A bicycle odometer (which measures distance traveled) is attached near the rmmx* ps50c5rarpe1u; gx. +nx:jls/wheel hub and is decsxj:0x/ urs ag; m+.rx1n*rpm5pe l5 signed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point;6intr-fpk iuv9(l5j. q jrb / on the rim have radial and/or tangential acceleration? If the disk’st5ji6kl; bf n.9/ju-v iqprr( 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 spr+jj - dnqg5ao7kcd pa) ,+7single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torquen5d/pd wi9x4 )qr ag2o than a larger force? Explain.

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

Why is it more difficult to do a sihbh3 8/tr)sx dt-up with your hands behind your head than when your arms are stretched out in front of you? A diagram may help you to / )bhdsxt 38hranswer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the ped ;,lwwa;p4.pmu5 .iif jbdeg (al 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 edp.gp,fij;( ;i wa .lbwm45ufront sprocket? Why?

Mammals that depend on being able to run fast have slender lowera a5to/6vsr:+0r lfvsm;qy7 g legs with flesh and muscle concentratesfv5t ;/sm rqgy60lv:a7or a+d 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. 8fj ag1rot6x;ir /nz ,3–35) carry a long, narrow beam?

If the net force on a system is zero, is the net torq4m+ajs ucks la/wih8(xc)i 1.ue also zero? If the net torque on a system is zero, is tcms)scw(8xliha kj1 +i4 au ./he net force zero?

Two inclines have the s1tssxx ksz/+ b g:*);qgvssz -ame height but make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at t;x tbk /sgxs+-v sg1*):zzsq she bottom be greater? Explain.

Two solid spheres simultaneously start roltt)p0dzsb)-vr3u ,a d ling (from rest) down an incline. One sphere has twice the radius )0,t-duzt 3v pbadr s)and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same m-0qs2hp+ip4u2bav x vass. 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 total kinetic energy at the bottom? Whic4+2u2 -shpa0vqvip bx h has the greater rotational KE?

We claim that momentum and angular momentum are cq; auil*;0 e iq77qvv 25vzpvaonserved. Yet most moving or rotating objects eventually p;;7a2v7 aiquv05leq *qzvi vslow down and stop. Explain.

If there were a great migration of people toward the Earth’s equator, how wou2zw3;tmd n zl(co1c:d ld this affect the ldtwdzc2z :c1l3;omn(ength of the day?

Can the diver of Fig. 8–29 dof-y 6z)mj hdwqja695it47my(bbb jt4 a somersault without having any initial rotation when she leaves the boardzm7 yhai djt )q9b6f-m4 w j5bbtjy64(?

The moment of inertia of a rotating solid disk about an axis through its x * 7wum.101dbaxah )ahxjrs5center of mass u*xbha)jh1xawar m.s x 0517d 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-kg mass in each outstz*zfok da0-5vretched hand. Iodk-zz*0 5af vf you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have thee47 r1x)aq4k *s qyoi,aa6gey same mass. However, one is hollow and thex4i,aaoe1s 7 a4qk*e)yyqr g6 other is solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle pointslx2r8a( -v c+gsudh yz3t5 vu56+ okth west. In what dl(g+5ytzoh5k u6r-ta3+8cv dus v2hx irection is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large free *tgew6hemp1/gpy;ip 0q6 9qqly rotating turntable. What happens if you walk toward the centewhpqt ;q6m i 6gg0q1y */9peepr?

A shortstop may leap into the air to catch a ball and throw it quickllauo2;c 354cf )zqiary. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs ;)5ri 2a4lzf3aocucq rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momeno53it x1fu z4ujrl*8htum, 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 helicojhi 8 *5tlruz1f4xou3 pter stable.

  Express the following angles ibh: t;q) l.,utl tq /zbh+a9win 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 amazing coinci+ah;zq,9y9wy /ehei9- s wqikdence. Calculate, using the information inside the Front Cover, the angular9;a-yw q,swy+qi/z hkhe 9ie9 diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the M; 5ymqf4t os1(nlzij3oon, 380,000 km from Earth. The beam diverges at a tmj3nlyf; 1z(s4 qi5on 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 iv id1y*lo7;( xdu,on 5hv* dbthe motor is turned off during operation, thdnix,* u5;7o vvyo(l *dbi1d he 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 xy;geivubilh;)8 . r/to another child. If the ball makes 15.0 revolutions, what is its diameter?u8gvx /b.lr)y;; iei h    m

  A bicycle with tires 68 cm in m hgk*z-nghqlv ; 35m*jn9e1h diameter travels 8.0 km. How many revolutions do the wheelmhknmj;5q h*v*le9 13 gzn -ghs make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate its m1 j*u4rezm5xp;aa d 8angular velocity in 4 pe5ammr8 1dazxju;*$rad/s$ $\omega$ =    $rad/sec$
(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– qsc7as(e5jm-q8ac. t nui8g338). (a) What iss8n eqgc 3cisqu5jm7a8a-. t ( 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 Earg 6 73uvfbxq owev)m99th (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear spen9xt.l;c d2u sed 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 particle 7.0 cm fropzn:gboogdw j;)e 397 m the axis of rotation is toe;wd zob7jg:)pgn9 3o experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 13i10fcslq c 9)f+l pa6zyl;n ym8+ w;af0 rpm to 280 rpm in 4.0 s.lpaiylc 9s0+;1fy ;+ fn a8mz6c) qwfl 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 radius q, p5iv 6j(l 4jiy zpm2;aho;j$R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboard the Apollo spacecraft put themselb ,)7e y3jvtjxjwx;1qves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotatj j 1wvxb3yje,;7xtq)ion 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. Thr a2p-0xdb3 0 w0 vidblc::au-g+qvjszough how ma-bvu 2dcsb +0vg0z-d3:0 :pwaqlj axiny revolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calculate v0mpo1dsr4pwqorli. 4p*g+0 v. no*v
(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 stresaklgy l jsyh+,b u4u8y fe:2nx0 :4n;jses of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions0a ey ubu2llk:4 :8 +xsjgfyjnn yh,4; 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 6.1/ie q+ha( c/+v+fctm1 :9lzpjc qrxj5 s. i9alfc1/jjr:+ /mt q+pqxh(cce1z+ vHow 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 15krc+ kz(mr l (wy7gfqe:m* 13g00 revolutions before it comeskgew m :k y*cr(3 17glm+zfq(r to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed uniformly frw9g,:y his vn:-g tgopl98 r,hom 95km/h to 45km/ggt,r- h,8g: nol yh9sw9:v pih 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 th8 xge(jy-ri8d /(aofrom 95km/h to 45km/h The tires have a diameter ofhg y88-d(tji /ae(rxo 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 ieb)6b3pba0ia) z3ajon each pedal when climbing a h6 )ba b )e3jpzaia3ib0ill. 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 N on the end of a7ji6e,ph /.3okn07n vdallse door 74 cm wide. What is the magnitude of the torque if tho/ j l3a,0n7pd n.elsi7vek 6he 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. 838)z 1izxae+h9b vnt z–39. Assume that a fricxazhtznz1v+ 3 e8b )9ition 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 ofae)cg r t, ag6 6n.pu:o2s;prq a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this rpr2apoune,.ga qtg6); s c6: system.

  The bolts on the cylinder head of an engine requ ty+p54f7-7t ant+m uc8uni5thqnc0a ire tightening to a torque of 38 ich0ctun au5 nq7 p+ t8fy5tn7-4t+ma$m \cdot N$ 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 radius 0.65 tz8c esz 5srcf/3g5i48 m when the axis of rotation is through its 85 z5z tre5gc 3ifcss/center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.k1o - cam5s0ku o( .qnmjf/-wd7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of 0aofdon 5-k1sqc- muk(.m/ wj the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, linoeef:8j ,pxm 48m6eeght rod is rotated in a horizontal circle of jmeo486em f 8p,eenx:radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating at constant angulff)tx id8/s.z ar speed (Fig. 8–42). The friction force between h/it.dzf8)f sxer 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 wib8 4w:d4pqdnqip( :of inertia of the array of point objects shown in Fig. 8–43 abowiqi4dwp(qb: 4 p :nd8ut
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen a+ ,zxs.ve1dis toms 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 correct rgyqid7 p )m--pate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if m7--p)yq gpid the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniptcnqe2 fl 4em18s7s1 form cylinder with a radius of 8.50 cm and a mass of 0.580 2m te f1pss n1l8c74qekg. 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, accellcu f5t5+ whi5erating 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 oxhi4mrhg jr do526q;9n a small 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 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate 452 ro;dhqi9jxg rm6hthe 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 evenl4ex5 ,bu+ f:fdt(aeu tually brought uniformly to t e f:x(u4ae5,dblf+urest 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 accelera.cbn2 uc xziwy,yaw)4-7 cl6xwo1t*ites 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 ty 3n*toaz 2nrlqsr )+a6hh.5bhe forearm, which rotates about the elbow joint under th 2l .rozty*6as braq h+n5h)3ne 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 loko4qvkqf;n,xke(4j 5 h-7 ebrng thin rod, as shown in Fig. 8–46.h4 5nexbjqer k7 -f4k,kov(q;
(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 consist 4;,bw zd4flggmd (h.fs 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 hamfj+f9ru+x du),fv(vp1 v/hm xmer from rest within four full turns (revolutions) and rev x hf1u)rx9upf+j+fmv( /,vdleases 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 inertiaw7r-t/b g 0dlr q74cay 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 deveq )/ cwh3orfki, *skh9lops a 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 rusi/ba d80 k4),- ifw*4lbh di ne8zplolls without slipping down a lane di-wd,hz /in ) kb0 a88lifu4pbes 4*lat 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Eaijwj4a(m 1p q6ktfq-n42z72 s.n ewfwrth with respect to the Sun as the sum of two a.sjw ( 2jmz4tkq41ipe7fq2- fnnw w6terms,
(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 of7jlq3g;y2t olf s *:qw 1640 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder. ;s2*gql:ywl7qf o3tj    J

  A sphere of radius 20.0 cm and mass 1-yos)gq 7j v- )x4mzxn.80 kg starts from rest and rolls without slipp-z ) xsm-o7)yxvjgn q4ing 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 ;;tclg hs1e+cnz 5*v my4i1b ion its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of tht b4c+c 1hvi*n e g1l;zis;my5e 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 end of a thin :osq) 6mnen+h string in a circle of radius 1.o)qsnhn 6m+e:10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grinding whein4qt * dfmoq7p9w, u3el of radius 18 cm when rotating at 1500 rpm? 4o39i *w,qnfqutd7 p m    $kg \cdot m^2$
(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 adt)zpddl6e azs55(6vf /( nithat is rotating at a rate of 1.3rev/s If he raises his f/ (a6dvlzidtn ) 5p65dz(asearms to a horizontal position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce her moment ofhf ;:qmoup6 htpl w1oo+ cu81- inertia by a factor of at uc ;ph h86oqmwf1p+o oul1-:bout 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 rotation rate from an y cgkgz5z3wp8l,4yx6 initial rate of 1.0 rev every 2.0 s to a final rate of 3 x54 3c8k z,ygy6lwzpg$rev/s$ 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 io6( ,sas;idz 8fjv 1wxts 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, approximatz;,si 6xd8jfaso( w1v ely shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a jxu88 p/ l( mpntra/q83/xdrk oh64trfigure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum of tsmb/e c u x7j)ytr;.la ,jig28ro,w*i he 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 8)wbj,-ii.) s4bkwxtyvcw0 orical disk of moment of inertia I is dropped onto an identical disk rotating at angulsvy8bk b-iwi,t0w4xc ow.)j )ar speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns a 3bbkyd( vn01v 2lt kqm;(o,kct 2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotating mer* *5e:sgudsrz ry-go-round platform of radius 3.0 m and moment of i*5s*ez gu dr:snertia 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 ymv, 2m+yxywohzcp q.0a2 6q8 angular velocity of 0.8mwp zcxv8yaq2h206 +y,yqo m. $rad/s$ 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, los92ory q1 8 p)h+nwnsn:w-piriing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what wours q wyip9)2rnn 8nh +owp1:i-ld 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 e.nscob ;zvgaay52(/ r;iq qv;xcess 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 0mnue :fkpv phdbw8d-iy4.- 5$ 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 n( )wzndkum)o-b. di c+j60docan rotate freely without friction. The moment of inertia of the person plus the platform is .0obk oi)6d)-w (j dnz +dmucn$I_P$ 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 kfy7x msl;po)z6-bj/diameter merry-go-round turntable that ibpoymzsl7; fj/ )k-6xs mounted 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 rollsb+b4ze4uvx5f 0 i9won( 1 j(ojvj ,vmw on the ground with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rop5+1e(jvx, j vj4o 9ofbmi wbzw(v0u4ne unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth suc-i d( wr4*h l:tfsis6y4v4n ieh that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own ev:f4i(rhi*s4l6-yw sdtn i4axis) 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 k* 4 g9.wlhcxlm/$s^2$ 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 sha. ghl)jzdb sa7.ms4 :cpe of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s intcjg.shd m 7zasb:4.l)erval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two soli2ew4m n1t -:unpid qz9d 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 0.01-:9en 4pq t2unwzim1d 0 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of the rea r xbm4o(8fkym -oq2 t /x6ui2wet,fo1r 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 mas4;y)ykb ) ex+ gjk/obqs 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 gye/+k)j ybox ;kb) 4qradius 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 auo-bptzj7.rw3v.u1 8k xv bq7 low-pollution automobile is for it to use energy stored in a heavy rotatuo v8x.qu7vjw. z3 kbb 7rpt1-ing flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to 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 illustrat gxk -oky4-;ades 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 horizalc05if; 6oa-m7n,aga*c10) .x onfjfhb u pvontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., we ignore frictionx-3d9p o3wrl i1jlev6a*k 2j;:cxjtg) about a hinge attached to a wall, as in Fig. 8–54. The rod is hel;jdt: i6vrxjl92k33jg*1o - lc exawpd 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 verticalxjx3ec( ,)v6cntu1 ho 0 8j6 0ausagqyly on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step o0ah()uce6j qsa,nxj 6 0gx8u1cy3 tv 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 amh9u b 5;izussmmx+hz ,j r/-; turn with a radius The force sr5h-sm,+i; hzxj/b 9zm;u mus 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 2o-e : uv636 lzl yfraf8nzhb.usq+2g rock into a sling of length 1.5 m and begins whirlin2+6y foa .zglelq-u: fr8h2 vu63znbsg the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a sor gao6k,+oq ijr7p9h7; v8prblid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, andr7i ap,;q+ gr 7jrpko9bo h68v with arms held tightly against her body.   

  You are designing a clutch assembly which consists of trai 5e.i o;j:e(ryks5wo cylindrical plates, of massij s5riya;:( .eek or5 $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 ra df0,vctexu47dn, g3hdius r rolls along the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the mend4 th,0df, vgc37ux arble 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 assyyjk- ibihpa )5- d-,eume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as6tk7/ jp 20 yy;zliyby the car reduces its speed uniformly from 90km/h z 2 i;k6yp7j/0lyy tbyto 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s