A bicycle odometer (which measu+kr i3zrxjjmg c76 )u:res distance traveled) is attached near the wheel hub and jxi)rg7+juc m:rz6k3is 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. *fymzlb8+qaa sz919nk2w04 u/slxs u Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the poibysnla xs0m4fzz 8/2us9qw* 19+ukl ant have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the angular velocit*n;bmm09ng t;l n/ nkky $\omega$ Explain.

Can a small force ever exert a greater torque than a larg4. z vp;9jvwmder force? Explain.

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

Why is it more difficult to do a sit-up with your hands (qt0fg j6x- ug1cl1ljbehind your head than when your arms are stretched out in front of you? A diagram may he u1gj1xlgl q6 jtc-0(flp you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel rcbry,ude qv25, w7-fand 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 harderbe, f7 u r-5,w2cvryqd to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast a;gk la 4gd.(1ju* ofghave slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mau gko f.l41 ja*;(dggass is advantageous.

Why do tightrope walkqfix32y*r x8y ers (Fig. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero? Ifr9xscceu4j l: 50-q8knqj h9h the net torque on a system 5q 4h:cn0 kcejsj rux-q 9l89his zero, is the net force zero?

Two inclines have the 8hjn++ yv4sv0ccon) zb+ z 1jcsame height but make different angles with the horizontal. The same steel ball is rolled down each incline. 0jscb+vj4++1 ozvh c)cnyn8z On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultb8m8s2wh w( ,ex vubpv0(w k 4olap0h3aneously 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 therxub0(0phe(h2vaw,3w4lmbsw po88 kve? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass.28sc 2c d0sbvmao cg4- They start from rest at the top of an incline. Which reaches the bottom first? Which has the greac0 sa2 cc ms28b4g-odvter speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum andqflg a2fy.a5:w1u/2q;nk zb z angular momentum are conserved. Yet most moving or rotating objects eventually slow down and sto2 f ulb2f az5qgqkz1a:.;/ywnp. Explain.

If there were a great migration of people to7 p i2eky- n z5zt1xxte;3k5lpward the Earth’s equator, how would this affect the length3 e ;i15teptkkn-7 z x5pylz2x of the day?

Can the diver of Fig. 8–29 do a somersault without h7dpk3be / j4ddaving any initial rotation when she leaves the boardejkb3dd4d 7p /?

The moment of inertia of a rotating solid disk about an agsp b1 0e+gm3ixis through its center of mge1+igbp 30smass 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 rotatinz9d8pq,jbwbkth(5 5 hyf0n- fg stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angultd, fh0q5y-9 5(kjw bfhnbp z8ar velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is hollow any, l 13dt1 yqdhz-02mdti2t io2 o)lnpd the other is solid. Describe an experiment to detydo tit1yhddtnlp21-zlm 302, 2)o iq ermine which is which.

The angular velocity of a wheel rotating on a hnij 0obr*rs9mc8-f8muor ,- vorizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the t8o8s mij--9m*b rvc,f0o rnurop of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating ,ixf c kl-+i5aqkh , 53,g(ebixo 8zgjturntable. What happens if 3i -,gaxxoe5k,jcb ,k5igf+8lqhz( iyou walk toward the center?

A shortstop may leap into the air to cnyhtd.5 6wa/ ratch a ball and throw it quickly. As he throws the ball, the upper part of his body 5. nar/tydw6 hrotates. 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 hel+tg qe a 86t0 xswf;ids0o2sr+icopter must have more than one rotorg8trqiods +ae ;6+sf020ts w x (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (a y5z2.gnxl-ag ) 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 Earh:h5 d yt1fi p*o5suf+sl4a )lwo/) lmth because of an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) uis5yo+ )*1l:w) tfh4mp5la h sol/d fof the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 330s z qzb uyv/zwqyqt5 346j-w80,000 km from Earth. The beam diverges at anuz-bvqqz4s 6 w3jwyt03z /y5q 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. Whenmr:/ii w2tz sf)1m:qodul53ene wb6 , the motor is turned off during operation, the blades slow to rest 5s/ :t62 i nwb),u:ere1qof limdw3mz 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 t ip4xx*j6l2vrgo*x m-/+aztm to another child. If the ball makes 15.0 rev p/4ja+v zolri6t2*-x x*mgtx olutions, what is its diameter?    m

  A bicycle with tires 68 cm in diamg0p8t6tr2, yl9yy qedeter travels 8.0 km. How many revolutions do the whe0,qgt9y 86yl tped r2yels make?    $rev$

  (a) A grinding wheel 0yy0 vtabm d.df+ h c4e4ua5d+(.35 m in diameter rotates at 2500 rpm. Calculate its angular vel 4yfdcuv0 ad+4d5. (bhtayme+ocity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.0 s (Fig)8 kymqi.d u3rsy. i5a. 8–38). (a) What is ths. ary3uk.5yi8dq m )ie linear speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of 79;nsn(qj nkv4j5 od pthe Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a poi m;/uwkft7 vndn76y +jnt
(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 centfdy5v (- qu.icrifuge rotate if a particle 7.0 cm from the axis of rotation is to ex u5d v-yiqf.c(perience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center fromj7 mtegho 1rq( (m7e7s 130 rpm to 2807h1( mmeqgor t 7se(7j 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 radiuse67 hm0+o dgme $R_1$ 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 aboarj -+fz;ml :aqto hw.p2d the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, th tf2j : m;-q+pal.wzohey 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 fj j5;kl 98zgjrrom rest to 15,000 rpm in 220 s. Through how many revolutions did it tu;g59kl zj r8jjrn in this time?    $rev$

  An automobile engine slowsklczj5f)i(i fad 7-j6 down from 4500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeedkg0c o+ym:-*dzlk(j2 j 7 1uwuexgc 1l jets in a whirling “human cem1 zu gw-cjyu(1e x7*k2 lo0+d:gj lkcntrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter acceleratbs:evz:np lm v(ql;-p3l yg/6 es uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have t3/z;y6 v-l p: slp v(:mnblgeqraveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/u :qsoedp40 ,dmin It turns 1500 revolutions before it comes t spu,qdd 0oe4:o 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 revolutionm 1 hb1 1hiezkh9,;tnrs as the car reduces its speed uniformly from 95km/h to 45km/h The tires thh1 ;kzbr1h1n9 ,me ihave 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 aipbq6xq0s5 8u/m / xo*nzq5jb v3l/ab:wdg +g s the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diame+l qza:g0g5/vup* 5/wx qbji6d 3msqnob /bx8ter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her s2 j 9*vdx.o s0vbj6btweight on each pedal when climbing a hill. The pedals rotatedvb.o6j jsx9v s2bt0 * 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. Whato)/7pdo( dp (z*tjksp is the magnitud dj(tp (po/koz)7ds*pe of the 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 the axle of the wheel shownv .9(zz auzm mx (9uabjds63x2 in Fig. 8–39. Assume that a friction torquex s2au(96dmua.j( 3mzzzx v9 b 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 ends of a 9(9g e3irrmarmassless rod which pivots as shown in Fig. 8–40. Initially the rod is held in trei r99 ar3(mghe 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 tightens 7n:q;9u pupging to a torque of 38u: nspq; gpu79 $m \cdot N$ 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 0.6je;wh;-id w 5pza2dk3h5j9i i 48 m when the axis of rotation is through its center.;9edjai3zwwjdhp ;k 5 2-i5ih    $kg \cdot m^2$

  Calculate the moment of ,)b3r l6h n4ntjkpkx 3inertia of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 3jbl )n3p k46kth,xnr kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram baljnff21vjo u+p:gy e82l on the end of a thin, light rod is rotated in a horizontal circle of radius 1.2o82 1fv:fjngp+2eyu j 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 angulro81a 1vx6qh lar speed (Fig. 8–42). The frictio16rq8olhv1 axn force 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 oymyu)pj1 ii7,,sk 6sbf inertia of the array of point objects shown in Fig. 8–43 abopjsii7 smub, k)yy16 ,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 atoms +by6 q :+gn1(gajm tbpwhose 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 cylicd /y.3nbv v e/-bm;sy/.jjpbndrical satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If thvye.j-v .ymnb;pbj/dbs/c / 3e 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 radius o7aa d-x ,zpzp)f 8.50 cm and a mass of 0.580 kg. Calculatepd )aza7,-x pz
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerating it from rest to u1k0ndc.mo;o (o rs8e3 $rev/s$ 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 s 3lw cu0./ n8ppxtga:bmall hand-driven merry-go-round and is able to accelerate it from rest to a frequenc .p0ngct3abl/x:wp8u y 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 eventualhs; 6n4tfs t3aly brought uniformly to rest by a frictional torque of 1.2fas hs3;n6t4t $m \cdot N$ 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. 8a9ghxb dm ::ocymcaj5(q 5; t,vv-2 jt–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 thrown solely by the action of the fol r4fw.4s84-hkqhdah+4 7f sry0v nkn rearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–4 hknwrf 84kd47v+.0l- h4as4f hsnqry 5. 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 blade0 yu;-8gdzjkb can be considered a long thin rod, as shown in Fig. 8–4y8djzgu-k ;0 b6.
(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 cyntfqikbwd,fa -3 x 2-:54vuhonsists 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 x+s ; 3gqt/g+zznmr9h from rest within four full turns (revolutions)mgg9z+ +/ qr xz3tsnh; 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 oovkh 6.6o g13 zqmdg+df 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 develops a torquh vkah,z lfahs 9a v2o24(n26aomm(s) e 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 slipping down a i2k8kt5kmk:bs, bd9b lane attkbm9kb8 5idks, 2 bk: 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the3n+xbz*v1gv9 jxvg1n 8qj1v z Earth with respect to the Sun as the sum of t1zxv3 g19n+ vq*bz v n8vx1jjgwo terms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radiuso:8l izr :evw3 of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it iz :e8w3: volirs a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg so6rk; ji a6kpxn910g/w-qqhztarts from rest and rolls without slipping6kz nx-/kw0ai p 69g;h1jqroq 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 vei9m c:cth 2lg 4texs..rtically on its tip. It starts to fall and its lower end does not slip. What will be the speed of t ic..: lt shxcet4m9g2he upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of hqd(ljbk)i5 dtgy*f yl 1900 pa 0.210-kg ball rotating on the end of a thin string in a ct9bq(0y0pgd i)h1 ljdkyf*l5ircle 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 grinding wheh/i (sfbj;+d wia6.ewk j8ao0vv7(xo el of radius 18 cm when rotatinoh ((k7 e./wjifd6a+v;xa8i vo bjsw0g 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 a (89q(bqv uwhtt his side, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48bt qwu(9(8qhv, 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 showkrciwr.q t3,pi:y93 )zq zc9fn in Fig. 8–29) can reduce her moment of inertia by a factor of about trzz,k i yqrqc3w9:i3.)p9c f3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increae,t.pfnq s roh6 ,d7*rse her spin rotation rate from an initial rate of 1.0 rev every 2.0 s to , .ptrqsr *n6 fode7,ha 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 arouzqn+t.f g;h(8g ,vrh dnd a vertical axis through 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 8f; nghr.tqzv ghd (+,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 spe3s1id fo 2i2kinning 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 Earta uno( 5i ys6 y-+ v5j(erewyvah7x)y3h
(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 momentykp3(g *d:ry.mivg c- of inertia I is dropped onto an identical disk rotating at angular speed :g.yg-d kr*y p3i(cvm $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns a1ofr mp uhxa9v1 b-1-rt 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 thegws 6u)ytiao5,r8a y zo vf(/5kliqs,2.t* pi center of a rotating merry-go-round platform of radius 3.0 m and mgk tpy,5orf o i t)5su a2 iqv(zy6l.aw/i*8s,oment 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 rotatg iemqdvvd (4v4 sb;48j;+ipding freely with an angular velocity of 0v b;4ddd( 8;sqejvi 4gmv+4pi.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 mug 3k7pae yg32nof (4collapses 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 rotationapu7f4 ogm(2nke3 gy 3 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 wip.of-m1z2zn x nds 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 khv( 1rvm2k*j4b)6z 7o3ghpq jfm6 g y$ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, that can rotate freely withogd9 * 0fpt9vqdut friction. The mot* d9g0 9vpdqfment 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 the edge of a 6.5-m hd0phr( hf pyp3s+;4c diameter merry-go-round turntable that is mounted on frictionless bearings fsr;h pph dhyp0+4c(3and 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 groun,v 4ov(g6dmdrw.qc8p d with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holdincq,dg( 6m 8dw vo.4rpvg onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces the .edy-,:+gqft6ahossr zv*( r Earth. Determine the ratio of the Moon’s spin angular momentum (abod-r6f (.tsa: qvs,r oz+y he*gut its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates frlbhq.x9du:eg1uj1 z j4 ;ms m8bh)o:yom rest at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder m.em:tn n a6wodw(7a-of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torqmn-aon7.mw :e d6ta(wue delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 k/s-d9gmsw y) gg and di)gd-9yg /sw msameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of the rear te8*hg 939rc+oqi gbx 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 7g + gr/ ehm.8 7tw,pjkvah9ilas 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 unifo/,meah j7k r.l8hi tg9g+pw v7rm 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 it to use energy storuq8p;r+4:syu pkb+mm::8df 1pbl v 4 peo1rsped in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to t pl dp44mb1;: :r:r 8kpefp8+ssvbuy +1moqpuravel 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 illustrab0/0o5a hogg2vewz, pz*x : jites 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 horizontal surface at speee d;9l6 et7q sv2nnmh);y5ynod 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 canu+1 ah *uzrj7w 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 of gravity acts at the center of mass of the rod, as shown. [Hint: See F7 j*+uwhaz u1rig. 8–21g.]

A wheel of mass M has radius R. It is sy5*: r(4retan* tznw qtanding vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it r e*nnty( rt5qwa4zr:* ests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speeduq ii0y9p lmjy 46l3rpa.s( :c v=4.2m/s on a flat road is making a turn with a c3m6 (ipjial:lur. s4 y9 py0qradius 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 /4hshv zxlu:piri93((;q;wwz 3 o otyrock into a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate o4 (;:o(wx/hytozvz wus h;3 iql9irp3f 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 th sjn5 536orneain rods, making reasonable estimates for the dimensions. Then calculate the ratio of then55a6s oer nj3 angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clut9wdwdl9;w9 y5uww m.k ch assembly which consists of two cylindrical plates, of m w99www mw;5l.u 9dykdass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 alongy o 9ecwj;)9c9dp+sd1mqcn f* the looped 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 without leaving pw cm9+qsd cd*1nj cy;99ofe)the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assume vs5-xsv7vkp. ewt9z2am(k8kif s0-b $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uniformly g ,eey2;, jj25isecb vfrom 90km/h je2 ivec,y;sj5 e2g,bto 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