A bicycle odometer (which58wm3 y ec4azw measures distance traveled) is attached near the wheel hub and is designed for 2735wwca 4myez8 -inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does *pw z; faztjza j-gv/vc0u221l :.dgra point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either compu t:;vzj /f 2w0vjcp*. gz1ad2a-z lgronent of linear acceleration change?

Could a nonrigid body be described b t)ceyv57z07x(9i;x (pnmrpu 8j w yyuy a single value of the angular velocity $\omega$ Explain.

Can a small force everefw2ey; x6,luu,he0z exert a greater 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 with your hands behind your hekr9.g+7vg 8v0 5cx l+m9l krjpb/xeug ad than when your arms are stretched out in front of you? A diagrag+e+k.b7vr0j x9 l /x5mg9gkvc u8lpr m may help you to answer this.

A 21-speed bicycle has 2n pj(rq6+a; rl y*hunjvvm9/seven sprockets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a sma9vh2 ;q /j6+m(nnuljrva yr *pll 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 i q :ni .p*bv)1t niwd7qdv(lex-8q/yfast have slender lower legs with flesh and muscq:itl7/pv vniqe w xi.8y -dd*1n(q )ble 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. 8–35) carry a long, ncf(jgm7ib im b9d*u*e-o y)te5z ,uf/arrow beam?

If the net force on a syss nzj(.h+ 8 2idm3sop op-8pmdtem is zero, is the net torque also zero? If the net torque on a system is zero, is the net z3i2 8nm pddsop-+mjh p. os8(force zero?

Two inclines have the same height but make different angles with the51ljk-( vsbt h horizontal. The same steel ball is rolled down each incline. On whibht1 vk5s -(ljch incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from r*(3rz5 gsp wblest) down an incline. One sphere has twice the radius and twice the mass of the ob rg(w5*3pz slther. 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 thd fnv +e8l.,r+)azvyl e same mass. They start from rest at tva+,zny er )f8l+vl. dhe 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? Which has the greater rotational KE?

We claim that momentum and angulauojvmk869 *sld u3+ udr momentum are conserved. Yet most moving or rotating objects eventually slow down and stop. ms39d*+uk olujd8 v6uExplain.

If there were a great migration of people toward the Ear,x ea , hwv+bi+w+ma4eth’s equator, how would this a 4+exb, ewhwm,+aa +viffect the length of the day?

Can the diver of Fig. 8–29 do a somersault wi-v d9t +mqf(/j/ znjrd4ojqw 9thout having any initial rotation when d/tfqj4j-+qz r(jm99 ovn/wdshe leaves the board?

The moment of inertia of a rotating solid disk about an axis th ha8l ks ),u*zz-koqb3rough its center of masq8zs,lau)bk oh3kz*-s 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 stg f3.a,v4v yh;arheq 4e(t baxi3: x5xool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, ox.,e r;3 3vy hh4vtaq4gi5xxa( f: bear stay the same? Explain.

Two spheres look identical and have the s ,-uo o7isjpgxrgh1 -7ame mass. However, one is hollow and the other is solid. Describe an experiment to deter opj 1gi7h 7u,xs-g-romine which is which.

The angular velocity of a wheel rotatz.uybsjn30:uf.5fib p-0nq) zq sbo(ing on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at thz u bs nq-05)bupsyoi f.0.(:3 jqnzbfe top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotatin4(v1sb,2 j6 ikm eguhfg turntable. What happens if you walk toward the center6g,sv 14km(ehibu 2 jf?

A shortstop may leap into the air to catch a ball and throw it ykm(xr8vc(-nkih j3g4dz m 1-quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice tha(g1m4c(ir3h 8myzv -kjdk -nx t his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular mo ss5ou*pnw.7ad ;aawn(. 2cp qmentum, discuss why a helicopter must havs(.d ;aapw7nap s oq* nu5.wc2e more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in rada9 da4yc(d w.ly9d5,im jjk+ lians: (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 coincidence. Calc(i u:pqdw-9qwfxm)6na i2)0hu r; iu lulate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon,l)iu :9qr hp26dqu(- ;wxwm0a u )iifn as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moono.7943il300r7ymz4q uqv c)cl je ruazpx z ;h, 380,000 km from Earth. The beam diverges at an .c493u q7zlmx zv ;rziperaj3 0y ohcu0)74 lqangle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blenddp2f 8k)q oh:aer rotate at a rate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s.akoh28f:d)q p What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5ofn 9l 5jzoc0q5hy- 6.fecp, f m to another child. If the ball makes zfqlopch. e0,5j5 9ny off6 -c15.0 revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many ovp+vr(vl:- tu*/ id/ ashz6nrevolutions do the w 6u-zalv:ih * +s(t/o/vnprdvheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate its anguf+f,mndt t+ bc;z6qf;5k0zx g yxpnk)5n 7u,slar fs5 by6gn+u;dz+5k,pf tfq c,m xn7zx0nt;) kvelocity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in s-dwaz46*po7 u*z0aop*hwq z 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1h sqw0 ud *4ao6pw*oz *az7z-p.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orbit around the Sun 1zzvnk )cz9pc85( lrz   $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed o-re ;1wxv* ntuesd6 +pf 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 centri3 o30w1sof +vb dxg wymh4 ys,vdkh79+fuge rotate if a particle 7.0 cm from the axis of rotation is to experien+y7xbs303d w oh4hvym v9df1 wo ,s+kgce an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates unc8monl:;jyc (iformly about its center from 130 rpm to 28 my;cln: oc8j(0 rpm in 4.0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radiu .tqeny w.cz/-s $R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboard the Apollo spacecra n4ynq2tixpt e x0d.y4hb)e-:ft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spbyt)hxnd en yx.4:itq e2-0 p4acecraft 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 22 *e rcawxq(d3g0o )eg70 s. Through how many revolutions did it turn in this timx3)rgwaq*eg e( 7 d0coe?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpz8y5o54kh/ mnmog *vq 6noqx9m in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeed jets intjt8 6r,8nqv0f: qp lte5nng1 a whirling “human centrifuge,” which take,: tt5pj 0lq qrg nn81evf8nt6s 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 skk+a1/a./tewo4j an-4/ 8p zh frfgyaccelerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled in this tio tywk 84zffk.hnj/ a/psgr/ e+a1 -4ame?    m

  A cooling fan is turned ofjdbw-h6rh,vf h l1l4a* 0py0r f when it is running at 850rev/min It turns 1500 revolutions before it comes to a stopv r*phd4l-y1 jhrb6f0 ,w0hl a.
(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 sp9p 16, ibrjhdfeed uniformly from 95km/h to 45km/h The tires id6 pjr b,1fh9have 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 i8+ m m)lgxle *,pczvgi0 v2p.sts speed uniformly from 95km/h to 45km/h The tires haveis0,*v l+xpm 2) .8gzclm pgev a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each 6r)sk 7k4et iqpedal when climbing a hill. The peda ekq4i tk67sr)ls 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 a door 74 cm w a cg 3ohnie(nhpxc4r6.npx,s+g4;9 qide. What is the magnitude of the torque if the for x4egs;rg 9c 3n.h4o ,6pn+nc( ahpxqice 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 swnco7 9vf4,l f-nl py;hown in Fig. 8–39. Assume that a fnf 9fv-w4;lo lp n7yc,riction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, a2oxxdw v3t* h8k 19dvsre attached 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. Calcula9v vd 3*8wsotx1xh dk2te the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine req3qp,0e6ibxy-g vsqe8uire tightening to a torque of 38 bxq ei0,-p8sve36 gqy $m \cdot N$ 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 10.8x n . 2xt1sc72u1f cc3o1dwmmg-kg sphere of radius 0.648 m when the axis of rotation is thro1x gm umxs2d wn7t1c co.132cfugh its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycled/ymqav m9w3s6.mgb ,:ofe6z 8wqb5d wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. Thed s .qwz :my3/bmqvm6a9ow5d f6b8 ge, mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in a horizont c05dv 8iwtj)gal circle of radius 1.2 m. Calculagvct )8 d50iwjte
(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 pv vsd1 ;u/6mzu1yupb31w- 0fbfpbnx .otter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and the clay b6- u fu10;3nvpb1xpbszvf ym 1.w/ udis 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point object.mr 0y 8x+sylds shown in Fig. 8–43 aboutly .s80r mdxy+
(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 whose total mass is h a(lx1vg*e2- 6tjeew e ic/y2$5.3 \times10^{ -26}$ 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 spinning2gcbhot2u ,k9 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, what is the required steady fouc9, g bkt2o2hrce 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 unifo g9o-4;o 7ap+u48 eyjuyvlqtsrm cylinder with a radius of 8.50 cm and a mass of 0.580 kg. Calcgy8- y;7oq4tavu 9elo j+pu4sulate
(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 rest to z7b mkmn10p a9d 5g)pv3 $rev/s$ 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 cwdg/ 3+dn/luh+,n r9ttsp7 qsmall 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 twopdn dq7lr 9t,s n3/t+ucg/+h w children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is e7ja 5a /nkageej *ezwsfa 44-.ventually brought uniformly to rest by a frictional torque jz45e*- a a /e4a7.wkgjafnes 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 bat (8lx gwh;ph-ll 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 ,g6 ngok 1du:ba73yfv of the forearm, which rotatd7bk: yong,6g 1afv3 ues about the elbow joint under the 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 lonxwzdh,j 5a* w- f*liibao4v)8 g thin rod, as shown in Fig. 8–46jzi* i8*xa- w,w h5bdfol4 )av.
(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 mfbgp34f2m x-nasses, $m_1$ 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 acceleratog (;6fepfw z+es the hammer from rest within four full turns (revolutions) anf6(oz ;fwpeg +d releases 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 inertia ofi jdd7w9n)z8n lh83c y $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque of 2 sv/ v5gr89cgh80 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls without 2jrr t1ejy)lyo4bwj *u/ef 2,slipping down a lane areo1tu) jbyl4/rjy f2*ew, 2jt 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with sx:vy9w:5o fr6c.ew7r xq 8sgrespect to the Sun as the sum of twex:w98 5q7y xcsrfv6. rw :osgo terms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 16 pq.a2.l9x 1tcl /pwji40 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? 1wx/. la l.jpc2 t9qipAssume it is a solid cylinder.    J

  A sphere of radius 20 hb9j 1k bm8v95ujb-vi w pp5;h+7xkzk.0 cm and mass 1.80 kg starts from rest and rolls without slipping p ;b 75kmi1 p5j8juvh bwvkb9-xh+k9zdown a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

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

  A 2.30-m-long pole is balanced vertically on its tip. It starts to fall anm.q1i-;.b(uoth gasld its lower end does not slip. What will be the speed of the uppe.ubi;m qhsg1(a. t-l or end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momeiq 2(2tr a9lau31xfhu ntum of a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at al(9hifr2 2 t1 axuauq3n angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentu)t2 sn9 qhdeoco(.g.am of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotatg)cotades q.9h2. o n(ing 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 si 3yx4dbt8vrg+ i l-c, bo.mw/qde, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms tx3+co4tg/8 b.,vyqil w-dbr mo 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 of inx4h)sny*7,:4cpwdikf cxf-,z hgk c:ertia by a factor of abou :*:)cw4hhxfsp-cngz c, 47,kykfdi xt 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 wa7iv(t n jcgj4)u1*vspin rotation rate from an initial rate of 1.0 re*wiv 1v ng)4(cj 7jtuav every 2.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis through its center at cvz*8qlwzb fprg7o*s1y:f 6 9 a frequency of 1.5rev/s The wheel can be considered a uniform doc8*:*b v1fz zrqylwpg6s7f 9isk 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 the clay sticks to it?    $rev/s$

  (a) What is the angular momentum zjwq-2cm4ho8 of 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 momeko+2r(,-r 8r ei nw tvngtmdot-rv;8,ntum 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 momea /ool 5m tqspn;75;)qb,ffnynt of inertia I is dropped onto an identical disk rotating at ao7o;man5;ns) l, bfyqf pt5/qngular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.8y9aq-oumvt* 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 jv m4)zcdvbw e 2:.0kpmerry-go-round platform of ramc4 dp .v0wz) kevb2j:dius 3.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-roun qbyrj:*c d0u:itg 0o6d is rotating freely with an angular velocity of 0.8 qir0j :dgb yt6u0:c*o$rad/s$ 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, loof tzx6m; 6um8 nm1( ug9hvlz1sing 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 1;9m1m6 lvf6hnuzt ( uxgom8zangular 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 excess0vink ,zh vhy;d00o(h 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 cf pri359t;hc:f)muq $ 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 platz q5, asv2iqt7form, initially at rest, that can rotate freely without fricq ,7zsv ati25qtion. The moment of inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person sta7 y y./fbe2lmmi w;wh64 9nqsonds at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moments.7qw 6 i2yoybh/mmne;4 w9lf of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of t z+w:bek*uv/w9;sgb9sxqr1tk 3 i t*y he 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+kqyt3btsx*9k/uw1r ;bi v s gz:9* we 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 side always pmwo. s.uosm06z4o8 a 4iq y.cfaces the Earth. Determine the ratio o wq40.m.oou8my coiz .sp s4a6f the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates q m9*ui/fagbveo + 1. :9*z7qskencrxfrom rest at 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 uniform cylig +8r4 jdq(umf/x l:zr3j *hmynder of radius 0.20 m acquires a rotational ra ryuj(zx hf/mq8jrgd:4 * 3lm+te of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of masss beq/ /bp;12i2gzsi h 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 speedi pzqg21 bei/ s/h;2bs 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 thsqz1i b/rdfd*5ao 1 ;me angular speed 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 times as great,u.m *971h,sndu f g4qgla*,+da jfi 7yelqn9w were rotating at a speed of 1.0 revolution every 12 days. If it wer.94iusj7awfyu1q9, l*+ e a mhlngg d*n,fdq7e to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for 76:dmu wl nq1wit to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 140w wu6:lq7dn 1m0 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 illustrates an2 8 f7uhv33 p16f duc4moyspok $H_2O$ 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 surf3r/b;qtpkfrc 80as7)6r tbb:)jk sqface 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 ignly-o; -nt(rf9(xiwn7lqr9seyr r n)/3l i)i aore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8tr-(f(o7x l3r;9i nrely )lisna/n)wryq - 9i–21g.]

A wheel of mass M has radius R. It i jv9s4 ,nw2wvwpifkfj8ik;-9f3um y (s standing vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where wykmwskv4v 3f( pui ; wj,89jf9f- 2nih

  A bicyclist traveling with speed v=4.2m/s op7eym dtxr +ld(kka02(jr0x 4n a flat road is making a turn with a radius The forces acting on the cyclist and(7 jrl x 4k+r(tpdxa0d2e0 ykm 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 -:ys;p59 f/arwf- /e dqjjkqld,r4 blx tgb040.50-kg rock into a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 r-gxr4l;-fpj yt/:d 5s/dabq f, br0qwek l4j 9pm 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 axykh3uun0j, ,s a solid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and wiuxukn ,h ,3jy0th arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cyl q16k c+6wu.k/p*(s qvq akntlindrical plates, of pq1w(/ +qulkstqc k*k a 6n6v.mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the looped rough track of Fig8ox:) yee(4rll yng, .yqm0ap. 8–58. What is the minimum value of the vertical height h that the marble must drop if it, 0 gqelm p8y).x4ayny (ol:er 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 d-:phny vh keim6e i,3v:m(mqo++emas7o f .1oo not assume $r\ll R$ h =    (R-r)

  The tires of a car make 8; d q,n 7hsgpf1y/qqj65 revolutions as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diamet6/s7h1qnqqfp, yjdg ;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