A bicycle odometer (which measures dis-p2k. jrbu, jdtance traveled) is attached near the wheel hub and is designed for 27-incp k-jb ,2jru.dh 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 on ta xcv8yg( z*p4he 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 either compy *acxp4(vzg8onent of linear acceleration change?

Could a nonrigid body be described by a c)/g1, b8rbs9:wzgzka3lolq w82)vzt)h s ti single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greaterdl 6d7g :giqwo/tb/9fc cu;tp678vxj torque 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 sit-up 2 2quznt kot3*with your hands behind your head than when your arms are stretched out in front of you? A d* 2kt3nzq2uot iagram may help you to answer this.

A 21-speed bicycle has seven sprocke- u0to,xhwbt *1zqfng: -xb4d ts at the rear wheel and three at the pedal cranks. In which gear is it harde fbxgou-znb*q01tdw4t ,x:h -r 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 slf ph(frf tiaod58u 3 y(k6 .tx/uvd/1fender lower legs with flesh a1fruhfxpv(/ut k f6o8dt/ (3 5yd.fi and muscle concentrated 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. 8–32pug x/gjoc6 -5) carry a long, narrow beam?

If the net force on a system is zero, is the net torque also zerow9 3xfvlmv*9p ? If the net torque on a system is zero, is the nflv*xw pv939met force zero?

Two inclines have the same height5v,xwz:ay bi 6 but make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.v,:yawb5 i6xz

Two solid spheres simultaneously start rolling (from wny 5v6j n:uut;xrzqy 2*cs() rest) down an incline. One sphere has twiceu(;) t* nsz 5ucnvrxjy2:q6w y the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. They s w:v1le8 h/ fkk izzrn4w;-.uxtart from rest at the top of an incline. Which reaches the bottom first? Which has the greater znx/4h- :lzirfewu; k. 81kvw speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are con7ue+xz 6jp3gkserved. Yet most moving or rotating objects eventually slow down and s 7+xjezpugk 63top. Explain.

If there were a great migration of people toward the Earth’s equf n1 g;xc4f3ekator, how would this affecn ;3c1fge fxk4t the length of the day?

Can the diver of Fig. 8–-vbfttb(: 4ikvp/f( d29 do a somersault without having any initial rotation when she leaves the bt v (di:/fbtfk4v-(pb oard?

The moment of inertia of a rotating solid disk about an axis through itxx;3krzw.; r3fa jm* ts center of mass;w *zmr3a;.rfx jtx3k 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 outstre)7yqfct*9/ cmg0f;h :ou hbwjtched hand. If you suddenly drop the masses, will your angular velocity increase, decreaseow c*/fq) h:ct ;9u jghbm7y0f, or stay the same? Explain.

Two spheres look identical andlrlt7j0n9m3l+ vk e -h have the same mass. However, one is hollow and the other is solid. Describe an experiment30 kej ntl7rv+lm l-h9 to determine which is which.

The angular velocity of 4(/9ztgtmz1 z nh(( 8ayibvkna wheel rotating on a horizontal axle points west. In what direction is the linear velocity of kz8gn (ytz/n9biz(a( hmt1v4a 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 frev(av ap(7q-f tely rotating turntable. What happens if ya7(fv ( vqta-pou walk toward the center?

A shortstop may leap into th ji9 tvlci owy4370gu0e air to catch a ball and throw it quickly. As he throws uw0lt 3v97i igyc0j4othe ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the ) inpzoiy v;k m fd28-zmmm)x5,ra3 ,glaw of conservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second po f 8imgyd-)zr mvz,x5 mmkp3a)2 n;,iropeller can operate to keep the helicopter stable.

  Express the following angles in ra-gpv.nc ,.fkpz y/o.zy3p +c kdians: (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 coinciden5 w7xmgf 5-nxqce. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen og f5n-xxm7w5q n Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beam d;:w*ymm /ug b :7.pgrjdlmi/i; sfw)piverges at an a)/y / :d.js ilb: wg*mm;prpgfm;iw7ungle $\theta$ (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 rpmb whmws) ( 1r9+7q he.k c0x*5-jsawd-rajpj w. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow as drjmjewac-s5*k)wr190p hb(+q7h. jx- ww down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to anothegu aqfy( u-5z7r child. If the ball makes 15.0 uuaf5 gz- 7q(yrevolutions, what is its diameter?    m

  A bicycle with tires kcq273x/uln p 68 cm in diameter travels 8.0 km. How many revolutions do the wheelscxlu 7n3/2 pkq make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calcwa8x3b.kilf y,av1 t (ulate its angular velocity vb1a8,w .x a(3tlfiykin $rad/s$ $\omega$ =    $rad/sec$
(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 mak20c(0m2 uak f l zcpmqf:voeqr-ku ;0,es one complete revolution in 4.0 s (Fig. 8–38). (a) What is the lm(;plk mqeua2 o-zc20fv0u:cf0qkr,inear 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 (hi;wtn+rc 9-ql o-v eorbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear spe/h4bcw/ f dj7ded 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 fro f8i+v0uv ykr 1zzr+vdm(klr*bkb(: (m the axis of rotation is to experience (mvfi:ykv1z0k*zv+ r l 8+d(rbb(ku ran acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its cent(;x4)d;; (ha4jpb ) qbbvem vidzq:wter from 130 rpm to 280 rpm in 4.0 s. Determine v q tw4j: zh4bi;b(edq(;dp b) ;vax)m
(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 wsnxnh; gfht ,; jg a:1av5q+iq 4b6n1$R_1$ 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 themse79ccg8ofc3 lxa/ xp-1 lw;loxlves 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 7cxcolll -xx9 ; af1g38cwop/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 unzs4 pd kdg*x:v7+ nl,oiformly from rest to 15,000 rpm in 220 s. Through how many rez*d:4 vnolg7+ kp,xsdvolutions did it turn in this time?    $rev$

  An automobile engine slows down frbr71hvtf2 ck14sli //msnfa2om 4500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for td*3eqe9 or7/5 i koiojqlt/ 09in vp*rhe stresses of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final speed. 3*roq olr e//ioi9*ie5n qkv097 tpjd
(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 0w,ehh0g5vbayb (/uxmc.4pu 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the whevyh0uehg4/b0acb.p mxw, u5( el have traveled in this time?    m

  A cooling fan is turned off when it is running at 850 g:ip6jw i9qp7f4z; lgrev/min It turns 1500 revolutions before it comesflg4z6 ii9jqgw7 p:p; 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 revolu4 gmtwk +./ju// fczertions as the car reduces its speed uniformly from 95kmrjm./+/c e/f4 k tgwuz/h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car red e bvmfi)s 9fxwd.:yfn/ttg/69oqzp 8 k/p8i2uces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.8.8pwed6vx9n/) / fgikfy iztp 9m82fb s/q :to0 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 m55 wkiqc:cp y 5jsz.x2wdn40weight on each pedal when climbing a hill. The pedals rotate in a circle of radius 15 :0j.5y5d2 zwck4snc p qxwim7 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force (d):n pf6yak uma8. +dpg;rqd+oya4 fof 55 N on the end of a door 74 cm wide. What is the magaamp fn6y guk)yd d(; r4op +:q.8da+fnitude of the torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fig. 8–39.5 sq8f-lm d8beml.1 - chek/e5ydp-fc Assume that a friction torque o dy8celh - .1lc-epes-8dmkmb f/qf5 5f 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attat i02n/mauu 2jched to the ends of a massless rod which pivots as shoatj22/un iu0mwn 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 system.

  The bolts on the cylinder head of an engine req+a xnu qc/t(j-p;a.8lj yt fmfw4) o4juire tightening to a torque of 38 cm (4o -alu )x+.jqw/8nfttjf j4; pya$m \cdot N$ 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 2izwmblt + tq+mc:(x3 radius 0.648 m when the axis of rotation is through its 3 cqmx b:2++mt(ilwztcenter.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicy tfj5v9t0wx/)*p2f 2wyoxt 5) qc 4wjkltm4vycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The massk2tlvcjtxt)0j /xpww yqv 52fy* o)f 95 mwt44 of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, 5 jqzoa*el,n(1 4j1kcyuu2qzlight rod is rotated in a horizontal circle of radius 1.2 m.11zujok zl4 5acjnyq( *2equ, 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 whee s fgs. iyp1h5ovd.7,gl rotating at constant angular speed (Fiopsfsg 7i y,5 1dh.v.gg. 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 arr8,wl1p oxk 5mray of point objects shown in Fig. 8–43rl1o x,pmk5 8w 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 two oxygen atoms whose6 jd6: vxs1igp 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 satewkq z5xb 66yu-llite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a maqby65xz-u k w6ss of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform c*kf7b wttlyu . r;n14pylinder with a radius of 8.50 cm and a mass of 0.580 kg. Cal fwpt7 t1;ukrl4.b ny*culate
(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 res7a a8qlvwz (urc8.xv: t 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 han1wu 5(bsy u k8ak:)x wpfd+.ewhne+1dd-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 the torque required to produce the acceleration, neglecting frictionn e5)y+w pa 1.kefd kduu: hws1bx+(w8al torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is eventually brought,jvicbv*q ,5t uniformly to rest by a frictional tor5 vqtibv,,c j*que of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a 3.6-kg ball ,(re b(wz3sz3nu repddrk3zp:c0 q8/7fn 3tbat 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 solelc g /wkvx*vx3-spj-md5:hm06c w r z.by by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is m5wc r dv 6g-xcp:jw/h.3m 0bvxs-z*k 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 in Fiort8nb s6a5yh /anr)043v yo kg. 8–46. 3 0ar)t/b rkyo5syon 68 a4nhv
(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 cott1/zxf5 qu :hvow7+xccv5 -6tf8p op nsists 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 re5l us( h2q5xutia (2jmst within four full turns (revolutions) and releases it at a speed of 2u(2a t2l5smujhxiq5( 8 $m/s$ 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 momenttpges g2( s+ (xvcs.q( 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 i,ja 5b qlh/jsr./ns:5 h8z,mx q2vdstorque 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 rol e y; uc0e9p/.-segxqfls without slipping down a lane at 9/;gcyue0e. xq s efp-3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy o nk/x81+ nucj ;y)n ruu:*xzkof the Earth with respect to the Sun as the sum of two termsk+) x knz1nc 8y;urnj:/xou *u,
(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 4te t8lino w( b(99y0k yvxm,umass of 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 t 8, oy(m(4ntu9wx eiby09klva solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from resacg.yg(mkq97kj w *g.l 8zg g6t and rolls without slipy6mgz .gcj7gk98 lg.w kg q(*aping 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 vertipb(5 1o8s;g/: oyvzna okexl1cally on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of/z;1(by olk1nsp:ovx 5ao 8eg energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rota(wp; g6h3gkmf7ti f+ tting on the end of a thin string in a circle of radiuh w +k;tiff73mg(p 6tgs 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum( 7dgwv/dcoc/ of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm wcwdg/o vd(7/c hen rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotating at a rate kah; u5;:w.7cg b0)pqfo7k 7rwwf huxof 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases to 0.8 u7x.p o) 7 ;uw:fbkawwqkh; c7hf 50gr$rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) upn, cn 85bk9glo0rv*can reduce her moment of inertia by a factor of about 3.5 w,u5l n98*0ngp vckro bhen 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 x*ec8:y cpms 9p1mlg6 increase her spin rotation rate from an initial rate of 1.0 rev every 2.0 s to a final rate o8ppce:s * y1xmcm9lg6f 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 throuc(,bh36ami6 qf.fn v lgh its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the ce6b3 a (m,ff6qlc .nvhinter of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater ow9 t36jciry9spinning 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 momentumw*2 krmxwm nkj2- 2(dc 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 moment omr-meay8-szrn x +0 .jf inertia I is dropped onto an identical z0me xmajr8-r .+- nysdisk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2i p rwq7;(brj euwe6m/g 2q09v/zjek49nnm-s .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 ab-hrnm86y iz4x( , clqt the center of a rotating merry-go-round platform of radius 3.0 m and moment of inertiacimrn-x6 b 4qz8,( lyh 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 )v o) a7 /vxy:hfc,zywan angular velocity of 0 h)v 7x, yo/yc:zfaw)v.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 wrk9e mubjww-1x)(l43 ud7mo a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing3mu)9 eu4wowjrl1( wk-m 7dxb 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 excess 6kv ov08 8l 0d2flvhxhof 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 7g rodr5yprdz6ybiw37w;h(*$ 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 rotatlsj;4g5a5 * ue55k rybduv.c ue freely without friction. The moment of inertia of the person pl5g*ev4j 5ubykru; u c5a d5ls.us 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 stan g;9 ubx9z b78skoii)ma,quh4ds at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of i9zuu sx qa bb;)h4mo,97k8i ginertia 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 tn3 q ( wa buhlu6.nt3.x d7zc1dp0i-5l z(xclohe 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 length5d l.i c3b n7l. xxq6ct0ouhw3(npl1dau -z(z of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces the Earth.6xc8y d v*rsf(p8 as v.6og, rs)kgt8g Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the l6ydvp ss f s8g86ct*g) k r8vo.ag(,rxatter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at z, gol7mpwh b7v33/ty a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a w-s a*bjnu5q7pck7frpgt 27z0u- ; og uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor. jprp;n27* z0g f t -a5cqubkougs7w7-    $m \cdot N$

  (a) A yo-yo is made of two solid cylindc2ss ns4uz0 *mrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mas02musncz4s * ss 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 from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed(js,pa+eh82usid t 8s* cje9o of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 m7sal+.ryc *i 8spa-xtimes as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform spi *sr- lyps+8m xaa.c7here at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a ldfy,9mw ks 7 wpqvmkx8078h/q ow-pollution 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 diameter 1.50 m and7kvy 8 w7kx, m d9f/wmpqhs0q8 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 illustrateb wbyi5 8g.ngg6/o d(;un iqq(( npb3ys 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 oq6//1zjkm3fl sfav u -.cy. ywn a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., we ignore frict n0 bjpv p/v.xbca;1a0ion) about a hinge attached to a wall, as in Fig. 8–54. The rod is h0bva 1vp/cp.x0 ab; jneld 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 standingi5nz**-nfi ;,wn -zqv8g qv bm vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step -*b g -,qi5v;fq mnwzz* 8innvagainst 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 r/lj.jrn7+o ;jb */zvflon-jis making a turn with a radius The forces acting on the cyclist and cycle arlr+f/vb.ojo- 7l jzr*nj ;/nje 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 i5uvbr* -lq2q length 1.5 m and begins whirling 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 toriv bq 2lrqu-*5que come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder 4vavs ajfz)94b8ly s3+j1l daand her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretav341) zy fda9j4+l asslv8jb ched arms, and with arms held tightly against her body.   

  You are designing a clutcw1b8dttv p+ w ec62)twh assembly which consists of two cylindrical plates, of maspwte16 +v)wt8d bwt c2s $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 radi0 u*j1log+ano (ee jll4i0cm4us r rolls along the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if itu1ejol+ l*40j40g maeol ci(n 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 nx7;hzox:o7 qcu4jo dk- -gk n3ot assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revs.y(dwh 3 e:mb (usmcf m8u3r7olutions as the car reduces its speed uniformly from 90km/h trs 3: dbucf sy.8 h((w73uemmmo 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