A bicycle odometer (wz sgjo688 qz2shich measures distance traveled) is attached near the wheel hub and is designed for 27-inch w zog868 q2szjsheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a poinucfhbkm n yo29 ;o;v79t 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 magnitude of either component ov;y9ucom72h o b ;kf9nf linear acceleration change?

Could a nonrigid body be describl6 ,ejtjwstx//i vfcs+ d *z53ed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a gn22al)z+ u hijreater torque than a larger force? Explain.

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

Why is it more difficult to do a sit-up with your hands behind your head tly. d.027 yujaj6c xphhan when your arms are stretched out in frontdh2 jx0acp y.67l yu.j of you? A diagram may help you to answer this.

A 21-speed bicycle has seven su/ mm3 ob58h(. yx9zwrx2k1iavxtle 4prockets 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 to p3(amk hum/xtz.yexb9o 4215v w rlxi8edal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on bei+390 iopwjfxvng able to run fast have slender lower legs with flesh and mupjwo fiv9x3+0scle 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-/2vi rgdh 1na a long, narrow beam?

If the net force on a sy6yuj6zc6 0 6tn+uvwn bstem is zero, is the net torque also zero? If the net torque zyjn6v0w6bn c ut 6+u6on a system is zero, is the net force zero?

Two inclines have the same height but make different angles withwi(;x wq.qrc5k n+p7 ft7 pe0o the horizontal. The same n( wiq; 5kotpq+7r f.ex7wc0psteel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultlu, 5scf mxk79aneously start rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches t,79 m fcxlk5suhe 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 sbt.)yykl4 e3r ame mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater toy t4rbk.3y el)tal kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are cozozox,3 * ilqx9 *2mqsnserved. Yet most moving or rota x2x3s o*mq,ol*9izqzting objects eventually slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equator, how woulbrk88 xrkk6id (j)w8:uwj,zv d this affec krd6zwv8 8rw,bik:k) xj(uj8t the length of the day?

Can the diver of Fig. 8–29 dftiy: k-a9,s6n5lw m-7 naqfno a somersault without having any initial rotation when n: 6fnnt lm- -fsw,5ya79aikqshe leaves the board?

The moment of inertia of a rotating solid disk about an axis through its center: ctjn* (l*dxp of mass isctp (n:** ljdx $\frac{1}{2}WR^2$ (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 ; k 6bfstebq8+on a rotating stool holding a 2-kg mass in each outstretch+ 8f; bqetbsk6ed hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have t m/i8bc(xc3 iphe same mass. However, one is hollow and the other is solid. Describe an experiment to determine which is which. /xm ibcc(83ip

The angular velocity of a wheel rotating osiut o oiq563 ud+nd(/n a horizontal axle points west. In what direction is the lineario/n+6qi(u3s 5d t uod velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating turno4;g niwo8 *m dxuei, *9he*rqn+bc1c table. What happens if you walk r+c eoewi8n nh*91q*u;b dcm xog*,4itoward the center?

A shortstop may leap into the l7ua; m650o 3gi g,dqcpw x(hsair to catch 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 him0s;3(c65pgu,l qh w axo 7dgis 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 helicb8 b+ri3sliw5opter must have more than one rotor (or propeller). Discuss one or more br8wi5 is bl+3ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radidl9 jzmzys (b9bo+4 y+ans: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincidenc zb1 49yzwcvq-e. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun 4vzz9-ycq bw1and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,02ytiq42 +lfick iswhj -5k7t 200 km from Earth. The beam diverges at aniyi 425fs2icqk7- lhtjw+2kt 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 re8:l,gyr c *z9t uons,otate at a rate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 , 8tenlygz *9:urosc,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 to1 6:d-y a f xxzsmarr;emr:66d another child. If the ball makes 15.0 revolutions, what is its di; xm: m:x fs6 1ze-6ra6adryrdameter?    m

  A bicycle with tires 68 cm om5z rw 7tpa;qv/ig09 in diameter travels 8.0 km. How many revolutions do the0/tg p;wo 9m 7zaqir5v wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter roto1*lk 5dx532yv m kzf1 bihfp(ates at 2500 rpm. Calculate its angular vx k53b*1omh5vfk1 (fiz yp 2ldelocity 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 make6a+vp ca qp (t.7v9rxps one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m froma6q pp+7v9( xtp.cr av 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 (khbbbcrw7f*82;y by (4oh 2oa a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed w;h-yer2l*-i.x sv b8,dsnwqof a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifugew 7gv72p:3 snpz 8kivc rotate if a particle 7.0 cm from the axis of rsc78 ivz pn3pv2k7:g wotation is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel acceler-.itv 2v+o wxiwvd 4w2ates uniformly about its center from 130 rpm to 280 rpm in 4.0 s. Determ2vwwvwix2-+ odvt 4.iine
(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 radius4k1abr kvo+.)l v,mc x3.cu df $R_1$ 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 Apouh+j/z )uhbu /llo spacecraft put themselves into a slow rotation to distribute the Sun’s en/ zub uh/h+u)jergy 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 accelera4+qigm+uro/*ec ru ,w tes uniformly from rest to 15,000 rpm in 220 s. Through how many+ gi+ ,rr4owuqucm/ *e revolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calculu vlcme,,o;q6.d9of date
(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 hin6a0y( 7rayhp0u 2liaghspeed jets in a whirling “human centrifuge,” whl iayap0u(hy0n7 a26 rich 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 accelerates uvna-u9 an9ib8 niformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled8 a-bvniu n99a in this time?    m

  A cooling fan is turned of; ;gaig uaiz b8(4if,if when it is running at 850rev/min It turns 1500 revolutions before it comes to4a; ;fgiabu(8,zgi ii a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed unifor,+q8 awupyq *zmly from 95km/h t8pa zy+*qwq ,uo 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 6 zduh)ssp78qf ou z,ziz 45+m:;vxp/l5 revolutions as the car reduces its speed uniformly from 95km/h tom u+4z7hx is)p ;l/pzz8uf:q,dsvo 5z 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 bike puts all her weight on each pedal when climbingzcadu-t ) :y4m/h7jic a hill. The pedals rotate in a jd taicu-m4: z/) ych7circle 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 ozs,ftnxv :- i1n the end of a door 74 cm wide. What is the magnitude of the torque if the -1:n z f,xsvitforce 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 show.+k7x snbwle:x ytg2.n in Fig. 8–39. Assume that a frictil+2exgtk.sx.:w ybn 7on 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 *i2ug+vo/3ew 5ak7wi zhfa24iwy *cz pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude o4c2zve/wi2 uz*7 y awi+f3*gkwah5i and direction of the net torque on this system.

  The bolts on the cylinder head of an engi-wz,s :d-ylfbne require tightening to a torque of 38lbd--,fw:zys $m \cdot N$ 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 0.6b0) 6jymik.uu,ijps:7h8c ei4 o r4 wt48 m when the amjp:oc740)6,i.sbi8tiu 4eh jkr y wuxis of rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in ; mn,e5 hqmc qn3zhi66diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hn6 qz 3 c,n5hi6mmqh;eub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on tnu*q2zqqxkf 6it/zfe0 c7 4+/g l6hwmhe end of a thin, light rod is rotated in a horizontal circle of rafzzgq6*w0q6 i/24ntmx/7hlcqkfu e+ dius 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 potterr1gq*;dsvn+9m kn rpqj. .pg/ ’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and the clay is 1.5 N trmdrj.*;s/p. 1p nn9q +k ggqvotal.
(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 ot,bw,i.-vq9t,tq p jc bo(l;e w,dud 5bjects shown in Fig. 8–43 abou-d;q(. u jwe9dbo cqvti t,l,,,pw5tbt
(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 ygbrmq 74r1 w4is $5.3 \times10^{ -26}$ 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 cyl2 +pozev 77dryv m-7kwindrical satellite spinning at the 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 of each rocket imd+ve7 77yv - rwkzpo2f 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 +6:t6ihf ghuco ;zw( iradius of 8.50 cm and a mass of 0.580 kg. Cali6g u+hi( zcf;o htw6:culate
(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 zxl6 2gm;c(fobqefv3s-s+1l 6vv kz8 bat, accelerating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven merry-go-round an na2m7/52psu:sduwdw d is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to prod/5up2u72 ws:dwsan dm uce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is eventually br mivndt5+(q+/c1z)vqyv k5 c+zc2 ej bought uniform +z5q)n bm5v+2(qtzy+icjv /vce dkc1ly to 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-kg ball apa(e4glq,x +3p0rt 5z v8g daht 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, dcm38wipu,b*;gk h w /which rotates about the elbow joint under the action of the triceps muscle, wmk p3ub/8g;wi d,*ch 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 n1 nomdjizh z8,g8 p/i-6 d6hebe considered a long thin rod, as shown in Fig. 8–46/mipoi, zdeh8zdj hn66gn18 -.
(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 l /toou o7a(xhzz64b/1 lqkd /consists 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 -phb6iw r 14c*.oevzpu m3/ ymthe hammer from rest within four full turns (revolutions) and releases it at a speed of 28m1*y/ rmch-6ebuvz w4oi.3 p p $m/s$ 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 mok2pwooy)3c ll-)6zxc s vn2e.ment 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 dehz,ylo0 /u,s-9ooyl bvelops 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 23 ako-ys3(s5ap d96j be tibmradius 9.0 cm rolls without slipping down a lane at 3ss ap9ek35moi(db -abt6 y23j .3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with ; es 9f.d/cyncip*9 sfrespect to the Sun as the sum of two te s .sfc9f9py;c/dnei*rms,
(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 mudw s+ rjlj 7w..2)zrwuch net work is required to accelerate it from rest to a rotation rate of 1.00 revzw.srw 2)l7jdr+j w.uolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.8.t +bnh+3a4toau q)sx0 kg starts from rest and rolls without slipping down a + 4a.thu tnxoq )sa+3b30.0 $^{\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 and its:1 6+fwii(rnkw i*nk9/2yy f my +xhfq lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservafw+rki9f2ki(qm ny x+ 6f/y *n1:wyihtion of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end of ab8q9 z/xle-kp thin string in a- xl/ekqz9 bp8 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 grin4qvh fzia g2 6n pa*(+(u89eywtxa -doding wheel of radius 18 cm when ro( z dwtqgv*6+e4fhn-2(iapxo9aa 8 uytating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotating at a rate o+8 vn1yhufdm97 v.od uep(zh0f 1.30(o+7mvph f1.d yuhu 8de vnz9rev/s If he raises his arms to a horizontal 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 reduc4l) b2)2o5 avghojjai e her moment of inertia by a factor of about 3.5 when cha2ov )2l h5j4oi a)jabgnging 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 a0gire87bq:f)xx . ko18s kwi qn initial rate of 1.0 rev )eqs:.08xqfo71gk 8 xiwkb ir 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 aroyu/mt1r fucy) .s. c-ogv6ga3j9 vxf 4und a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considc/uys g3ay-1v xm6 r)f4uocjvf.t9g .ered 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 rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular moa)of(oz)v d9,-lqpju mentum of a figure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular mome8,o h:skb.j oentum of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is dropped onto an idehd 4;: u qh:mjh5pobi4g*8mkw ntical disk roh h mpiom b4 ;4*5jhd::wgk8uqtating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns aosc)xii 5;ky0g xg3rpc,o,fb t/s +)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 the center of a rotating merry-go-round platfouro(c:cz/j 4 mft.rh /rm of radius 3.0 m and mo .tzc/hrr/ ojc:mf4(u ment 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 r0q.gwf w djj8 d;fy mj/o/cp8/otating freely with an angular velocity of 0. /j8j m d.gpqjwow d8c;/0ff/y8 $rad/s$ 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 collaps4/5vu* fo rpg+m-r8(zw yzay tes 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 rotatio(f4zurt *pyv/+5 8wyr- a zmgon 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 windxv-sm r,;mef 3s 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 0i1:.tuda 0+, mjewaf2 tx uqw$ 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, w*ip: cjw-uq3initially at rest, that can rotate freely without friction. The moment of inertia of the person plus th uc qi:jpw3*-we 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 personlwr:5lecyc,hhpt5 slh )nd*-+k-num,6a7w m stands at the edge of a 6.5-m diameter merry-go-round turntab56 khdlcr)tle u5w,c-,n h:lh-yp7s +mw *man le that is mounted 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 imo xab 6mtt 0ti9,i-2end of the rope 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. What length of rope unwinds from the spool? How far does the spool’s ceiabm6t0tixmo 9-ti , 2nter of mass move?

  The Moon orbits the Elmxvi* 26as e;arth such that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (sml6 ia ;ex2v*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 frw2apsjbx84:f46twv0a jp 9pcy lzd;u0z1lu: om 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 of radius 0.20hc88l bj8 y18j+etc un m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque d+yn1 eccjtj8ub8 8l8helivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disksxaw/i07yh g;c 2auc7l , each of mass 0.050 kg and dil;u7yg07acc/ xa w2ih ameter 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 therv)1x54a d/9qshf9 hbkmv a* n7o vfi9 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 sv6,z 6 a1cguvcize of our Sun, 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-quarters of its mass in the process, what would its rotation speed be? Assumec gv,u1 6azc6v 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 i msj+opop4*rwp6,ixm2g 3, xgs for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of p mi,g6o,x+wpms*ojx2 r3 p4g 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 an9 m x)+lkbrlbrw5 a3z,-9s0qt1 rptud $H_2O$ 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 at3 d2eg2itj fj6 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 jk.,b9 +j t:eu -xcuwya73d7tstk 9fl length 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 acceleractk t3t:. d779ulexws -y9 ,jjf+ aubktion 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 iv9i gx.5+jlyio+lte :s standing vertically on the floor, and we want to exert a heijl:v5lx9gi.t y+ o+orizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed 0jh:wrqm4xnab5 p+;wf k8a/jv=4.2m/s on a flat road is making a turn with a radius The forces acting on tpw+f h4aqj; /ra5km8 n0:xbjwhe 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 i(wlb:m*h :ylg7c w2s-kqfd 3fnto a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it fr7fwdkg3 w2(s - :lcy mlhfb*q:om rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s (;0eok sxc3v2trbd:c body as a solid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. 3d2:0b e vr(ckot;x csThen calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch asse 6f9p/j -nz lgp1ji3*xd.j,eulc v o4ombly which consists of two cylindrical plates, of mau*z egj3o-914cd/fnjx l pv.lio,pj 6ss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 alo,cvg( dip+3py)moq7 c73l vzu ng the looped rough track of Fig. 8–58. What is the minimum value of the vertical heigc 73lpi yz,pm(guvod) c vq+37ht 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, but do noi/-kf2rmv yc2t assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions a6v n2342fmpdnt 2urwt i;+hr* e6m mbhs the car reduces its speed uniformly from 90km/h to 60km/hit; etfmh6 b3n*nrd+vhu42 r6mwm22p 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