A bicycle odometer (which measures distance traveled) is attaf*lc :witbxp*o* /s7 eched near the wheel hub an sbto7*l:cf/*ip w*exd is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point pif0tkalw r:*q(zn -,on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point a(0f:ik *lpwq-t z, rnhave radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described b22h,k2 j6 yt aqto-j; q8ssr; osn2kyqy a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger forcs+e1q,m su uj*e? 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 y9njj +j omqv/c)jv-) 5a9srt bour hands behind your head than when your arms are stretched out in front of you? A diagrj vs) m q9-t+vanc/5j9j)bojr am may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the p( ua73-h 9a embt-btwpedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a lhm t7pu9-e3a( tbba-warge front sprocket? Why?

Mammals that depend -aec3ays.fi 3,p uz b-on being able to run fast have slender lower legs with flesh and muscle concentrated hi-- i,s ucbz.fp33aay egh, 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 (Figs,zodp 0 u7oyzyv(h/+. 8–35) carry a long, narrow beam?

If the net force on x00 ffb 11sw/ xfeu:wea system is zero, is the net torque also zero? If the net torque on a system is zero, is the nets/fef0bwu0x e 1x 1w:f force zero?

Two inclines have the same height but make different angles with the 33/zcj/tzzn*q uk 2ihhorizontal. The same steel ball is rolled down each incline. On which incline will the speed of the bal3 2/z/nkut* izzc jhq3l at the bottom be greater? Explain.

Two solid spheres simuoyy; p)46 sbh *kxopo,ltaneously 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 total kinetic energy at t;k yy4 hxobs*6)po, ophe bottom?

A sphere and a cylinder have the s+ s/s3ba6v)9wfz w-xivco6ux x uom4)ame radius and the same mass. They start from rest at the top of an uwv3-f w)9xoi usc4)vm+ x s/6b6azx oincline. 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 aren /uh- vw;l6sk conserved. Yet most moving or rotating objects eventv-6nh s;kw ul/ually slow down and stop. Explain.

If there were a greatrcg4o9t6ikefqk ukv; 21 s1 *t migration of people toward the Earth’s equator, how would this affect the length of the dayt6vkei1rquok 1ct fs*; 492k g?

Can the diver of Fig. 8–29 do a somersault without having any initial rot .o3 p)fd h+p-v z(tqubebt26zation when she leaves the board?) bb3tzphp .6-qoztu (e2 fdv+

The moment of inertia of a rotating solid disk about an a boq8vn6 n3z;vxis through its center of6zvn vonbq;83 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 rotating st(/xg72gyg)jdqs uyh:dfu p6 ;ool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decreas6;hdux7/2(p uy:sgf gg d)yjqe, or stay the same? Explain.

Two spheres look identical and have the sam+.zsrfyv b:ps r:(+w fe mass. However, one is hollow and the other is solid. Dryp(f.:sv rbwz :sf++ escribe an experiment to determine which is which.

The angular velocity of a wheejgpcb qpue8x ,9e*1c 8l rotating 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 tqp,9expb uce*1c8j8g angential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rot ) 85poark:.ogsj;wwnating turntable. What happ8 wko gwr.5)npos;j a:ens if you walk toward the center?

A shortstop may leap into thek /y5z8ucu tu2e/:m xk air to catch a ball and throw it quickly. As he throws the ball, the upper part of hiukck5 8u/m2tux e/zy: s body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of ae/ p.;ei) c ykcy52ykd4yah- sngular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopt s5e-4)apdc kyc/ hy;yiy k.2eer stable.

  Express the following angles in radi2 l(c j44h dcngp;x0r;qaowc0 ans: (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 becausezlvp5*g/v v1 q of an amazing coincidence. Calculate, using the information inside the Front Coverpv1v v*lz/g q5, 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 Moon, 380,000 km from wdp*:4;q4po;zfz p.n jp jw(yk ki7-oEarth. The beam diverges at apjyd* po.in fwkjow47:k p;zzp 4(-q ;n angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. When the m5fnaq 03sn/c7bt7icsotor is turned off during operation, the blades slow to 75tb7qcnnfa si s03c/rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a,c(lv7n qx ug2 level floor 3.5 m to another child. If the ball makes 15.0,un (qv7l xcg2 revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many *z)akpac2 y+ilgs4b;/i ns/ srevolutions do the wheels make;+ 4 a/p)giszaksyic2/nlsb *?    $rev$

  (a) A grinding wheel 0.35 m in di1*.zhkv vvs a(m c02a12yimfavh j6/ n2 net,pameter rotates at 2500 rpm. Calculate its angulf26mvh(0a2 js 1*vi h n/peka2.amy,vztc n1var 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 md/n aah-ixmm 4 1v4-wuakes one complete revolution in 4.0 s (Fig. 8–38). (a) What is the li maai-nw m44h vx1/du-near speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orbit aroun ud7 m zq0fh3p36ldpj/d the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a po1e48z hruej-x ,aj1 2i)luqre int
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.0 cm from the axis of;zux:gu2e+vyd n8 o3m rotation is to experi vuzd+2 ox:egmn;83 uyence an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly90ptblrlw+7- x4 zsu l about its center from 130 rpm to 280 rpm in l7 +r9ux-l wsb0ptzl44.0 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 radiu1/gedd 4k cgfjfsk2/e7f9 l6rs $R_1$ 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 M.d0qe*r p6 kvfoon, astronauts 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-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determinedvkp6 .eq *f0r
(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 rem*15aos 2jfnqst to 15,000 rpm in 220 s. Through how many revoa j*fnqm2s5o1 lutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500*ben:fb/ t9:n6 e hymt* .hbxl 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 flying highspeed jets in a whirlinl nlv-mp 66gj0g “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before renlplm6jv6g0- aching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 360 rpm k 1wim xxustot6 qul+m*960o) in 6.5 s. How far m1o qo)xuus*it0 t kx69w+ ml6will a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned rf(l5u5unq)5 s4:npe1f b daboff when it is running at 850rev/min It turns 1500 revolutions before it comes to a sto5 b(npbefa)u1s nl5fu4: qdr5p.
(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 rel i+2iiws0s32s y ./y a6rvuq3cpsre9duces its speed uniformly from 95km/h to 45km/h The tires have a diameter o+ep6 3w02y2 . 3suilssry vir/is9caqf 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed unifor3e n0bm /mz .m2fa;vydmly from 95km/h to 45km/h The tires have a diameter of 0.80 m.2nm m.e; z0a /mvb3fdy
(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 x3+ alvk- rb: y6psg2leach pedal when climbing a hill. The pedav3:+x6l-lb r ska2pyg 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 wide. bo y .(fyng(92pg 1iuhWhat is the magnitude of t h9fig p2o(ygb. uy(1nhe 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 t8ir1d 2ppr1d vc hh,)o3ua5qkhe axle of the wheel shown in Fig. 8–39. Assume that a friction torc 5k uar8p13vq1)ho2hp, ddri que of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to t z4sy9lxod;l: he ends of a massless rod which pivots as shown in Fig. 8–40. yldx; zl9o:4sInitially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening tog 4qb* 24p wd26o72awxgixrme8/x xc v a torque of 38 24xqci v 2w*x/x47edr2mwgg obx6ap8$m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of radius 0.)5z.qawlr1dp 648 m when the axis of rotation is through its lraz w1d5q) .pcenter.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm iuz*u )n6k+ -o ;0+feyeuwoutmn diameter. The rim and te u 0o+) -uk6ouyuf*tw+men z;ire 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 d.0vzq i0, o 9rvzzhh;1yem+bthe end of a thin, light rod is rotated in a horizontal circle ofy bhohzi,d m vz .q1e+z;0v09r radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wh1xoyf(lq;ky0 5a/hji;e- m tv eel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and the clay f ;k ato; jmyl5q/v0e-x(yi1his 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 gert 3e1f+g1tinertia of the array of point objects shown in Fig. 8–43er31 tge g1+tf 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 a 3opc) ;iyw+la ,iqmz-j sn1)ensm0 f/7*wfostoms 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 cylindricalt8g-u 0ao-nhf 6j/ vse satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the s-n0a6/sgv u et -8hofjatellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 cm and a mass .l 8*nig zqho,of 0.580 kil *ngh.oz,q8g. Calculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, me3e bzlq(qc4yob./; 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 sr0px2cgbh;3 mocu b;*mall hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (e03b2 gc;x;cbhrp* omuach 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 eventually brought cp5jl3;8k77i jw u4kkb2cf p ouniformly to rest by a frictionakp7k iu 24pkc73oc5 ljbf 8j;wl torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelera evf a0j h075ewli5vs4tes a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the f4v wx; 4dr2)xor5opyaction of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball wro;rxydxv 2p 54f)o4 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 rod, as shown in Fiph b3mapz8j em,8 wf57f ga;9f:kte;ug. 8–46.kmubg;ete7:aj f85za38 pp w;, hff9m
(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 con: 6d b(pgslfy :+pcb7bsists of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates )zwlo3 c1b(ic,6lbd 3fyt6 o :po rq3ythe hammer from rest within four full turns (revol z31bf)b3w: 6o, r yilctply3qocdo6(utions) and 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 iner :92i*yes ,k c0o3zeyzl3y mogtia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque ozkpf * . 4dwtob+/3;uhvmjgnu */bdw- f 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 9.0yd rfv(:3r)+z hin uubm86 +f/bhzn)i cm rolls without slipping down a lane d))ry(mn6u:i+ zrivbbf3 u /hnhz+f 8at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the st * yy-ge1q 3:lj5r dbm*kg7fgzn(i)xum of two term *3 edq7 tyn:g mr)yzgx(jl1fi*kbg-5s,
(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 radius of :6jnry)rf la d7(k2is7.50 m. How much net work is required to accelerate it from rest to a rr :l )y7rjfs(d2kna6iotation rate of 1.00 revolution 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 witho/hq;lqovpg/xbm5 ), ftc; a)iut sli; c/x ofhg)bqp5t ;m)/a,iq vlpping 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 starts to+hxfnpm9-+c ,k u vzv* 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 g*mn z u+k,+chv-x fpv9round? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momenubda2 *yzcz1.nso b, krf)d*4tum of a 0.210-kg ball rotating on the end of a thin string in a circle of kzyan*dsd1*),br2. c zfu4boradius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical,d:(jv br +obk .sniffh8c39m grinding wheel of radius 18 (d83v,hf+b: 9oc r fmsi. kjbncm 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 qt, -qn +s-cdbat a rate of 1.3rev/s If he raises his arms to a horizontal s- -n ,qct+bdqposition, 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 zi0jb( )ircsgt;*t m(moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If s;btc *sii(tj(mz0 gr)he 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 6ye*bxb 85z(vry azs 7rate of 1.0 rev every 2.0 s to a final rate of 3yz6 e sr8*bbyzv(a7x 5 $rev/s$ 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 fred3he- q(:0.hcc/l aiaua :apuquency 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 shaph-qaac/ ea:ch uaid: .l 0p3u(ed as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spcuw/xg i4c3y ;inning 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 mqa8eo2 0s i .l sidhfhq*d1l1(omentum of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is dropped onto av7w r.0tfz (vqn identical disk rotating at angulatwvrv7 qf .z(0r speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4 +v 1fj6c7, xarzhk(u k$rev/s$ 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 mr9 4o/-ugqm lxerry-go-round platform of -9 um4qo l/xgrradius 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-round is rotatipk ze ,lz,;7(rha fl:xng freely with an angular velocity of 0.8(:;phlx7a r zfz,l,e k $rad/s$ 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 eventualng*;x fqww c fl2id16,ly collapses into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest in2 f* 6gnxldwi,c; 1wfqteger)$\approx$    $rad/s$ (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 winda.jt(o5 xdy82md d go/s 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 j 8rv .rpb0vp;novvlri66d 1($ 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 platfor h02k8 b t-o,xwsw+7jpcv2y9yap5tl bm, initially at rest, that can rotate freely without friction. The moment bv,tp 8os c72hax0t95- j kp2yl wby+wof 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 stands at the edgsrcc*q8 b0o5 fe of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia of 1* qcs5of08rc b700 $kg \cdot m^2$ 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 enwo ui(j/jw8 j/-sezy5s n k79ld of the rope lying on the top edge of the spool. A person grabs the end of the rope and 7wn kj 8j(yj /w/5l esu-isoz9walks a distance L, holding 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 thaty r1/jn+ hop; z6y6peix ic2m; the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the lattey2mx 1n;i6 zpr+ eijy 6cpo/h;r case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate +* -zj2fhrrsa 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 uno5s dng (6nxxa003nvnb a4.b lwy:xlu :p30bpiform cylinder of radius 0.20 m acquires a rotational rate nun.bswx 6n4 05l(a300xdabpxgv:bnyl :p3 oof 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 mass 0.050 kg and ry/ qm1(a,egr 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. 1rya,(/g erqm0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, hpqh/0v ks)9vkhr2h . kow is the 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, werem3zor;a/mi7z rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all ti;m/izz3oa 7r mmes, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution aut 2c(c 6m4jqcyj.syp t1omobile is for it to use energy stored in a heavy ro6csyq1(pjc tjm4 2c. ytating 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 illustrarf q;bv).kn8d tes an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) so bh p60.*8luu5oey jis rolling on a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., we ignorhu2r 5ga++7 ltjbrbj v/w0; le8tq9iwe friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and h;2r8u7l j jtqr e+v5wg9wl0/a t ibb+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 a,t-m6 gy:a whradius R. It is standing vertically on the floor, and we want t whg:-amy ,at6o 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 h

  A bicyclist traveling wiv.llde93:v5 ru6diq-f8a j obth speed v=4.2m/s on a flat road is making a turn with a radiu9fdovqe-il:r3 aul685 jd . bvs The 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 rock intucfs c s-i .l/bw1j7eewm2r3 :o a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle abo:erl72suc es c w-i .mbjfw3/1ve his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thm9rwhmom brm 4. -wnm t3b2i/7in rods, making reasonable estimates f3bmm4rht9-m w/o.72nm wbimror 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 which consists of two cylindrical plates, ofq l9-xjhjha(rd:x9r7a9xrl 9:h(dh- rjq7 xj 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 ro4ivxsyf ev zl.icek016- o35vugh 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 highest point of the lixeeys zvoc l iv 516-v4.03fkoop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but gmc c q-8,ut,, 9p+q nbxdamu-do not assume $r\ll R$ h =    (R-r)

  The tires of a car mknp(1o ;wg 2a-79 jfe)lvzv smake 85 revolutions as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) Wen (wfv9jlz) - ;1k7sag2vopmhat 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