A bicycle odometer (which measures distance trag(lxy5mn; k4aveled) is attached near the wheel hub and is designed for 27-inch wheels. What ; gaml (kn4x5yhappens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular veloc*/ yvf-:hbutxrzhb 8*ity. Does a 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 magnvbxhf* ru 8hbz:*y-t/itude of either component of linear acceleration change?

Could a nonrigid body be describe axmm2) .1f76tw gvcvkd by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger forceu3 n ,ilk k)2hu*1zut jzfwqvg7b ;/;j? Explain.

If a force $\vec{F}$ acts on an object such that its lever arm is zero, does it have any effect on the object’s motion? Explain.

Why is it more difficult to do a sit-up with your hands behind y/ts pw jqza7*byd,,b rl 6a(8four head than when your arms are stretched out in front of you?d6y,ftab p,7ql r/jwb (saz*8 A diagram may help you to answer this.

A 21-speed bicycle ht5xl4id* y1 ql a91ylqas seven sprockets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder tyixy9dalql l1q1 *54to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower l 0j fscwo cbb.qst 5kd)t l44-i21j-faegs with flesh and muscle concentrated highb4lssbak wfct5. 4j-i1)q-cf tj 20od, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 89f-mad:)bke) bqx y5j–35) carry a long, narrow beam?

If the net force on a system is zero, is the net to 3 xrwfyw9dyl8* /ht,njtvg41w+hd1 hrque also zero? If the net torque on a system is zero, is thw3 g9yvw /r1yh+h1 w,t8* xtf hlj4ndde net force zero?

Two inclines have the same height but make different angles with the horizontd17l rs5 abf65tvw1wa al. The same stee5at715w6 f dvawlr 1bsl ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneousd/jro)g znk;z mcl1jhu0,v6 iz 5m:6sly start rolling (from rest) down an incline. One sphhz m6z)jjus; 0 rdm5,c:k6/gizvo nl1ere 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 the bottom?

A sphere and a cylinder have the same radius and the same m1n:m45efa*7 hk 0coo/a zw 9soi sha1aass. They start from rest at the top of an a *sacfa1:1o9 ok/ao5 hw 4ns0z7mhieincline. 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 07k ;x zt ,yd3e:viyjcand angular momentum are conserved. Yet most moving or rotating objects eventualli, zkcy3jx;0e dtv 7:yy slow down and stop. Explain.

If there were a great mn9yln b -w0hqm5ia, j.igration of people toward the Earth’s equator, how would this affect the length of the day?.0jnbnmw 59hlai- ,y q

Can the diver of Fig. 8–29 da,k7s:k-unohk 8n:e2k 8 b sedo a somersault without having any initial rotation when ,k k-h2u:se ko7sn8kena:d 8bshe leaves the board?

The moment of inertia of a rotating solid dio/l48pegbef* -om .t furn1,jsk about an axis through its center of masmr pfu1n.ot 4e/e-f*glojb, 8s 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 rotat qdqr(*yrdwvt fw5: pbh-4+ k;ing stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity in4wpt- drqf+(*h wvd;:byq 5kr crease, decrease, or stay the same? Explain.

Two spheres look identical andf 7w5 z dqpe/ii32kn3j have the same mass. However, one is hollow and the other is solid. Describe an experiment to de327j/fdp3n qzk5wei itermine which is which.

The angular velocity of a wheel rotating on a horizontal axle poin rg4u;cunf+))v tswg a*1ys 8nts west. In what direcu8 ygs4rw* )t;nav1 )gfc nsu+tion 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 stanx xpydj 08i:d8ding on the edge of a large freely rotating turntable. What happens if you walk toward the cent:jdxd8p x8iy 0er?

A shortstop may leap into the air to catch a bal,dn4.pvh(g x oa0hzw.gy j1u( l and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Exp0hav hzwgg .x(d4 .ojpnu 1y(,lain.

On the basis of the law of conservation of angular momentum, discuss why a heliu:k2 fgt7)pj.ijff( l6bebe1,vqvz 1copter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stabflf 62.)k(vf j1bvj eipt 1:eq7 zg,uble.

  Express the following angles b* hek-,cidbj 696netexw2k+ 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 ql ( :tzrsjbh ,pb75,)ms( vznan amazing coincidence. Calculate, using the information inside the Front Coverq: (zpsj zmt)l, rn5(hb7s,bv , 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 mes u20tg m8k0the Moon, 380,000 km from Earth. The beam diverges at0ge20mtms k8 u an 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 91zrbl*f qtx4 ho g0n50osi0m a rate of 6500 rpm. When the motor is turned off during oper94g100olbo izmth 0 5nr*xsqfation, 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 a level floor 3.5 m to another child. If the* pj.6ac4m9ke ww6.fs mayf a, ball makes 15.0 revolutions, what is its diame ,aa6fm.cj w4py6e k.sw*maf9ter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. Ho w7txh w-//lan 5zgn(yw many revolutions do the(/n lz7/ay-ww5thgn x wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 250kfhi o d.gb.b3a +i91at:auw00 rpm. Calculate its angular vd+f3 tko9w a.i i.bbg0 ahu:1aelocity 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 qkxpg20e0r0 -.gr0 eqyl-tjd) 1crx ks (Fig. 8–38). (a) What is the linear sper.1k0)2ryr p cq0 k g0xeteqjldg0-x- ed 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 around thtw ;mfi*+n3 fg-yozu+e Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a pis9ir9z .t x3foint
(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 iy:/ mgh) 92ix/fd hc-;oukkesf a particle 7.0 cm from the axis of rotatiog9h f;c -ixkmh)ekoud /2 /sy:n is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel acceleraaylb/ 43c nmiu dg4(/rtes uniformly about its center from 130 rpm to 2803(bun/ adly4mi4c/ g r rpm in 4.0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radius fvgdb 4 6 ng84aa-mgb(mu4-o h.fcu: m$R_1$ 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 spacecraft put thek98l3i56kjezdwwq p)q,l1yl mselves into a slow rotatiojlw3 1),8wl6lq ekp9y5kziq dn 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. 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. Through how lxypjkbqh -3d5h 4zsk /*8e-m many revolutions did it turn in kmpeh 5*jqs3y-zkb x-8/4hld this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calculb/cg9xs95z ewn- cl0t b-m zh*ate
(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 jetk9f cqwz3:h:x s in a whirling “human centrifuge,” which takes 1.0 mkfcxz: h: wq93in to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 360gvyj8w6sv)hh(kh 2w2ewgr 1 ) rpm in 6.5 s. How far will a point ow8)g 1 s)gj2 6why2 (whevkrhvn the edge of the wheel have traveled in this time?    m

  A cooling fan is turn,2pp fhf+o0h fi3 mnuyn41d4 led off when it is running at 850rev/min It turns 1500 revolutions before it comes to a stop. mifynn3f 01+fp4,hdp 24lh o u
(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 spjkz 8xn;m9u )reed uniformly from 95km/h to 45km/h The tires havjrz 8k)n 9;xume 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 gu7i0ki*6 8z kk4tvtpreduces its speed uniformly from 95km/h to 45km/h The tires have a diametervkk84z7* 0g t t6ukiip of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bi*(zc9hjckcei 1wm)*9ejo rbm, 3*e jmke puts all her weight on each pedal when climbing a hill. The pedals rotate in m*wb jmk31rj*i*9 zh)ecc, eeo (cj9ma 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 of56 *vl4x* mev3 wer0cwvn6zyf a door 74 cm wide. What is the magnitude of the torque if 566e wnwe* y 3xzlmvfvvr40c* 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 torqu zhnry 1s37na5e about the axle of the wheel shown in Fig. 8–39. Assume that a frictio3 yrn7n zas15hn torque 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 the f)y owo u-z7j.ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is heljzw -youf.o )7d 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 ofb*pyb -v5f23lhz7tlpi4.a: dk l+e hm an engine require tightening to a torque of 38 l abf* 5pmlkvh24.ld+-3hpyie 7 b t:z$m \cdot N$ 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 i *rroq4*ah-zk nertia of a 10.8-kg sphere of radius 0.648 m when the axis of rotation is through its ce*-4 rqkr hz*oanter.    $kg \cdot m^2$

  Calculate the moment of inertia of a bnsvxmozyg3/ ;r(1iw: icycle 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 ignoremyxg(vs/; z1 r3:oniwd (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is n/q3kpk otnh7b +,rm 1rotated in a horizontal circleo1mr k7k n+tq/b,h3 np of 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 zg/)ylwb fa:ew99vq3 a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and the clay is 1.5 9gz)b:w yeaf9 wl/ q3vN total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point objects shownsn,y)5k0 eb2o2d y tyj in Fig. 8–43 ab2)5dbkoesyt ny 2j, 0yout
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen atoms whose total mass )tn:a vv7a ,o6ul2fhx 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 thr69u ( 89axfoxrjfmb/sx 6 z;pe correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force o9;69b zou(frr /ax6j8fpxxs m f 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 8.50w eq/pt8gpa3p -mekr1 vkf avd*.k348 cm and a mass of 0.580 kg. Cam/af8tvk k 1a8* qvepp43-dr ekp3w.glculate
(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 fro)yuczr f)d+)gd1 s6 kxm 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 ablegewq5o y*; wg/ to accelerateoq/ w*g;5geyw 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 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 eventua jmf7b8jugvm9z 6z6.flly brought uniformly t jf fmjb8 96zvg.uzm67o 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-kgtmf/,hyz: g k6 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 action of the shv3p(;a8qak e g-gh,forearm, which rotates about the elbow joint under the action of the triceps muscle,h evag8gs , 3q(;apkh- Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, as shown in hv-vx/ waqi -.zjd ):sFig. 8–4:iqvsw/ .a)xj-- zhdv6.
(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 ofcx q ;jom91fcrp7c/4 j 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 hamm.zw7 aki nubs -6,cd,her from rest within four full turns (revolutions) and releases it at a speed of a. id6s-,c, hk7nuz bw28 $m/s$ 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 v*ad2 +h* dwuexlc 3( ol.p-itmoment of inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine d jw( mw0/pn;d 9c+jfn lhx8/bpevelops a torque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls without 9.gqvbw5 yazkqj2* c 4slipping down a lane at 3.k4bwa. 2zc* q ygvq5j93 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to oqwnl rk::,fu 9-d3hb the Sun as the sum of two terms,3 wh:-urlq on kd:9,fb
(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 7.50 m. Hwpd33e rwq 4nf32khq;ow much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder. ; fpe32 3whq4k3wr dqn   J

  A sphere of radius 20.0 cm and mass 1.80 kg start klt(n4 geqnsbl,y qdo2 ;605ws from rest and rolls without slipping down a 30. ybq0n2 n owtsqel;l,6d(5k4 g0 $^{\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 tiph2h ft-nz mt68 iwl)i/. It starts 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: zw 6tilt-m2fhh)i8/n Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg balyqjtq7 3 vn8*rl rotating on the end of a thin string in qqy7rn3j 8 v*ta circle 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 gri gyzmu6p:q xnjjk+ ,2dv-zs)0 nding wheel of radius 18 cm when rotating sjn2,jgqdy- )z6:kzu 0+ pvmxat 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 ,m13il 2nqt gyc8dya :00gpz8ujuue /, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal positi8y3j:a, gqznl2tgp0/mcdi uu 0e8u1yon, 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 on;:yun3p 4j e00 c -y.v8ppkdanafe7gbe shown in Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changfd8ayn7. aevkb00npcup;4 ey-jpg: 3ing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial rate ofzy :a ww ojt:h)u-o*3i 1.0 rev every 2.0 s toow ht i j:3z:u-wy*)ao 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 itdzxo gu* iyb )qtu+7v*it k(* 0bz.1vws center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotati1txtz(wb0*yo u7vibkui*dq+zv g*.) ng 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 qeuo-g qy5 4x.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 momentum of the Earth7.)1 tbelas8z7wp dky
(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 id/+uvc6eq)rg 3nr5r y1yoq,r nxdd,;v entical disk rotatingv o,n+xg5e;/ ry,drdr v)qyr1n 63uqc at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at d/kjzm*z8+m nzy w8t- 2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at t lt6+ isnj( dd5vd-l:she center of a rotating merry-go-round platform of radius 3.0 m and momen6 sddv snj-+: li(5tdlt 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 mertuncfuw(f n.k f ;(+8ury-go-round is rotating freely with an angular velocity of 0nuu( n f8ufkw(c+ft;..8 $rad/s$ 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 hal; npd djc2u86oa5el 4agey/*n f 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 rotatio68ueg ndp*4 aoj5 2a/;ndyce ln rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in ey6czvye svp,8d :4 5idxcess 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 ag1fn-k3qjig f x7ff2pd;39 g$ 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 platfo/ bg2tyv4ze2ld5 pvw 4rm, initially at rest, that can rotate freely without friction. Th44wlt/y bepzd52v 2gv e moment of inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at t(6p/w wm1 vuz5ywcl) nhe edge of a 6.5-m diameter merry-go-round turntable that is yw )1pwzm 5c/6unw( lvmounted on frictionless 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 e3qv /0 ibjk(+wvk v+k qcl/au;nd of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance ivv0+cqql u3(a b;+ v//jkkkwL, 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 that the same s n59; ,3lyxaee mk6exiide always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its ownxe ;iekey3nma 659, lx axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest ,/zbngdhl2t8qb5. j2fht gm* 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 radius 0.20 m pk+ 8hjo **c 3,jdi8dnx(xqixacquires a rotational rate oji i(, hc*qopxdx 83xdn8j*+k f 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 d5rpo(p9w . kcaiameter 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,p9p (oc5rwka. if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular spj a j27gv6akh4eed 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, were gl 2rdi17 wvl-rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radii7dlgv -1wlr2us 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it55 eba jsd -qg2 z,sir0.qcrt8 to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrijsca25dezi0 .8grq-s tq r5b,cal 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 illustratesh-+ kwm c egezg3i/n-) 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 rolling on a horizonazi4)+ ma6x hgtal 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 len;n l kq9is g5;/muq fek.kdg 7*qeqm3 9m;p*lxgth L can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At 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 q ;eeu*kkgq;m9g.iq 3 pml5s7q9/ dfknl x m;*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 o zor(;n udn2upm)2ad6n the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has heighto r)p6nz d ;uu(n2md2a h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is makip hy;o-1qx j1sju28x p;pupm8ng a turn with a radius The forces acti;ho pmpqpyu pu8xxs1jj -81;2ng 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 into a sling of lengtf4tc5bq09g a pp-2dqkh 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does a90 fbpc2g- pqq4k5dtthe torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid xezw2 jxhqg.b1zt3 x m/:tel6/: gtx*cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with armtqz x 1tm:xg*wze/eh.xg/2x 3 6ltb :js held tightly against her body.   

  You are designing a clutch assembly which consists of two cbktw /7 -fprd6ylindrical plates, of bfd r-/6 7ktwpmass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 r(0hktm,u b j pwo.+xr su-z1fgtel/74adius r rolls along the looped rough track of Fig. 8–58. What is the minimum value of the vertical heig.bo zf 40,1jk7 t+(u/l-ws h xtrumgepht h that the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, bhib(;ccu 5xo4 ut do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the lx(t92ua7 hlzqlevnz+h ,7t8 car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) Whahenvlt,z q8z 7u(2+alx7 lh9tt 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