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 mag;umg,yr(e6q5hc0gf1 v ;c ck tnitude of the56c0 yh;q rv;e gcfmukct,g(1 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 s8 p8r1ts)x(lbplav)vw * zg+ry-h glg8ot :r:hown in Fig. 8–39. Assume that a fricti:1gt*lrla- zl8) 8(+rxgrvv:)8s b gtopyw hp on 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 rod whic(-)+p xqm*ycxer2 ebyh pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate tx+)ep yc2 e( -qm*rxybhe magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening to a to.pbcck t8 xrwa 7w2+*qrque of 38 w7.xpq28rc c*w+ ba tk$m \cdot N$ 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 sphe :odi m03vep 15a9xisure of radius 0.648 m when the axis of rotao9i5:dp1xam 03vs eu ition is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycmzjb: 5u7- rivv8 tdfxszb*,42tt gx*le wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignored (wx fb*x,m -i:szbz5t 4 ur7vj*8d2 vtgthy?).    $kg \cdot m^2$

  A small 650-gram ball o6kqc 7dovi kg :r,zna qs6)t.3n the end of a thin, light rod is rotated in a horizontal circle of racdo:z6)n7siq 6rk.q3tv akg,dius 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 rotatini0 tl;2v bsj au+x/f6ug at constant angular speed (Fig. 8–42). The fr 6x l uiavu;/+bjstf20iction 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 the array of point i ;+)6tgv+vlhg9d.jyaw*x r dobjects shown in Fig. 8–43vwt; 9.d+*gjgrla)ixyhvd +6 about
(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 th6*dhx1k( * uu g6xaviwo 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 at 0c fl0lps*7oq the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg 0fl*po qcl 07sand 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 with a radius of 8.50 cm andsuf1i,5; vxmmi3q/p +s 8hvgr a mass of 0.580 kg. iq3/v1fmmx+s5p; u g sh8ir,vCalculate
(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 from +8tf f;mnl3 mu*jqh d-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-rog2l.gz6ii u mk+;1phjund and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglzu62 i jlp+;kghgm.i1 ecting 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 ,9glq,yt5)p(a zxaij and is eventually brought uniformly to rest by a frictioj)q,5 zy(paitgla 9 ,xnal 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 acceleratesns,x+ahmz5:pd6ip p1bcc /j; 1rj*je 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 -e*h qfl8vy ;zby the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45h y8e*fzl-; vq. 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 k y2-r mpa2oiwtuz hl4782+k:qwb i7f in Fig. 8–46. wp-78aykt fh2qrb+4l :ok wi7 u z2mi2
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine consists of two masse8o2x/u8 2ia+vrspu yjs, $m_1$ 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 rest within four full turfz ezj.e /i96p wx 1bbaz17u-nk6mlg(ns (revolutions) and releuwjzzmkz.lxen 6f-pgb97 b61 a 1(i/eases 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-475j 7ecbh nxuzhv5b a moment 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/j..pxc v we6q torque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls without slipping d(xdrd1 m. awn*jt43nxfxs4 l(own a lane w x4ax(nm( *d1j trln x3fd.4sat 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the glu*0o-evg- 3z/xidt4s3bm; ,nn m inSun as the sum of two tz*dn;stnv b g, 3n/xi3l gm4 -imue-0oerms,
(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 me :r02sg -gzotv6+5- qdhqhha*gf /kcass of 1640 kg and a radius of 7.50 m. How much net work is r 62q+-dq trc ez-kh* /ag0:gghvo5h fsequired to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and massyfclsir81lr144h)bwh kk4r ( 1.80 kg starts from rest and rolls without slipping down a 30.0 fi4y( l4 hrrhw1 b)slrkc148k$^{\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 facu-g 0duenky9 x+s r*;ll and its lower end does not slip. Whnr +g9c dus0-u;ek* xyat will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the ngkde7s78 5ovsm/r5we;z l+aend of a thin stk/5vs87+m7rs nw eolea d5g;zring 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 momvx rt*zdbjyk)gf 5j)4g6 lb)6 entum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm jf4rtjk) d6g bv6b*l5gx)z y )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 rotz6ego xo a1l6l8we(t.ating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreaseo l t61a68w glzo(x.ees 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) c, iitgf:frxs:/6gb -ban reduce her moment of inertia by a factor of about 3.stf6bg:bi,fgx/ -: ir 5 when changing 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 from an in9ar,kr8o0z ,r)i z: u(hkce paf ;tyi;itial rate of 1.0 rev upry ik8thf,, ; (oka ;):ae0zr c9rzievery 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 )oo1 t *hevt;m:q ao-ffrequency 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 o-oottmfv o1 *q)h;: eaf 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 fiwg-2 fu 0u*-bg,l (3eiqmp hatgure 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 moment6vgzciy+ .r9 ohgnc(5l 2bf-bum of 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 momf .a(0dvkl rl/ent of inertia I is dropped onto an identical disk rotatin vaf(lk 0l/d.rg at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4 q4j96r eqr1/nuk 8b xp$rev/s$ 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-xw,.2 *wo6j34ak-tk ay n*dhcqg7c yago-round platform h6w3c7*aaw o2n*y , -x4qdj .tk ckygaof radius 3.0 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-z 10 k t,tj;o/pfmzio+qia( .x2bn1uigo-round is rotating freely with an angular velocity of 00tx+qi/ jp koz,o 1ai. t1(zi2fnu;bm.8 $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, losing about half itu0 9qzb6zj oe3j8(pbms mass in the process, and windi 0u96qz( mpjzobbj3e8 ng up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in excessb5c5a4i +rchc 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 7c ok1jlh yw* 5+*kk q yn4e8;.pvvdls$ 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 rotatdzc gv:ait3v )a,:n6 ee freely without friction. The moment of inertia of the person plus the platfon,vz3e:ta d c)aig6v :rm 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-j mc5hru 1+-xigo-round turntable that is mounted on frictionless bearings and has a moment of inertia of 15cj-i mu+r xh1700 $kg \cdot m^2$ 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 ro rs-oba2 qkl:)pe 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 slipping. What length os: )qo blkr2a-f rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same y e2m) .m n,7sfa.mazsside always faces the Earth. Determine the ratio of the Moon’s spin angul)zem na 7m.f.as2s,ymar 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 rest at a rate of 1(n .w x.gl ,el6 2k7pjpwq-urz 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 sha5pj2 j4*pt k/f3cv aywpe of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest*fyvj2w43 t/pajp5c k 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 disks, each of mass ( gm52lkoo04 -btjgzh0.050 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 cog o0o4mg(2hz - t5kjblnservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of t7a sgyjo o/6or0gihnw :s ;83rpeol5;he 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 8.0 ticwfcz n, de;4z+f9 0xpmes as great, were rotating at a;ef 0,wz9ccfpd n+x4z speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollu5ngc+oo obd:o8a uu5) tion automobile 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+gno c85oodu b)a5ou: 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 illustranpc yx0v /9*qyvsayj2/ t,f* qtes 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 (hoog) yx9pk ac7lho-m -3;3qiidzp) is rolling 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 d)bk :,tb5zl *fgdj u,o;ezb*M and length L can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as inuz)5lbb ,ze; kjgodf t ,bd*:* 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: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on the fl gazet0g3j+-wb3tao/ oor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The swt3z g-t/abo0jg+ e3a tep has height h, where h

  A bicyclist traveling wdcmg b,n 2gt 4iv-oi-:ej*qc h/46e-wsxxv.iith speed v=4.2m/s on a flat road is making a turn with a rai:2nv-idqsxb4c-je vwech/4-oi. *g6gt,xm dius The forces acting on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock if3l6jq ++ghpjnto 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 o3 +fq jhl6jpg+f 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 u fv,p n07:igirods, making reasonable estimates for 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 vi:fpngu 07i,.   

  You are designing a clutch+upr7/ i 8wmoj assembly which consists of two cylindrical plates, of masui+ jrw7/8pom s $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the looped rough track m0it9.pj oq8/z ,ofu7l 7efudw;mm1 nof Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highetozf7m0;9fl jwniq o muue7, /p.d 81mst point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do 4h pn1jx r19/x oac1iynot assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed 7 3yz rb,kox xx8*a)snuniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleratiox* k yx7)8an3rzxos,bn 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