A bicycle odometer (whicn4iu c98iub1*l j bt*zsp+2hg: g(phfh measures distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happjgpp1+ :hni t*4gc2h*9 u u sfb(izlb8ens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant008o .ujfmn3 dmorm3l/r que1 angular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the poi.n1or8m0j0r3oel q 3 u/mdmfunt have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be descri lnmf57hm8-.py ylgw11w z r3qbed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque thaz1n hjv85 (etfn 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 yjn*eor 7g 6z9four hands behind your head than when your arms are stretched out in front of you? A diagram may help you to ansg *o67 9fnrjzewer this.

A 21-speed bicycle has seven sprockets at the rear wheel and v (w:pxyfpr8)atoasj6*/ s5a three at the pedal cra a58tx*)jp/y (asrfp s:ovw6anks. 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 large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower legs with max 1df :ijmd)c -yvo 2l-x9d)flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distributioc ) 9maldfd:-dxim2 vy1- )xjon of mass is advantageous.

Why do tightrope walkeysuy)r sxgb5k +3s-w* 3pjbs5 rs (Fig. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero? If the iu8dyed 9tq s1/;pki1net torque on a system is zero, is the net fid11ukitep y ;q s98/dorce zero?

Two inclines have the same height but make diu6-.q*c; p loditi+pw fferent angles with the horizontal. T. iq dtou-wc*p+pli;6he same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from restp9r*jd r: 3vxa) 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 p v9a:dxrjr*3kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. They start fr (f f..7t+zv8i hv llo6:peqzsom rest at the top of atz+e.vhf7 8sfipl v6olz:q. (n 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 angular momentum are consedtp00qysl1l 9 r::t3l b ow8anrved. Yet most moving or rotating objects eventually slow down ::tw03byp0t9 rlls 1aq8o ndl and stop. Explain.

If there were a great migration of people toward the Earuci )dca0qvg* 3.ijrbof. +c k;x)1dhth’s equator, how would this afi)cvj.u0 choagdf +;r qcid13* .kbx)fect the length of the day?

Can the diver of Fig. 8–29 do a somersault wi 7lc w p:tv3,x+u0vxdcthout having any initial rotation when she leaves tvcxvw:0 7xp3u+d ,lc the board?

The moment of inertia of a rotating solid disk about an axis through its cente .ie -6e-l rl:9juoijxr of e69r:ji-j x-eilul o.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 1k)uderzdlc52y ,bu, eu1ih 45v5oy nrotating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity cy ,)ihu51z 5deornbdey,lv4k21u5uincrease, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However,1r01jo5 3 ,z7tio jf,a/x /rb jgcuroj one is hollow and the othe,37 0jcojz5/ r1i/rogarjj ou, 1ft bxr is solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotatin/be p (rym 6f6n+bbodj,scx. 2y,zfd)g on a horizontal axle points west. In what direct2s f6b+,bb( )dxdyp o 6rzfn,em.c/ yjion is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a largl8gwr av g9,9lwtw:c4e freely rotating turntable. What haw,lglg4 w8 9tvwa 9:crppens if you walk toward the center?

A shortstop may leap into the air to catch a ball and throw it quickly. ga fvj .611re q31dduq7h6k *l qu:imdAs he throws the ball, the upper part ofdud.vh36aql:u k 1q7q 1m1ieg6*r j df his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discuss why a helidvyh a ;wj+8t:kvqma6) v,4hgcopter must have more than one rotor (or propeller). Discuss one or more ways the6vh+48,tm;)q:dv wha yva g jk second propeller can operate to keep the helicopter stable.

  Express the following angles in rclmhj3cn.f*k ) dww:j: /gfsvh)z/.n adians: (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 becausy* 8 gzdatu08ce*k(bc e of an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radi(0 t k8*dyazecubg c8*ans) 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 Earth. The beam diverges o cpmyu(, b03lat an 3lcym0pbo( u ,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 motor is turnek vka+max -10g,jt(3 iuwdg.cb0 .ppad ofkpm 01gv0 3gkad.tc+-,ip jbw.xau a(f during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on+m1mw 1kyzhc1 jqpmd6.jc2( h a level floor 3.5 m to another child. If the ball makes 15.0 revolutiomh 1z.c ch( d k+qmj11wj2p6myns, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 w .uwla-:7zvrey r2egxn/;x 2 km. How many revolutions do the wheels vl:2.re-7zxux;awwey r2gn/ make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2zb0xuwzdr(y kv+105e/( dhln 500 rpm. Calculate its angularlvz(1k w0yxd0+n/zuehd b (5r 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 41 ;ewv*cyungu gk60f6.0 s (Fig. 8–38). (a) What is the linear speed of a 0gu1y n ku6*fgvewc;6child 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) ixn+/:xa6ee n8zx l nw76c,5ntrf2u oyn its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear spe2d*a 7i yedst:ed of a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.0 ge 3 et ;)ynhf u+uqv,+ev+c0rcm from the axis of ro u;3eh)re+evtyvf gqu c+0n +,tation is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpm to 2 jssok;p*1-xt e t6:sv80 rpm in;ptxt*:k6 e sos1 -svj 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 18afnphcqi7 o2 :my:7hw sg.web gc70k.xk2ns $R_1$ 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*cjuecia)xe0 *2zj (w spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerate ei0xae*jwzc*j (u)2 cd 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 re25cf ju 3ymlh4st to 15,000 rpm in 220 s. Through how many revolutions d 5yh4 l2mufcj3id it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rr;da wc6 9f t5b34j ynxoclxtlb;h*:0pm 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 h ac k3he+gq +:35oxfllighspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before g3+a :fcl3oq 5lh+k xereaching 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 acceleratessn/ n sm+ed*x2nm5mc/ 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 tim52nm/scm+nse/d*nxm e?    m

  A cooling fan is turned off when it is running at3 pghgs8lm)nl :bz z)l. cb9t) 850rev/min It turns 1500 revolutions before i.lg9s)t )zz mhlgb:nl c) p83bt 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 aseqd 9s ;2j5mjf the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.8025;dj sfm qe9j 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 ct y 1xev*hc-vag: 67aspeed uniformly from 95km/h to 45km/h The tires have a diameter of 0.80v-:cg 6vexc a17haty* 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 hercx;at5 n6atw0 weight on each pedal when climbing a hill. The pedals rotate in a circle of ra;t wta0a6n5xc dius 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 on6x;8w 5arvzs z the end of a door 74 cm wide. What is the magnitude of the torque if zzv8w 6 a;rxs5the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fig. 8–gfo 3+7w 1ogy jzbmj3rac-9p( ;*n dkw39. Assume thf(gpg;a*n w3k-zcj +ymo1do 7rbj 39 wat a friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are v1+z70 ztdhm)9xrw n fattached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizo1nrv0d h w7fz9+x)m ztntal 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 to a to:/i:brxzl1 rg; ,er4j;wk qum3,gv;yp u 7lpnrque of 38g:k7x l,mn:4g/;wp;bvze 3q1iuj;r r,up lyr $m \cdot N$ 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 oi94mllgzbr5* n 6a6gkdqoh57f radius 0.648 m when the axis of rotation isg56g5 ln946 kma*7oihlz dqbr through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicyclcj 4zk.pes -)qe wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignored)q4kj -. pcesz (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotateikgai92j(f n47g k5q68gh kkdd in a horizontal circle of radius 1.2k kq8ing 4gk(96ga k57fh j2id 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 wheel rotating at constant+3 sihe fel:yytc 2i2; angular speed (Fig. 8–42). The friction forcc ;2iey+3fles iy2: hte between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment ) ab-xy ka+tj;37j whibf hw6m ,wj:l(of inertia of the array of point objects shown in Fig. 8–43 abo kt)wbh+7-(,w;ja ila j3xmhwjf :y6b ut
(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 at7(1v-dtuyh7ink9cs u oms 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 correct rate,9m(8fu* knb sv engine9fb 8svm k*(nuers 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 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 raf4*2ic)k pz) znx6cq pdius of 8.50 cm and a mass of 0.p)2z 4)cf6*zcnqk ix p580 kg. 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 batt6srji)xpjo6 i/ n t;9v-2 hip, accelerating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-drivuc1y 5ho97j-cw.qfjp.pc c /nen 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 massjc.hjc7u cnq/ 9.-5 pf oyp1cw of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is evenycok u-jbm:r1x 6v 4*ftually brought uniformly to rest by a fricrv*6jk:4-ybufcoxm1 tional 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. n1w3kls2n;ldmb aep u1 )/a3ogd5k 4t 8–45 accelerates a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrgh+ mhpdn:2-6 ;g urdmown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is-d+6: mg;un dg mh2hpr 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, u;4s vqcavecv 02ra3y ,*5dudas shown in Fig. 8–46; dvqvccvdsuy0ae2ar , 35u4 *.
(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 consdvl5 0uustew2:t 7j, d (uiprvj.q *+zists of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer froi1k6xcrbg ,y3gk tssu1u(c h;6 eq3b-m rest within four full turns (revolutions) and releas3qhrgb6 ,tk11s3 (ixbc6yk u-s c;gue es 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 of 9n4+xzrzcx:n (xe t8g$3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque ohx-( h5q,ahjsdk k.7mf 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0.qo:3e tx,a3c )v3 gn(ki ofsj cm rolls without slipping down a lane3tg(ajo,s. :vc)ef i 33onk qx at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy oyie++svqv a+(90yflacr:mnv ( r*ql 4f the Earth with respect to the Sun as the sum of two termsm4y0fa*+qv++ clr :vyr qe sn(lv a9(i,
(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 vy 2y(gnl:yr5e3nkd)a mass of 1640 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.(: ydlgyey nnvk)2r3500 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from dwor4x9qb z(g7 tjdj9 ,4 f6dvrest and rolls without slddzdx79qb wj(vf9oj64, gt4r ipping 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 starwwi-aisb8 y- :ts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of en-- y wabwsi:8iergy.]    $m/s$

  What is the angular momentum of a 0.2g+s : u4 (u,vukim5ymg10-kg ball rotating on the end of a thin string in a ci,+k(guuv s4muyi m:g 5rcle of radius 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 grindinj +(i8;ju) 1fouqyt vb9)ydgfg wheel of rjtyf y98)i v u+;1gobqf(j)d uadius 18 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at hzy,24t8t1j8f ;*prixe gnz hevumj9 0is side, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the sp4j9xgtfm in*evzr80 tezj,ph28y u;1eed 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 re3s;q 9weoy,idduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations i,ise 3 yoq;wd9n 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 e buiyx rm61g9 yp2m6-her spin rotation rate from an initial rate of 1.0 rev every 2.0 s ip9 mrmeg-26yy61bux 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 th8r d 0nj ze)q lro-cdfvj(;xl/- ma;z0rough its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the vr(n /mcfrd8d;j -la;0o)qz exzl- 0jcenter of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular mo3ous3nuh -,e)yjgv)2 mudtn 8nrb/7w pi;ad,mentum 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 momen xa:ow3qt v-rph0o 67ptum 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 an o0r6.)lh -ekkgwcjt pb:j-m 8zmb x9. identical disk rotating at angularbjz-:6reok9lj .t-p w m.h0bm8)g xck speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4rm qqqtpjv. nqa717 xsjq)-)*a x9-hs9+ohmu $rev/s$ 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 cuja 5kr8 /0bm*ye*sf9 tf+ooi enter of a rotating merry-go-round platform of radius 3.0 m and moment o*o f*5 uk8tsmoe a/b+fj9 i0ryf 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 ro2 , dw , 4wauxnd+i4dr103u/nomdxwvstating freely with an angular velocity of 0.8xvuumdo3sn ,ddw/w,2ir 4w and+14x0 $rad/s$ 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 half ith27 9/ohiwa1m dwo6 *4rnadhjs 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 ratejw r/awm7iod16a 2h4*d hhn o9 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 exr;3/ qyv hieg-n x83/by xu;vfcess 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 l5sc.5bp;z rn k4mfps08q1dh$ 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,g c0e :-)qg74d eb; mz,kb)aaf ti*xkx initially at rest, that can rotate freely without friction. The moment of inertia of the person plus the platform ise *a xd) gmt7f kx):,;-bgk0e cqza4bi $I_P$ 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 diameter merry-go-round tu zala8 fc r.nmihf)6r- wh(;2drntable that is mounted on frictionless .hfhr)z a-c8aw2 i( mf6;nrld bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of the rope lyin 93b- aigt yzvxyg 0x5-id7b2kg on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spyz 3iag5b- gdkt-0xvby2 7i 9xool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces the Earth. Deterb:x3k z;xgvezqej89vs9 f ,.l mine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbibxz .fvs jl:eg,e;8kv3 x9q9 ztal angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from restn8f41tdkdnnm+- 5qny s,6f e v 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 radiu h cze6l7z8k 6bzt+9qss 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constacskl7b68 hz6 et9z+ qznt angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical diskl-fohy60yf zh *ite 4/s, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050ih4ltefz0hy- yf6*o/ kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the an: d,;zudo h z(uhfawb )2h88amurn4 .xgular 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 time-z henbokt;)):m6 iegs as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, andk))ogb6mti-e h: ;z ne that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollutir5 1nm 0won*jdon automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such1jm5nn d 0rwo* 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 illustratt8jx+wx5 62kxe,ota qes an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rollinga8cyvwnm,v rjz56v64 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 lengp+z*cna*i2 8,cexvrn+v)x lp th L can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as nc*i *xra 2vpplncx8z+ +),v ein 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. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on the floor, and wrrq, m8l7b 0rs3buit )d3.jpve want to exert a horizontal force F at its axle so that 73mr)rt i3pvub rq,djl0. s8b it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=bfcpr8see+g b6 / :a/y4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist a p 8e:cb/yagbf 6r+s/end cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length qwi7)7cnuq )yn)kqt) 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating itunkcq77) yqn) )qwti) 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 az/v 2o.ao iec6k+2ym*hn ;sv qs thin rods, making reasonable estimates for the d;+.k2/izme6vyo *s vnh2 aqcoimensions. 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 cylindricalces:3pk 5d-js plates, of m53dkssj: p ec-ass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 l/v amlz+9)o+sqn*5n wjm1uq sooped 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 highest point of the loop uw 1*oa l/mss)vm 5 +n9+qqjnzwithout leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assumpng kve8kzf37:nmsal6,q y9r :;asoh, a 3me (e $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutionsal,eoe7yjf ), as the car reduces its speed uniformly from 90km/h ya7j lo)ee,f, to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s