A bicycle odometer (which su,mg du5)lqykri7y)(ia:9l measures distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. Wmai5 y7 r,li)9) (gu:u klqdsyhat happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotate4pk5gm 8nsh kj1k)vk *s at constant angular velocity. Does a point on the rim have radial and/or tangential accelerat vhnk15kpk s*4j8)gmkion? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the 7) c/yv)))c,zjtqzui0e w r wdangular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force*;uzh w ukd-m9? 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 yod-caz1.rx.cgnlz0:rh u ). iiur hands behind your head than when your arms are stretched out in front of you? A di-ir z0.la . :rci.h 1)dguzcnxagram may help you to answer this.

A 21-speed bicycle has seve*;nst (fs-et cn sprockets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a la ft e-n*(s;tcsrge rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower legq6 kcas apry2*p2q ul3. m ..v5xgln0ds with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis0 r.gd*nyca qx.qul2 s3 v26p.pl 5amk of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrowkh k*x u;3uklzr+x33c beam?

If the net force on a system is zero:b33g 7jf *r ihv8i8jh /utskyf;5x qo, is the net torque also zero? If the net torque on a system is ibo/ i:;f5v jh h3ur kxg7q*8fj38 tyszero, is the net force zero?

Two inclines have the same height but make different anglc+te h,yyxs2( es with the horizontal. The same steel ball is rolled down each incline. On whichec y+y2,xths( incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneou;2ki y8xr bwcv2 xq f- k1;-0dc1jjmywsly 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 there? Which has the greater tor;c fyjkw wv;-1d21ji xxyk-2cb0m 8qtal kinetic energy at the bottom?

A sphere and a cylinder havy*3fg* qdk0 :o4qvxdce the same 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 the bottom? Which has the greater total kinetic ener x vqk 3y*dg0dq4ofc:*gy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most moving6f 3hzjhafcn)1v)xz h )e/ 3mx or rotating of /vzzc6hah) 3n3 x1j)mx)e hfbjects eventually slow down and stop. Explain.

If there were a great/.nie/n e (xl89ppzpc migration of people toward the Earth’s equator, how would this affect the length o.(e/ c/ npnzp9l8pexif the day?

Can the diver of Fig. 8–29 do a sdx -u,+e ko,aix)q+-jbrtz r *omersault without having any initial rotation when she le-o ar* rx)kxj,dzb t,+qieu-+aves the board?

The moment of inertia of a rotating sot yvc2p1q-e) 67piyol9k.y ejlid disk about an axis through its center of mass ivyeo2p y p)l1.67ickj9qye- ts $\frac{1}{2}WR^2$ (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 eac9y ekdmi8 y+t9lm(2swh outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Ex+ey mi(y9m9dlt2k 8swplain.

Two spheres look identical and have the same mass. However, onzt u*aovg 9.zwlz06l(l1vd4ue is hollow and the othe(4d1zu zvvt0a .9 luowl*l6zgr is solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points west.*g z3zuv(cy 1:)qhug 78zeup v In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top ozec zu3(7) vuy*q v guz8:hg1pf the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on qre.i/m)/qbkv d) b8afvl9+k the edge of a large freely rotating turntable. What happens if you walk towv+9/))medlkrk.fba iq /b8 qvard the center?

A shortstop may leap into the air to 3 (rqu sh(,ki6v p.vxicatch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in vi(,q.xp(srh 63uki vthe opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of ang(fh vn(j 3ljgj1yw .aqdq5t )3ular momentum, discuss why a helicopter must have more than one rotor (or propeller).gflyt1 hj aw.j)53(3qdqvn(j Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in uwkyx; wl4j3.6d2 xhk(droj * 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 amazinv9r .w l c2gblundrak6ka 9 k4sxe/)66g coincidence. Calculate, using the information inside the Front6 9)v6ul.knw6kr 2r9exga as bc kdl4/ Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moonq1+e;vigrk .q z96-5ts k jtpa9-wika, 380,000 km from Earth. The beam diverges at an angi9q-gza p -kit51rw t9e6kj vk.s;q+ ale $\theta$ (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 the motor ius2 n :31ecdhhy5fj*ms turned off during operation, the blades slow to rest in 3.0 s. Wf2s3n duyjhm ec :5*h1hat 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 child. If the ball makexis :du6 9,d ptukf2k(s 15.0 revolutions, what is its dk: px69st u,uidk2(d fiameter?    m

  A bicycle with tires 68 cm ink.p74hq :rzg 9s-19a rxhsyxe diameter travels 8.0 km. How many revolutions do the wheelh9 e -rp9 ysxaz.k7h:rx4q1s gs make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates atzqyh3ql (1 )1r0p xeyt 2500 rpm. Calculate its angular velocityxpqe0y3l y(q 1 1trzh) in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-roayvd 35brj0p8qs.79 n gnug1-8 jf wnvund makes one complete revolution in 4.0 s (Fig. 8–38). (a) Wha8 nf0n. p83j n91bu grgqawdvy 5j7s-vt is the linear speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth /h-gw *5 u5hwozn us iw);y:kd(a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speedbjah2 .-t(u rkv4kt8,3ekt rz 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 centriksvfw,3c r(ne5r: yt0zb0 6oafuge rotate if a particle 7.0 cm from the axis of rotation is to experience an acc c6ae3vobf5 wn z,0rt(:s0y rkeleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its cen l2ex fcob42-iter from 130 rpm to 280 rpm in 4.0 s. Determie c22 o-4bfxline
(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 radiusqxcib x*3 /brbu m 2ks:+y8+rs $R_1$ 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, as8.jn,np)b, cwnma 1 iuz uda8;tronauts aboard the Apollo spacecraft 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-nn8aw p u18ca ijdb zn,.u);,mmin 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. Through27doqyvfw1 2 mq ri y.t0k03 n/i2udfi how many revolutions did it turn in thi dd 37yv/iq my2fu 0r1nkqi o2tfw0.2is time?    $rev$

  An automobile enginexnj3ng -7jad1d gr97u-3m4rhp / yrb w slows down from 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 z/nqjx p*mm++p ow9 sv,v)6cdfor the stresses of flying highspeed jets in a whirling “human centrifp/ q6vmnz )s9mow,p++xv *djc uge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from ws9n3k6zdsdji86 o: hp oag3;240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled in this t8gisks d6o hpao ; d33:9znwj6ime?    m

  A cooling fan is turned off when itxx-w 0k-wbexxkwcb36 0m 92 ts is running at 850rev/min It turns 1500 revolutions6w3w 2xxs- -ctk xe0m bb09kwx before it 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 reduces its sbf8-rqo c 0qb:peed uniformly from 95km/h to 45km/f0 rq:cbb q8o-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 revo3gg;jgcnd 2 (gx;prm. w2z8n zlutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a dignn (2 pz;mx;z32crgwjg g. 8dameter 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 ridinht4*era4q4b ca es xc; hsa32-g a bike puts all her weight on each pedal when climbing a hill. The pedals rotate24xh4 a-s tr4chas*aeqbc ;3 e 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 ofsz( 73p d;dfiy a door 74 cm wide. What is the magnitude of thdy3 zds; (7ipfe 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 ts(5q 4v ldhv3efq;br) he wheel shown in Fig. 8–39. Assume qhev;5vr) 4q 3bsfl d(that a 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 atez +u/ d)5gwajtached to the ends of 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 uw )dag5zje+ /system.

  The bolts on the cylinder hee y6 8sfp0wu82 u/4mzx3t5n knntwc6 lad of an engine require tightening to a torque of 388u3x fnl2 ntwuky5mnw64pzt6se c /8 0 $m \cdot N$ 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.c2*co.n7g:dgk xckg ab 8vj a8)8 a+xs8-kg sphere of radius 0.648 m when the axis of rotat)gd 2xk ga*8ao+s :jv k 8ca7gnbxc.c8ion is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.x vh wn*eh4,/b7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be /, bwv4n xehh*ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in aw i- lm7 vzk 2hwwmkft *6fmlm2.8bb/1 horizontal circle of radius 1.2 m. Calcula 2ibb 6wkh7tzmk m.12lwmflv8f-w*/ m te
(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 *2.aa spm .e,k4tkn ddat constant angular speed (Fig. 8–42). The friction forcea e,.ntp.da*d s2mk4k 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 of5vey(h /hr hr1 point objects shown in Fig. 8–43 e/hh rrv15yh( 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 whf 3h9km b,ghax+9+f 3lb grmx2ose 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 sa)1b/jc 0tz *tikri tq:tellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, whac/ 1kjqb*0tttzir) :i t is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 cm and a mass of 0.n j6 tgyw)d o,2cmssy72r;ld ,580 kgywmys), 2dsd 62jontr, ;l7cg. 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 swilk-9bhv (:ykqbqfp95k 4:ame -0cq qings a bat, accelerating 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 mmvz(s)lhi*o.e54rl+qa,bg6 m d o .emerry-go-round and is able to accelerate it from rest to aimblh( o4rgvea + zl).*5de. s6mmq, o 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 frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm ia v (2r0+sngv3f s+93rb-ysub wk6 cnfs shut off and is eventually brought uni(3f+cr2 g uv3b0as6snvfyswkb+ 9nr -formly to rest by a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. bwmc +1p)1cxiu ;i )hozpi *3r8–45 accelerates 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 o rbf -ch gs;;(u,3xfois thrown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscl3-o,; fhsocrxf bgu;( e, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rocsz:w 1xk9v (pkik8;. jdrj)l d, as shown in Fig. 8–46s:lpxvj)d k ;ijwc.1 kz8k9( r.
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine consists of two masses, rd)kln2 cy;j. $m_1$ 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 accelera(3 jvs8 asez5mtes the hammer from rest within four full turns (revolutions) and releases it at a s3sa5e mvzjs( 8peed 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 oruy;a 1h fi (glpmzoq51b g,1*f 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 developlw1lt9/ cej 5ys 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 rollsg4v hd9s x4vbi mz8x42 without slipping down a lane at 3gsx 4 2 z8vhib9m4x4vd.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun 61c 3 v5pqdhtraz:s 3yas the sum of two5vcp s1r h3atdz63:qy 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 tj1fnxj+ zx wf;.c ,u.mass 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 a soliw.xjtxj,. ;nuff+ zc1d cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 k8m.uxlk/g7tt(hflmtv i /: 6f g starts from rest and rolls without slippt tm/u6 (h/klf. tf:igm 8xlv7ing 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 s ,),0nsil3trq3rv fmstarts 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 fqins, t r,0r)3m3slv: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end1ev- h0 ixwtt *6d6cef of a thin string in a circle of radius 1.10 m at an angular *texw 6v601-d ftecih speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical:- mq**xzy adn grinding wheel of radius :x*m-dz *yq na18 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 at a ()j3 hzd*8s4qcqb gxlrate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decr)3z4bgxs qqlh* (cd8jeases 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 hers/exe /fez5n7 moment of inertia by a factor of about 3.5 when changing from the straight positione/5fs exze7/n 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 initial rate of 1. -4q .vw)7wvokbll5 omgnw h+30 r w7.5ow lqo4lk -hn3m)+v wvgbev 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 r9;) w x f7yho8pd:oesaotating around a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after t)o :;a7x wspfo8he 9dyhe clay sticks to it?    $rev/s$

  (a) What is the angular momentum o2by7 wn )+e r,qtof3ja7iwfw2f a figure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momeny(z.p1:+q/do uolnbmr j/am2tum 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 cylindy9wd+6j kay,pn4fk4 rv26f ji rical disk of moment of inertia I is dropped onto an identical disk rotating at angular speedv i4f+9y 2nfky4, 6a6pjj rwkd $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2b4r0b0 ntd1b9o7ft dt.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 ytv*p -mz a6r.merry-go-round platform of radius 3. zv6am.p*tr y-0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an angular velocity of9zy wfqop(i-4 0.8y(o q pwf4z9-i $rad/s$ 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 eventualwauc ;fp:r8 xr,h73drly collapses into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its ex ;ra3hr pcrf,: xud78wisting 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 of 120 ajylyrvhqi i1/x dfq+rnw*cd 4l: (;q7d 66g1$km/h$ 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 4 fu/b aszd1c:/ug2ku $ 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 atfonruro*y7g.9 4 i/we rest, that can rotate freely without friction. The moment of inertia of the person plus tr.wgun/49r7ifyo * oe he 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 .-an hrt 3a gin*mmc-0/ w2sueedge of a 6.5-m diameter merry-go-round turntable that is mounted on fricti 3am2* -0crtn maw iue-./hnsgonless bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on c 0u:stw6pp0wthe 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 sc0 pw:w0tp6u rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same on-5cz51h rmeside always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own a 5hrozec1mn5 -xis) 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 vt qvvm ;7rl.15bxz )hm/$s^2$ 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 shapen) yq5al/7gfxsq mgu9d4t 4t: of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular accqagt5f)t /7g9xd:m44sl qy un eleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made a*ke5m)ptmj:r4v t s/of two solid 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, ifrm*/:se5ja vtk)p mt4 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 th5n1- u(n 8pq0zgogb/d a1+ba/kfnq wle 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 pho.:0x blzf m + +iqz/0yut bk*c8sx*mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 iych+bxb.8o *zk: +/zfqs x*00pu ltmdays. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for al(c(u6d8 p jwks2 my)n low-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 and mass 240 kg, and should be able t2m(lj)c n k86w(supdyo travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates an e2w b; ,lvr.5n;lw+0buvedah$H_2O$ 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 surf4ncbl :vb :p dn9sb4*mjbul*2ace 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 a9hah7 yj6j dj1nd length L can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At 96d17jjayhhj the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on vc,* 2y-b- ho at:dx(b tl6wnwthe floor, and we want to exert a horizontal fotxcy btohv(nw 6bd:a2,- wl-*rce 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 fla ahx3syt(/81kz -do j i6k3jsdt road is making a turn with a radius The forces acting ok sta3y ds8xkjhdjoz63 (i-1/n 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 slin39gtm 24 twafs-yq pcz9 qrit;h/e4) ig of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is it)q9yz-m34ep9t whqc 42gfa ts ri;/ 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 soliob7yh n6jv*;bd cylinder and her arms as thin rods, making reasonable estih b ;j7*yvbn6omates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly s u+ap7/vm2ek wode- 9which consists of two cylindrical plates, of massa9kms7e d+v /-oe2wup $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 rol*plmb) h7sht* ls along the looped rough track of Fig. 8–58. What is the minimum value of the vertical hbthp)*7ms*l height h that 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, bf62. 3(f ;njks t:sj0d1vmb /cft sdnjut do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its8*)mxyz5o)lz ngx8rq speed uniformly from 90km/h to 60km/h The tires have a diametzg nlyx ox8zrq)5*m8) er 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