A bicycle odometer (which measures dis)e1 /f6svvbs: ox7 aqw*c2 y1l7xmdcytance traveled) is attached near the wheel hub and is designed for 27-inch wheels. Whx / 1qwc)y:*svx bf ev67d2yml7acs 1oat happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity+l7 f,nmsm6 ukvhuln-g+n qw;w y :9n4. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magn;u:+nvhlwu-f6wmnm+9n ,l y 47g snqkitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the angular velocd;z/w 9gjq:yz 00hqneity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger forceqb+fs+,1evmv tz86 cs ? 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 youf*zqe73 61vkrmstpul-w0 vb3r hands behind your head than when your arms are stretchedv01rkbv-mwtf psl3u 6 q*ze73 out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the pedal03iasb 4fm2vu 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 pedal, a small front sprocket or a large front sprocket? 4vua 3msi20fb Why?

Mammals that depend +xfcv1z0g /u84 t*bm6-qd lo7kvfteo on being able 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 d-t 6 bu7otvlxmz4f08 v*f ok1dcge /+qynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a lgwraw:0 cb8r*ong, narrow beam?

If the net force on a system is zero, is the net torque als*w7),ya jwnsxe j( i62vii6cvo zero? If the net torque on a system is zeij( yxwe6cijv,v)s*nw ia672ro, is the net force zero?

Two inclines have the same height but make32)f -3g k0kp lszf+d spjx:wc different angles with the horizontal. The same steel ball is rol3c f)wj-0 zlkfp3x:s2 gdsk +pled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneo9z j2scb,ys n+usly start rolling (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 th yzc+92ssjbn, ere? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the sakm7q3m j 0gw:ume radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at tm0 :m3jkq 7gwuhe bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved.o /mjkv;(*a3aw /hsmt Yet most moving or rotating objects eventually slow down an3 */ at;(a/j smvmohkwd stop. Explain.

If there were a great migration of people toward the Earth’s4a km nbf1kcj(k60d/c4eh (gz equator, how would this affecd(4nckbk6j gkeh(0zfc/ma4 1 t the length of the day?

Can the diver of Fig. 8–29 do a somersault without having any initixmi u4dj4-ynjt-)0 aual rotation when she leaves tuai4y d-)ux4- tj mj0nhe board?

The moment of inertia of 7rnwt-3fai m )a rotating solid disk about an axis through its center of masir7 mn)wt a3-fs is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2-kgw2.ixoa,cw,qyw t 7e+lxw4p, mass in each outstretched hand. If you suddenly drop the masses, will your angular velocityww+q.xewol i t 4, 7yxp,w2,ac increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, o.3lnv lqtfmz*wc5 + a.3(j ncnne is hollow and the other is solid. Describe an experiment to determine which is w(z * j cn.q.wn tf+m3vnla53lchich.

The angular velocity of a wheel rotating on a xqm5 zyc:u2+v 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 line 5yv:mc2q+x zuar acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standin1rd uez-0 jb)mg on the edge of a large freely rotating turntable. What happens if you walk t1 buz-e mjr)0doward the center?

A shortstop may leap into the air to catch a ball andgrm( i1ga8 vtq6 ar6;gd:b /cm throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quicklyg 8a r:mq r(dacbim;tg61 g6/v you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of rq.fafzyq1 w 474zt (s :9y,mzidmh/u angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the he4zu4,it97:zwfzys rd mqahyqm. / 1f (licopter stable.

  Express the following ang -a xaev, 6mllo*vb6in:u1fq;les 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 bece )c ze u4+.hlb(iw+ugause of an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (i g euu)cle4(z+i.+w bhn radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The bewbg ec.7evdo5jud3fe25o29h am diverges at an angle vh 9dcw5.o u3522 feogb 7ejde$\theta$ (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 zd b8oz3 da1jwii0)yl -fb.+o 6500 rpm. When the motor is turned off during operyo -d03 i8 aw.ijbo1f zz+d)blation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another chil7ea w5mkm3 j0t d)be6td. If the ball makes 15.0 revolutions, what is its di)a be05 k63tjwe mdt7mameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. Hojz * lk+-bjsrg2yj.f3 w many revolutions do thz.r*sl- b+2 yjjf k3gje wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at uph0 x q5j e::g-qxka:2500 rpm. Calculate its angular velocity i-xu::a k h0gp:jex qq5n $rad/s$ $\omega$ =    $rad/sec$
(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. 8i+/0b(hwbh 3xr oe .x0s ,llsz–38). (a) What is the lineablisrs b (.x+ 0ozxh0lw ,/h3er 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) in its orbit around t+mo uio fyrs-2p +1ay0he Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of ae(r-:4n+n bd+z yaooq qy0fd7 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 k7xc;ihc2g+ 0jpxa (na centrifuge rotate if a particle 7.0 cm from the axis of rotation ish;+n g(jxa 7c20kpixc to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel acceli.d l53yg gki-z 5 jjv-su 9aw.jj6+ljerates uniformly about its center from 130 rpm to 280 rpm inkju+wj asz-9 -5gvj.3il jjy5ig6 ld. 4.0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radiu l/(fhd hb sg*q;f3a5j8mbx 2s j7cw8cs $R_1$ 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 spa-tkn:v) d1sy0q9d ytmcecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thoughkt:01)d9mvn y s-dtyqt 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 ur : (babd(a3zmniformly from rest to 15,000 rpm in 220 s. Through how many revolutions d(ada(b br3 mz:id it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Caxf x2z, 1x/fjzlculate
(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 highspeelyoc4 h5om9 0wb-8bl zd jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete rwyl0 9458hoz-blbc omevolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 360 rpm in 6.5vy) 64uv2 40hju/9kp ykt9vwtr n:uat s. Howta/wvt ukyr9 t2:h09 v6nukjy p4u4)v far will a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 1ugt55qlc n5kym5smup7264 z)pmo x . h500 revolutions before itqz.p5) g725hpm m5n c utkms45yoxl6u 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 reigms/ jfb9*kt. 3iky -duces its speed uniformly from 95km/hskik*/ jf-i3y .bmg 9t 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 revol txn 92. )r8pa; p68lwfc4(tdf*clynh3p hv vgutions as the car reduces its speed uniformly from 95;c4yp3prg8t( l2 fp n) nxfv* clwtva6.dh89hkm/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 on each pedal whr.t: hr8s rajzc;lgy ;2+ak 5qen climbing a hill.;k:.+ ljr;2azscrht g85 rqya The pedals rotate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 / ydsrs6s.1-ibh xdx2 N on the end of a door 74 cm wide. What is the magnitude of the torque if thed./s x sib1 rxd6h2y-s 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 abo.hc l7(bk8qigut the axle of the wheel shown in Fig. 8–39. Assume that aih7bgkcl8q. ( friction 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 whicftqhgb +)uv8ob d5 k83h pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and directionkgod 8+ t8 hv5b3fq)ub of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening tbo q/ d97vi*fmze-(b r)5 3vueyf9xyzo a torque of 38zb * -r xm(qbuv5ed9yi3ozf/ f7 y)ve9 $m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of radiu0q1vw,9cr 8glu.cb um s 0.648 m when the axis of rotation is through its center 8c0r mgwuv lc.q9,1bu.    $kg \cdot m^2$

  Calculate the moment of inertiauvmk8z 3w52dq of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 2mz3q8v5kd uwkg. 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 rotm e 8t91bhklq9ated in a horizontal circle of radius 1.2 m. Ca8 91bq9el thmklculate
(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 atj:q )aywj +)0zx5z 6twmijn ft86eba6 constant angular speed (Fig. f6 axna) tw8z:6jwtq50j+ e6ijyz bm)8–42). The friction 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 objerrh)h84zkm-wgh7aw lr 4 xz17cts shown in Fig. 8–43 about r7hlm7za1hx-r)gk 8 4w wrzh4
(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 qzkmy oa513z(9r /dduhq5+:cr2 hjcooxygen 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 the corr6ltg3jouvp+ c- f) f/ ffn/xa1ect rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady for-afnlgju+fcfopx 63tfv)/ / 1ce 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 uyr4hrubz7 0g o+xj:b-niform cylinder with a radius of 8.50 cm and a mass of 0.580 x70r+ubro :h- g zyb4jkg. 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, accelerayr6 c 0/ b.4dfmxomce+g-xgn 5ting 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-rouj 81( fv:jq*-ad*dc/h xhevpx nd and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Asv 8 /aehd:hcxjpfx vd(*1-*qjsume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and isxq)qz6ds u3m rxm80qp;i6f1aku) ng7 eventually brought uniformly to rest ;1xskf3z)0un)6m6m piu dqaq r8q 7xgby 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) nc:l gk+,hsj a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the action of the forearm,o.j+)+ vi8 n bxp,hwb86+qw0vtwq)i rlc p3qr which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is at wiv8ox+qv h.+pq)0+6 8l ibn qcpr)w j3wb,rccelerated 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 e1loc:a1mb l3in Fig. 8–46 1c:3aelo bml1.
(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 c-(iobfvoq( xje:ox7;wp m49tonsists 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 from r0g i91y. /hz 3c)hd;nkzt wsgvest within four full turns (revolutions) and wti/9z.z dv30c)ys gkn1 h;hgreleases 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 amo.ol 434 -5l1r2l odr rorllh 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 torque of 280. o(f9tud c(dg $m \cdot N$ 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 slippq43i, w x5.l/cdw.k fy:uwbww ing down a lane/yci,fqw:5wwwwd x3bk . . l4u at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respsb*:p(fgrd/hrw4(+ g dpqix 0ect to the Sun as the sum of t4 w/grhp:(*x g prd0f b+d(siqwo terms,
(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 of 7.50 m. How much net wohmm x/5g;u .ho.pu yz3rk is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a smhuoyg/5mp uh;.. zx 3olid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rollst5+x wx ,37mp0nwsr7gi bi3ie without sb,e x3+g r5 m7ii7xw i0pstn3wlipping down a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

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

  A 2.30-m-long pole is balanced vertically on its tip. It starts to )l4wpmogwf pobh+g;1w 37lug r8kje*n/ e 4v7fall and its lower end does not slip. What will be thero1lpbwh; m+/o4gvkp*g ef l7 3w7eng4j 8u)w 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 xsm49y3ay,v/rk ; tvxrotating on the end of a thin string in a circle of radius 1.10 m at an x /a k ;vmt4sy3v9xyr,angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momenevwsym*3u d8 )u bvu0 t8i1m45loi :rjtum of a 2.8-kg uniform cylindrical grinding wheel of radi8 el i:um3u5 rs0y vvi4mt*jo1u)d wb8us 18 cm 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 a)lg0tq6i(ym vl) lkepwd1*h )t a rate of 1.3rev/s Il6l)qivg* lt)y wdm1h kp)0(ef he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce her momedh) gpf *vk8j3 xea. )(agrc2ynt of inertia by a factgagv 8kx(* .c ph 2a3rfy)dj)eor of about 3.5 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 ro;eub-ene5dwu()m w+a:)kt b etation rate from an initial rate of 1.0 rev -ekn de;umba)tw(b5w: )e +eu every 2.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis through its ce)ea,zm4fc vn5*s6ug4 o kl8z8a gv 5ccnter 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 of the rotating wheel. What is the f8cmvluc)f56 5aa ocz 84zvnks*,4e ggrequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skav 05,u)ow dgeae,u i5lbwr 5)qter 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 uk9: mp5 dofm2ad+kl(f 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 cylind.r )e)b(hot zprical disk of moment of inertia I is dropped onto an identical disk rotatinohz r)).e (tpbg 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.cnf 7-gsw-a/r h485s bqwiey04 $rev/s$ 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 merr 7 5wn: ;9ppa8 pq:fao(xkpoeqy-go-round platform of radius 3.0 m and moment 7 e p op5:np;f8xpkaa:oq( q9wof 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 v,6/ z( .ykqj4nb hnhilelocity of 0 nq/.bnl 6hyj4ki,(zh.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 itg5 g*fwju8d +ql-l +ajs mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nea8jlg lqwadj-uf*+ +g5 rest 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 or:jt oj *za7do1bkp : f)-t;cuqu57z sf 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 mg35j a k+bqvd8,a c;l$ 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 withoutxw f7, /li yds9e 75 q5ali/5wgpoq8rs frisxs ,7 yw5/i 5g5/qdlqr e7wfi8p 9aolction. The moment 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-iv3td )w *ce:8clw ud 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearingwc3 et:vil8c w)u-d *ds and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of the rope -zm(zimxz 5g*j*(kf u 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.zj-zuf*zx(i g5 (mk* m 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 Earthl3i rwh h+2pt*griddi wk34+0 such that the same side always faces the Earth. Determine the ki3+i4d htl* 30hpwgir+d2wrratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a uzng dm*s/2*trate 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 cylinderjsy 4++sdvv ;a;(+f cutro i-su+4r vz of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleoz(ut+ syr+f -rjv4;+4+vcisvu; a sdration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solidpe/m9ebfs/0- 3 grt hw cylindrical 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 the yo-yo when it reaches the end of its 1.0-m-long string, if it is released f/m- webr0sh 3ep9/ftg rom rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of the rear w-fh q l)w ny0i5i1p*fcheel ($\omega_R$) 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 rc u (/cvehh g9oio*76Sun, but with 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 uniformi o( /* ve67rucoc9ghh sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution auylzlz 9hc/p3 r 1aadfy /987hh9hbe7ltomobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, da9 7a h7lec lh 89l fh/ypbyh3z9z/1ruses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustratesnr;/ 9jzvorhco d4w,qlza33: an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal su-4b n**lbbboet(mu ; hy q6yn3rface 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 lenhus9 m.rv 2mz7gth 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 t mrv2u shm7z9.hen 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 sta z4yx sq58bhejhban 16 +o:7nznding vertically on the floor, and we want to:b781bj h 45q zsxhzenonay6+ exert a horizontal 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 speed v=4.2m/s on a flat road is making a turnw7 fu pqp:9j:qo8 tw/ wkaq9-x with a radius The forces acting j7-qwq9p9w 8wup:ax/otq: f kon 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 into a sling of length 1.5 m and begins w/we1ws+wy(p7 6q5q ely az3x chirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where doexw37y/ayp1qq6+ezs wl5 c (ew s the torque come from?    $m \cdot N$

  Model a figure skater’sks5v+t-hxq-j8 caq0 n body as a solid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a sp- +hn5x q0skc8t -aqvjinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch awr; ndt1(q/sfexikpp4s7cba ( f3.)h ssembly which consists of two cylindrical plates, of massdf3)7ih cpk;4b1s/atfer w (. xs p(qn $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 of Fqn7 7g s+yq2moh d8+0h(nwfe,zr o3zlig. 8–58. What is the minimum value of the vertical height h t8l2 qdh(e+,osq 3whzofzn nm+70 r g7yhat the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assu lvgx7io6j5x u:mu*ql .)4 nmime $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as th/b -rel6c 2hhfki7.b,qogsc/e car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acc/cgf hb/oeih bs6.,rk7 qc2- leleration 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