A bicycle odometer (which measures distance traveled) is attached near the wheele4n am8kkpi +t,kk)1h2n5 qzs hub and z4,8k15hnk 2eq kp)kiats+nm is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angula,4 zz47bdwoitr3x 6w yr velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocit4oyzx,w3br7 4tw di6z y increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be j +gd94jhsv+ger -p.t described by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger foru32kks.ulux/ ez4*nc -2c kiece? 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/*oye 4zvp7vt 3ts-b p with your hands behind your head than when your arms are stretched out in frontv3sbtoz7vy-*p t 4e/ p of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the pedal w l(u u.x meg09m9jpnzqr/63n cranks. In which gear is it harder to pedal.9l wj 90mm qug(rn p6uxz3n/e, 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 abld75y zlcfde ;6ps.c0s e to run fast have slender lower legs with flesh and musc67zdlce fd;s0 .p5cs yle concentrated high, 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. 8–35) carry a long, narv gjmd0vp70/ :vk1wxsy+ ss ,k3 sk/kvrow beam?

If the net force on a system is zero, is the net tor r 73mygmmf:skph18s)que also zero? If the net torque on a system is zero, is the net force zer mp1r:73m)g hy8sfks mo?

Two inclines have the same height but make differensxaap mkiid5x q,1ns9u *b:uj,0m 4f7t angles with the horizontal. The same steel ball is rolled down each incmp1mnuk*s ai0 ijafbx7s4,q ,dxu5: 9line. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an incli krmgb4d2q lry2 y,. :ex84wbine. One sphere has twice the radius and twice the mass of the other. Which ire,gq 42x.rmd w kl4b2y:8byreaches 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 radf/;r,d* *ejk5hdxkk)d w+tie 3kt3 f sius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bott rkdk ;xek *jfk)es f*iwd/t35hd3t+,om? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and ango o( 5ix-jsm2hs5y(jv9, czz aular momentum are conserved. Yet most moving or rotating obios szx( jm, zj295a 5y-(vochjects eventually slow down and stop. Explain.

If there were a great migrati7 , ;3dmlpt dwt.v4dkuon of people toward the Earth’s equator, how would this aff t.uvd43lw7t mdk,dp; ect the length of the day?

Can the diver of Fig. 8–29 do a somersault withouts,6 igvm3i 5jn;sw69bzzwh5 a having any initial rotation when she lea6w j; , a6wzivs sbgzi5hn935mves the board?

The moment of inertia of a rotating solid disb.y h*rlu j-drs/0.a wk about an axis through its center ofh lda0*../ u rjwbsry- mass is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool.e9pb8ol lfq*v- m o+q holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular vel*vmll.p98 efq q+ obo-ocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is hollohprieevv5qy ) 4gl. g;z- ,mx5w and the other is solid. Describe an experiment to determine which yge-5 i;m,e gvpq4).zr5lhxv is which.

The angular velocity of a wheel rotating on a ftsbnc(xje +9vte1.i5wmo4 y:nm 2c +horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the to49on+s(xtc i2mwyb fev+jen m t:51c.p of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standinv o81+fu 7kiv:5pi eqxg on the edge of a large freely rotating turntable. What happens if yo7kfi1x+:pv e iq58u vou walk toward the center?

A shortstop may leap into the air to catchrevlp-2z z e7g54r/kc a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips e-l/rzz75 2epc4gkvrand legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of consez8z0 erwe9,x6en9l lz/d j2te rvation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propelllee ,xdl ze wz6n8tz0j99r2/eer can operate to keep the helicopter stable.

  Express the following angles in radiamiyk.*lc6rm a* z g5e)ns: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincidence. Calculate, usxp 9t1 uxfx kc1)cmb5h3w51fxing the information inside the Front Cover, the angular diameters (in radi1fx1 9tm x5 )5 w3upcfcbxhkx1ans) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km19ah xmjluy5/cp5pxcgg 3o8(j. b bn6 from Earth. The beam diverges at an ao x5b/8p1gg 3lc x(mjap5 9jhbnu.c 6yngle $\theta$ (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. W9xfcc hf 6w-v3 juo;-ahen the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the jf6o av-ufwc9;-xh3c blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If tj2k2* e( kbm hnm;-mg2ee-njfu ,kgh3he ball makes 1ke(3gk2 km2j- ue ,*nh n2mf-jghe;mb5.0 revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutionom dz+9ppp5 :d2 -lr ij*xxdvo-s14chg *xhy3s do the ox341dh*v-:-2 smyxij+dp * r9z5 pdlcgxpohwheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calc 5* 5t2wu:.imtk dtugvulate its angular veld5gviuk5t 2u:mt *w. tocity 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 i sf2 3a0e8w0ztznmj7 lxecwxnk-;o4c/4k 7rmn 4.0 s (Fig. 8–38). (a) What is the linear speed of z20/c4arkte7n07xm4m sx l8c-ojwez fwk n3;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 orbitf;dhx 9b21 fvv around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a poinrd.ka,o*9 hmy t
(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 centricwzly*,4 e m8ifuge rotate if a particle 7.0 cm from the axis of rotation is to experience an accele8ei,lyz 4c *wmration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelera 3:hqdabbe0h 6tes uniformly about its center from 130 rpm to 280 rpm in 4.0 s 36eb0bdaq:hh . 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 2nq;77etdd, umb )p i4z4-nuvft:kfh kbo12qs $R_1$ 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 oc lr-c x-zut3wj4j02 put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinde 3jor-c4xw-lt0uzjc2r 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 acceler- . jtb oslo8,rx74f6vm7czpyates uniformly from rest to 15,000 rpm in 220 s. Through how many revolutp-m, ozcr7txy7so .8v bfl4 6jions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in+45yb. ,cfzh 7d6v*f3ccnfr q+aw -j9um el qm 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 flyiuvguig5 0 l6si ,p*x3vng highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 completevsix*g0uul g6pv3 ,5i 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 cz x57nntli,, from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveledn5 n7l,i,czxt in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It t d1 )iswj:ybd*urns 1500 revolutions before i)yw1 dsi:j* dbt 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 revolution9hu h4o2lp9f,:6egxnf q7 xkls as the car reduces its speed uniformly from 95km4kx6x:phgu 2h o, 7l9lfq9fen /h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car red2 /s*vrz 8dpzf80d utya h.2wquces its speed uniformly from 95kuz t8w0*.2/zd p8fr yasvqh2dm/h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bikes2c((j2 pk.pv2 lehy z puts all her weight on each pedal when climbing a hill. The pedals rotate ivc2jh.ks 2plz 2p ye((n 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; in1q/te2 j: u9xqwib 74 cm wide. What is the magnitude of the torque if tu: n1i i;xb9/e q2qwjthe 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 thw6tw0tvyi cm xzef;2kq*//5gg;sma 0e wheel shown in Fig. 8–39. Assume that a frict/kf x2;gw0qieagzw56tym t cs */ 0;mvion 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 ends of a massless rod which pi jfzk0y0*ou)ivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position zi 0)0k oyuj*fand then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening9tjplv+ .4p3v tb)vst to a torque of 38 +.9bst)p tpv3t4j vlv$m \cdot N$ 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 c3wrri6 vx d0zw/xe/).648 m when the axis of rotation is through its6 xwer wr dvi3x/0c/)z center.    $kg \cdot m^2$

  Calculate the moment of inertia ).v)qiod cbq0+9rfmp of a bicycle 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 (why?)qdcpqvo b 0 mirf)+9.).    $kg \cdot m^2$

  A small 650-gram ball o y0n/xxgl(ojkae.f6: ,hxv; p n the end of a thin, light rod is rotated in a horizontal circle of radius:pxel6jnva;kh .0xfy o, / g(x 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotati ju ,tzpmy,4 ;4dei6q3gk8znfng at constant angular speed (Fig. 8–42). The fric p uy4k4j6et ,;dg3z8,zqfmnition 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 of inertia of the array of po ( fwa kotbf9,)gye. lfwh4 g4/0-lhioint objects shown in Fig. 8–43 agl tgoebw ,0w4l/f 4a)9 hh.-ok ffy(ibout
(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 m6ubj2m wx67jfass 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 spinnig(y s/n 6/utmhng at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of /y/6mhtnu( s g3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 cm and a manvxhz2kpb g96 +nrl6/ ss of 0.5x6k9/2r6nb vlgnh pz+ 80 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 bat, accelerating it from r8wmijp7rw x (1est 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 na+l91 kc0o uhi1lktg(43z64kn;1 myni y jafand is able to accelerate it from y ngaa1n1lnc 9 ; ho01 yzj(6m4 k+uki3tl4fkirest 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 one+ek0vi m(4* hqr/ oeff and is eventually brought uniformly to rest by a frictional torque of 1.2 no/(*ik +mehe04 erqv $m \cdot N$ 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-kgl* lq0/q i cirldg1;m- 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 forearm, whichi)(hb8;3fb a ,76 qi dcpvzvjh rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from resthcb6bh3jvq7;zf( av )pii8,d 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 lojdz lhc eq:s8xr3* l;hfe do+j4q0,d5 ng thin rod, as shown in Fig. 8–46.;seh0ejz d48f*,:olq 53lrcj qd d+xh
(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 co)p c ):imyijxzc (7 gp)twh1o.5wnhs6nsists 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 fr6jzi 2 ( irsojtc6,0zpom rest within four full turns (revolutions) and releases it at a spe,rpt0oi26 cz6 j s(ijzed 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 2pn k v2pzq 81ge08cfk$3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops nte 9x,7x 1(jls gjptn.r4l. 1ektj/ca 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 cjyxp3d/ tirpa r6(e .7m rolls without slipping down a lane rx yiarppt3e 7 ./d(j6at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Eanp+ob ,ieltnq6p4 kd/07 /vf dldw.s3rth with respect to the Sun as the sum of two tet3dn+/fsp . 4p7qvl i6n,e ld/dkowb0rms,
(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. How much d yqh-4 7wwv2snet work is required to accelerate it from rest to a rotation rate of 1.00 revolution per y2w47w vsh-qd 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 4mz4i6ek.f 1cusgh7q1.80 kg starts from rest and rolls without slipping down a 37s1m ecfi4h.qug 64kz 0.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

  Two masses, $m_1$ = 18 kg and $m_2$ = 26.5 kg are connected by a rope that hangs over a pulley (as in Fig. 8–47). The pulley is a uniform cylinder of radius 0.260 m and mass 7.50 kg. Initially, is on the ground and $m_2$ rests 3.00 m above the ground. If the system is now released, use conservation of energy to determine the speed of $m_2$ just before it strikes the ground. Assume the pulley is frictionless.    $m/s$

  A 2.30-m-long pole is balanced vertically on its tip. It starts to fall andqhb+drjo1yr/5x: fvqc 9-k n ( 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: Uqrh dn + -c5qfox1v(kj:y/br 9se conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the./l;pahw8x 5ahtn 0 jb end of a thin string in a circle of an0l8 t hab/;5wxhjp.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 wheel b, uo5+nrx 7ygof radius 18 cm when rotating at 1500 rpmn+,75y xbugor?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotating at y ytyyy: c38o2 g )lpcok1sse.34jxo.a rate of 1.3rev/s If he raises his arms to a horizy kyyj..8y 1pcco23ot)y:g4 3xls oes ontal position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce her moment of 2p/+yoi0 i.+qmytk u g-xz9ouinertia by a factor of about 3.5 when changing from the sx9+zuio-.tgpiy0yku q/ 2m+o traight 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 incd*f7phc( e x kn4x-w2k(4mun arease her spin rotation rate from an initial rate of 1.0 re4-m (dnxe7n*pu 4ac(fx2kh wkv every 2.0 s 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 through its cyxk*c,gw p+ i-enter at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0 piy* ,+kxwcg-.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 rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular i(g9 2cd1 fq4m2pd po+j:blxvmomentum 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 momentum of the Eahwubd hho:nld -i) 7,ddx v)(9rth
(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 ontpa7t a3hqc g(bwd5 b-7 lpv2acw(+,hfo an identical disk rotating at angular spfl q(bwd3g pa-wbph2acc at,57+(7vheed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at+rzuc/ff 3 ;cs 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 the center of a rotating merry-go-round platfic*a0l3iv 4zpk r+3p porm of radius 3.0 p4p0rl3iz + 3i*kp vacm and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an angular vel3 qw8nuzw5v j2rq lv;)s)xqu/ ocity of 0.8svx/ wq5quw2r lz8quv)3 n);j $rad/s$ 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:vhglg )* u.fh;jj o:r, 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 momentuml:g.rv;h:o jg u)f hj*, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve windu4) kvwaz n:7xs 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 gojr/rv/ )yb86+2(lu lqdu) utyl9tx1 qen t6 $ 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, tha): om;3xncy vmt can rotate freely without friction. The moment o3mo yv n:);cxmf 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.) iyj4sgtf p/jg) 8x5 z39f k;0cvbtej5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a mome;y)t9 /f5z cjif3)kgx 4p b jgve8sj0tnt 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 -grip;im/rfaiqd( *4 lying 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. iradfmipg*/ q -4i(r;What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth s2dsro+ k (uak3wv9d 1u5b:cz auch that the same side always faces the Earth. Determine the ratio of the Moon’s spin o9rku acku:25z+wds 3 ( abd1vangular momentum (about 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 from rest at a rate of 1 -j-6 1cth 8 jl2uf5jnndgki 4rm/$s^2$ 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 c ah+jnyh5kh;:uxyj (p; 8rxy/ylinder of radius 0.20 m acquires a rotational rate o;rhyjxu ay5nhp;(k/yj h: x+8 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 2hvp 94zmq9poof mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of envo94z2m 9p hqpergy 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 re bf t,o *7uj0.boqtn ,jfmb;0d8re a/par 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,pmlo95 9zei/oks g* *xs(w-k j but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-queopxkkwi5*-l m9 j/g9 (*ossz arters 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-povy 9w:vn7fy y .ws+fq1llution automobile is for it to use energy stored in a h+ qwsv7y fn:yvw1y.f9eavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates anb:+c hio k6:d6p fa.hf $H_2O$ 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) isk1fh r jc12pax-yro9- rolling on a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (iz4c*y p0ia kn0: *wk+rh,slqj.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizoqj: zi+0l*sp , 0wr*ckk4hn ayntally 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 vertira;4ti/zu mgmpiysx 1w /10p7cally on the floor, and we want to exert a horizontal force F at 4zp/1w 0 sryx;ip1u mm/gita 7its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s4 l 1)ew+m b4cbjndv*zf,b dc4 on a flat road is making a turn with a radius The forces acbc+m4 d)d vbwc41l4nfbe ,*jz ting 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 2ztio p xr1a;fg)6j +ca 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 of 120 rpm after 5.0 s. What is the torque required to achieve this tcj r2;fgxz+61a)pio 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 t9dm3-j, gw welhin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstreg3 ew-,w9jmdltched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindrical xe4wg) dy -:oiplates, of mass yw4:xi) o eg-d$M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 as(jj3hd4 pts-vd; 4o9s 4eh5ddihy amg)53w 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 i 45ep4g5;dy9jdv3 s()t3so-jmwa4h sda dhh 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, but do not a(ni1vs +oq.k+ayr hq 4ssume $r\ll R$ h =    (R-r)

  The tires of a car makzp2 v3,w, d8qn69q2vftzt u mke 85 revolutions as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each tiru9tpm2 z8 2,vqz6qntw,k d3v fe? $\approx$    $rad/s^2$(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