A bicycle odometer (which measures distance traveled) is attached near w eqv cfvwnw;7d3;:q4the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?74 3n:q q;e v;wcwwfdv

Suppose a disk rotates at constant angular velocity. Does a oveom/ bmnwl :) 3;+:qzc sapate;l0*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 eithe:z0 qala:p+ so3ble*; t)/e m;vwnco mr component of linear acceleration change?

Could a nonrigid body be described by a single value a sj3 4(hfem)fof the angular velocity $\omega$ Explain.

Can a small force ever exert a greater t q 9 bhp:cs+bwzugj .gp4w/.:dorque 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 wjr ;+wbwl)nzw: 43vqm eu;uxb 2gl:-with your hands behind your head than when your arms are stretched out in fro;:)wwmrulzj ;:vgex l32b+-wnbw u4qnt of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and thref-r ug onp-h13e at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear spu- 1hpog rf3n-rocket? 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 fastf)5)qp3 ywlu1jm3k: c/yj+suu4hhgc have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantayhlk) 3quuhyc):/j+fsjwmc5 p14g 3ugeous.

Why do tightrope walkers (Fig. 8–35) carry a long, nw 6x0n:+ivwxu arrow beam?

If the net force on a systf t1b g1xebg9* wb:je4em is zero, is the net torque also zero? If the net torque on a system is ze1 xgwbee g94*1jbtf:bro, is the net force zero?

Two inclines have the same height but make dix9q2s jji6t1rfferent 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?j qsi 21rtjx69 Explain.

Two solid spheres simultaneously start rolling (from rest) down an incline. O81(fbqgedr;a ap5f;0sjvk 5dne sphere has twice thrs0ep1a k f8d qg;vajd55fb(;e 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 m1ofcwr 3yn 7 y- ;ix,wb/r;uhoass. They start from rest at the top of an in,o;n 3w7 o;fc/h-byrwu rix1y cline. 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 angular momentum are conserved. Yet most movin /wg;0f8zb oxq.z,mo vg or rotatinqw ,zbf.vo8/ gxo;zm0g objects eventually slow down and stop. Explain.

If there were a great migration of people toward the Ear3;pu-azq -gfltx p fbsv;-d)) th’s equator, how would this affect the lengt-d qaz;lgvbf-;tu f3)s)-pxph of the day?

Can the diver of Fig. 8–29 do a somersaultsg8h 0 t pk.6t56pfxv vsipxnf* qki* o;6nm)7 without having any initial rotation when she leaves thsvn6 ;i8 xkxpf*0o6nm v qttfi6.p*7h)ksp5ge board?

The moment of inertia ofod - wcsdx .hx:r9px5 8boy-b* mn(2qj a rotating solid disk about an axis through its center of2xpn(x xo-.d58bd y:s wjhr-c9b* moq mass 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 rotatr ikd(* ay4pidkj5qca-8--o vk c05vring stool holding a 2-kg mass in each outstretchdi c5i5*apv j4k ckr--d- y ko0(arv8qed hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identic; :hdztr)t 5pgal and have the same mass. However, one is hollow and the other is solid.tdr:pt 5g)zh; Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points .w1x fnj:9u fhwest. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the fu1f9hnwj: x. 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 edg)+:+ti7.hr qlnf qjwe e of a large freely rotating turntable. What happens if you walk toward the center? wlhfn +q+):tq.jie r7

A shortstop may leap into +2avpyx t)td(w9nvg6 the air to catch 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 r)x pav g wy+2vd9t(t6notate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discui5 shdqq/ rel:7i77li 3j pl 5yzehfj864nus0ss why a helicopter must have more than one rprz:f5lel e7j /ihi6lq5uydihjns7430 7s q8otor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angl+ fmn+ ,qftti4mwt2;pk r.f4pes in radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincidence. Calculatedbh,z -+kwikq*(k6 ik, using the information inside the Front Cover, the angular diameters (in radians) of the Sun an(zd-,kk*i wh qk 6ki+bd the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beam 3l *tbm5/b ksqdiverges at an 3b/bq ksl*tm5angle $\theta$ (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 moto2i vetl yt1mw;:)su4 0 :lssowr is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the bladi wl2:ysw:lte;om 1)usvts4 0 es slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another qu-u 6hhxo i2oua6y(5 child. If the ball makes 15.0 revolutqu 6h26yh-oouxia u( 5ions, what is its diameter?    m

  A bicycle with tires 68 cm )4,j1c7mq y dvy8 ouerin diameter travels 8.0 km. How many revolutions do the wheels mamq7eor,18y jucv4y) d ke?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotatesh(r*.*lo yp9 pt1thu l at 2500 rpm. Calculate its angul*typ*lp(ulh 1trh.9 o ar velocity 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 4.0 s yg68s uggu8jfjz1as ;r;t4*h(Fig. 8–38). (a)*s 6s 4 ggu;ayf1h8 jtr;8jzug What 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 (a) r xyfscd8q .. 6:nbqp*ef0sl,y7hvw 5 in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a p n9 pkxftyfq7 )u87nxyxu-1e-oint
(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 ifrypz5;gzb mh8y g1d blv0lzxh; )f)95 a particle 7.0 cm from the axis of rotation is to experh5hzyx;bbf0vpdr;)lyz)95g18 ml g z ience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its ce sd,hc+ fiht)t,s862 r44k/zeel jecdnter from 130 rpm to 280 rpmh ,e4j+tec6 fss 2ed48c)k lzhi,tr/d 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 radiuddb7a,c;fi 03iu)v3sz w o,tvs $R_1$ 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 aboarx(o cznum:;a/t g*mp( d the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accuo;n ((paxg:/ m*zmctelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpm in 220 s. Throujl8 ..2vry egw+a*.:sqwtmf a gh how many :lj 8g+yawsmf .raeq*.wv t.2revolutions did it turn in this time?    $rev$

  An automobile engine st8ip k 7 pr6nnwm.+9uplows 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 for the stresses of flyingp1/ fn 9,iamlu/qikt9 v;--xnv wl5qmpd-zm, highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reachiv qmi919za,m-tx n kmi dfln,-5u/l/qwp;-pv ng 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 2d *xtpt0ne,yrlzx1(01un ox5 40 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled in tp z1dyx1n0tt5 ,0xnu*e r(xlohis time?    m

  A cooling fan is turned off when it is(tv 4g9,ngdq rz :i0os running at 850rev/min It turns 1500 revolutiono:9z(0 t,ng rqd4igv ss 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 speed um 95udz g6yn/yniformly from 9dgy /u96nym 5z5km/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 reduces its spe8/3y +b6nqsgzsk8j jw ed uniformly from 95km/h to 45km/h The tires have a diameter of 0.80 m. /3q bwjzj8ss8+6yk gn
(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 al*m(e8v roqt ovo ub,..l her weight on each pedal when climbing a hill. The pedals rotate in a circle of radiqbo,v.(vu .8*rmote ous 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 2 uarpktxsvc+ynx6aj87g6 .. of 55 N on the end of a door 74 cm wide. What is the magnitude of the torque if the forckx66r pg.uya.+v2 nsca8tx j7e 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 e3cr5dkiw z q8lrm7+dwa2u5+ about the axle of the wheel shown in Fig. 8–39. Assume that a fr82r3rq+d+we dlm c7w5 zi5uka iction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mas0r pxje15p2ely-3codc:v nj *s m, are attached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is hp yj 0p:rn5v-1e*djco3 elxc 2eld 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 require tkv3*.lo ndh- s:au,h nightening to a torque of 38 :3v* nk-d.,usolhnha $m \cdot N$ 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 dsjep o 3ye/v3),xl2aa 10.8-kg sphere of radius 0.648 m when the axis of rotation is thro3)lx,/3 e ds vypeja2ough its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diames6 o3 o8 ;kaydhiur1i,ter. The rim ank6ao8o1yi,dr;3ih u sd tire have a combined mass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a th(h6 1ye vop2xcg+vd2py4r r9 ein, light rod is rotated in a horizontal circle of radius 1.(y194rp+r y2gc6hv oeepdxv2 2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wh.k- yqkg i)tb9eel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and k)-ki9.q ygbtthe 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 iner.3kihj13:kn s7i mwtf tia of the array of point objects shown in Fig. 8–43 kwi 3kh.:i73tjsmnf 1 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 consists7;psl tk: lz:gvy sf, fu3n/5n of two oxygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the cow;difn s8 0+rjrrect 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 + 8iwdjr;nsf 0the 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 cmq-364 qgmk( au9zs/ /yvr+a(stkc ejh and a mass of 0.580 kg. Calculate s/-tq4a/ (3azyjg s+kmeq r(hu6vk c9
(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 frouul, i -1a*hq5tbdyr944wnf cm rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven merry-go-round and is eqfbw:538/fw n 7xvt-*xqgwg 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 f/wqg8gwq 3nv tb5-x ffe*xw7 :rictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is sh 9z a:nhd:(s yc,0gqesut off and is eventually brought uniformly tszyhg 9sq ce ad:(:0,no 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. 8–45 accelerates a 3.6-kg bat-8p anif(nrg0ab m4 8ll at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the action of thw u7vq1 +aeqkle - 4b4abs7p)xe forearm, which r-u1lp7 v+ b)eq es awbax74qk4otates 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 bqx4 mwm- 1.p nmq;pd6ke considered a long thin rod, as shown in Fig. 8–46. m mm-w pxk4nq1 pq6;.d
(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)rnc f3h6.h )arv. vi2eqgfx7 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 rest within four full turns ( 7hnaycp+9*kf revolutions) and releases it at a spa pc+*fn9kyh7eed 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 of1hd*h,qj5m zx $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque of 280it alxjfcj *rk*, d;gf8zd)rtqg ib4); mr;,3 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0i*14ebk. 6qex3won lt 88yczo cm rolls without slipping down a lane at 3.qkewb8* 384ox 6 tcey1.o nilz3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the v yiqzuji:7 m9x(:pw/o i;w55u-d osuSun as the sum of two terms, v5x yu uq :(5 -wizoji:9/7p;iwmsudo
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radiusa +-d a7qj sg6n/bq9jckl-y i: of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revol9b q/: -kjgl6 dq7+y-isn jaacution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls without ig04sn)393 wee :tox8ka s5u j44uqbc kil;jqslipping down a 30.0 qu sb443kj3ecqnu:508 g)wxjalksi4;o 9eti $^{\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 andno y,8g hg:6of its lower end doeo :6g,h8on yfgs not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotajk6+t73 uc8vfh 5kylr5: hmi hting on the end of a thin string in a circle of radius 1.10 m at an angular sl6fk8r :hjvmutky 75ih3 h+c5peed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grinding wheelatysq02pa, ffx ;+n)kp-4cbz of radius 18 cm when rotating at 15+qxks t ba 0ff)a z-yp2p;n,4c00 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 r4kr 0kv1jnl(w, v; xvkotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig.n kv,jw1l;x0v(k rkv4 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 mom d-uxtrja2 x.f9g7d/ment of inertia by a factor of about 3.5 when changing from the straight position to the tuck posi7 d9f-mxdu2jg ta./ rxtion. 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 initp.3.gtzmw je hois8dx / 5vq,;yghb (8ial rate of 1.0 rev every 2.0 s to g8pq;gixtyw5 e(ho/zs. vdb8.j,3 mh 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 a h4poxpnfk u t-xw/4 90frequency 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, o4x n4/ -p9optxhfw0 kunto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinning a klsv9hcg- b50 o)86kpbt, djqt 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum of the Ear k+wfy9g4+k2) sxga poth
(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 of iner-,0o kyswa9fqtia I is dropped onto an identical disk rotating at a s9qwok0,y-afngular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2. ptyhm- ,hzj:)ap43fsl. y8e o4 $rev/s$ 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 merry-go-round platfor )j:xlv(3zkvxgi5yrf(g(te gs* +az* m of radius 3.0 m and moment of inertia 920 5l yeftsv(xkzxg i)+:*(az3gg( v*j r$kg \cdot m^2$ 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 a q53d0)sem qd9 gj tm8e8wp8jjn angular velocity of 0.8gwj5e 8s qdm89tj)p j8 em0q3d $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, losing about hv kb9tc/w h86nalf its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would thvnthc/bw6 k89 e 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 utaruo9* +xx k)7v,zgwinds in excess 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 bb;9i; ps lrc1$ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, that can rotate freely withoi+96ba j/p*p d vv*r yjtb1px9ut friction. The moment of inertia of the personx6p+vp yt9jbvipdbr9a1**j/ plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diamestc;a1:b1s+ s *qmwui ter merry-go-round turntable that is mounted on frictionless bearings and has a moment of iner*s1 tc+;ba 1isq :wsumtia 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 rop(-8i3ifdwyec e rolls on the ground with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, id f 8-wyi3(ceholding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces the Ew)q wonbu+2s2v )+ kaqarth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, tre2uq)ovka2 wbqw+ +sn)at the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from resthanwc)8gc09,:yp/ qeh.n9 :5wc2iz pxvg piz 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 cylinder of radiust mlgnyy. civtu2+ +cm lw7ehe )2e5cke,621d 0.20 m acquires a rotatiemg.v +ec 2,et+lel2m kc1 wdunyyi5c 6h) 7t2onal rate 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.0y3i qhoo2 bq of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is release3byqqiho0 .o2 d from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of the rear -baoz-8 u n grnp333ydwheel ($\omega_R$) 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 wit 9-(f5)j hpur 0cyupomgg(r3wh mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the ) yjf mrgpwogcp5h0(u3ru(9 - 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 ,cs/(k .7jn zp+uaqg sautomobile is for it to use energy stored in uap gzs,+.j/ck7( snqa 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 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 z n5:*a*htwnib0e y 1ian $H_2O$ 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 a9e qk.kbkzbx( b)1 31z.rv tul 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 fricti c /bufatwzj6wpv - tknw;e/)ab4 +8a*on) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and ;n+ t) zac4-jwt fu/pa8 k/6vwbeawb*then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on the floor, and we x35 rok)t0 xo-2uc8,rwdqfxlwant to exert a horizontal force F at its axle so that it will climb ao 05,x - qlx d8rwurkcf)2xo3t step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat roadi b rrr-.li1h) is making a turn with a radius The )b.i1hr i rrl-forces acting on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg roq,(an 1qx,pyg ck 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 1,(q,x gpnyaqof 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her armssk4s y1o2zxqe s2 c*v9 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 with arms held tightly against her bodyo92kv4 1sq2*x s sezcy.   

  You are designing a clutch assembly which consists ogcd.qaznn6d- 1 . aq(ff two cylindrical plates, of mass.ndc -aa.qd(1zq gfn6 $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 loope2e- ngvfx: :z3ykpmd* d rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the hi*p mzfe2:d k-3:nvxg yghest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assume 7kefjp 5jhd1jdx8qga 49pf e2-w1 p)i$r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions apg9vm t56cim (s the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of tic mgp65mv (90.90 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