A bicycle odometer (which measures distance traveled) is attac 5tjekld(;7j fhed near the wheel hub and is designed for 27-inch whee ;te(jjkl57 fdls. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates vflljv)0,h2jl 92p g jat constant angular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity incv jf,j llvgj09l2h)p2reases 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 described by a single value of e6y** ((w3hm mwgladythe angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force1h5etv67fqb x zqlp/8l d r6/; bu,cqd? 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 diffi o lsm82gbt,e45wm38 edeh4f jcult to do a sit-up with your hands behind your head than when your a tm2h4,le3wgee df5 j b4o88smrms are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seve*wv0fnd )s+vg 0rzn*p n sprockets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In whiv*+dwgz* fr)ns0 p vn0ch gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have g v .+yqj hfudojkml;n*8fxx /g-9n:b5fz5x4 slender lower legs with flesh and muscle concentrated high, close to the body (Fi/xz8bxyg.hj4fdv o:k5 fmx-jnu l+ *gn5 q9;fg. 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, narrow beam?e;keabymg;v1r 9 8k sf3 h)xv5

If the net force on a system i-cy2 hqfa9b9 js zero, is the net torque also zero? If the net torquchb99qj2a -y fe on a system is zero, is the net force zero?

Two inclines have the same height but make different anglej3(xjg- t r0vx5lm ;tqs with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bvgm5q0tx x3j-j r l;t(ottom be greater? Explain.

Two solid spheres simultaneously start rolling t e.a,v1r 17 hxe/niup(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 there? Which has the greater total kinetic energy at the bottom?t1xv e/i17a,heu.r pn

A sphere and a cylinder have the same radius and the sq1fw0me0 m.uu+k yoq (ame mass. They start from rest at the top of an inclin+qw0 fm.(ke1o0 yuq ume. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most mhos8 0d7+ ta;h3 xxwkfoving or rotating objects eventually slow do ;xhwkts8+03doxaf h 7wn and stop. Explain.

If there were a great migration of people toward the Earth’sx5ea j ag6qkvb4p* c8a(5n8sy equator, how would this affect the lengths6xacb8q 8eg5k5a* v yj4(apn of the day?

Can the diver of Fig. mc2 m7tj6je.l (mfky)y 2(alg8–29 do a somersault without having any initial rotation when she leaves the boar(e aty.j2lmmcg7( jy fk ml26)d?

The moment of inertia of a rotating solid dii9 1cmc(d hdt:2zh d*psk about an axis through its center of mamd dd i129(tzc*phc:h ss 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 holding a 2-kg mass in eacr/pll +5irfpr8i;) kph outstretched hand. If you suddenly drop the masses, ipf5r li +k/p8rl r;p)will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identicala:: ,f 6 xqnw8levd2hf and have the same mass. However, one is hollow and the other is soeq6w ,lv f2f8anhdx:: lid. Describe an experiment to determine which is which.

The angular velocity of a wheel roto anzb*k/l/e6 i3 )c/co zfl;uating on a 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 top of the w*lkzo )nf;elc6/a uzb 3cio//heel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating turntable. W -fp mn2kjq*2t7 inci4 ud-c-mp9pi(zhat happens if yt( *9czpn-ii -kp7j2m4n 2 iq-dumpc fou walk toward the center?

A shortstop may leap into the ekt2d..jpk5 4e4uvhzdn+ gj 3air 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 willj5e4 zpd+teduk 4. 2h.kjng3v notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conserv8yhm-7kyu:v4ie9m d w ation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.-9e m4uyvkh7:w 8midy

  Express the following angles in radians: (3g4d ( mhlixo)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 coi86j1adfkhm;kz c5 r5- ucvl* fncidence. Calculate, using the informationfaj- fmuczrlh8k*d 6;1 5cvk5 inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,0n5qkv,fh d jh9uvxos f6li, cp/s5,(;00 km from Earth. The beam diverges at,iv;,59hk dhxvl j n ,uq(s/fs6o5cfp an angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm./y 2iltc wc dfxqn6*(. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the bladesqf6wi t.*l x cyn(d/c2 slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. Iff3cc(3kpfk(x vq3 v3s the ball makes 15.0 revolutions, what ivpsc k33xq3( v3(cfk fs its diameter?    m

  A bicycle with tires 68 cm in2j*vv*7o*dr9 map( ,z)tp vrr 3atyhk diameter travels 8.0 km. How many revolutions do the whee9m* )o(z vrvkar tjpy3ta*, ph7rv *2dls make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate its a4yu-a4io /qlrtb8 fjp gg*y ;80ows f4 0uu1adngular velocitf*-l18uda4g4gbosj/0yfp4qw u y uo8; 0t iray 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 revolutioivbl*f+o+ n2wcrx ho2a-xh(( n in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from thr -(lv x(+nco+ awh2 *ox2hfibe 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 orbie3fhtje o)5h+1hjs+rdwq+ s 0t around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed o p+za:,s, otgef a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particpzu3f x9- np1q +c5zeen9;ne nle 7.0 cm from the axis of rotation is to experience an accelerazn- u neq9pcf9z5 x3n en;1ep+tion of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly abo7bnld 0ae o bwe- gd7sw-36th9ut its center from 130 rpm to 280 rpm n06-e wb-ddltw3e7o 9ab h7sgin 4.0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radius gu0 zve nu 6t58lm:x(f$R_1$ 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, astronauths- z wtws5z2/, jnch5fyd6:fs aboard the Apollo spacecraft 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 rhjtww/h5ysd,zsncz -f5 6 2:fevolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpm in 220 s. Through ho*y u.pr i/xt85pwlt4. -qokdqw many revolutions did it turn in this time/qo*-.r5qiy dwu.t8kpl ptx4 ?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calculatp tr6wqu*w/ im8v.hk63d brd*av )it(e
(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 thenyrrs,x96g3 uln8e v, stresses of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its fin96yr e,x,n 3vru8gslnal 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 accelewt/ 5jmr1br 6:)nlp v3 xzk+a pkq/l8vrates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have tra/zjr3kpw /16l qvk5 )ltvp8mn +br :xaveled in this time?    m

  A cooling fan is turned off when it is running at 850rev (sqz m6dujks*7ok.ddqlv z)5 ks c9+)/min It turns 1500 revolutions before it comes to a sjsqkd )l.mq zsc7k9d) *zk6 sv(duo5 +top.
(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 speedvn fq.bs3+7kl uniformly from 95km/h .3v+ nksf lq7bto 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 as2*vh7o x b(s,xekud qnvk(+2y the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter ofxk2+, e(bs(vdkn v2 yhxuq*o7 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 climbqar3 xi0mjld0 06 pq,tcs /b99ewp5uting a hill. The pedal09w00t /muct 9j6s3dxqpi eb,lra q5ps rotate 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. What is thsu 2yi zcpn9uejj0+)6e magnitude of the torqy 6zjs+0jpu)c 29ei unue 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 shown in Figz 3ttbpr,ix; / (d7pzd. 8–39. Assume that a friction torq7zztxi3ddt,;(b/p pr ue 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 whi4ef we4c2t8k tch pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magn4e f2c tk8ew4titude and direction of the net torque on this system.

  The bolts on the cylinder head of an eng;4wkbgj (*e fsine require tightening to a torque of 38 ;j b ewgk4*(sf$m \cdot N$ 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 0.648ljj/4ki )qg m+t99kmx m when the axis of rotatkmx9 4)jjt 9qmigk+l/ion is through its center.    $kg \cdot m^2$

  Calculate the moment of ineh1;bu+ (*sbscfzy+jc rtia of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mas ;zfcb(h+c1u*yjbs +ss of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rot2e5-bfa+exz4ibg(v t/6r mhe5i1:x c yp9byn ated in a horizontal circle of rai e1+ch9/ bpvz:fmg xr5ebaty(n 6x2 b45ey-idius 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 rotcvou6af teb v,;5yu1ji(7 r5qcbt 1x0ating at constant angular speed (Fig. 6 ra0tx5y;evb vu ,7jio f5c b1u(tcq18–42). The friction 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 ar5,vcxyi urn/ 2ray of point objects shown in Fig. 8–43 abounyv 5ic2xu,/rt
(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 atomtp/+ep3, y8195d2 )yxuvpjq kdhgb ucs whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the corgd0dd 8k1e h6ffop ph5.* -sdkrect 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 requi-ep5* dhs0hg186okf dpkd .dfred 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 uniformj k3;vg.my4rm cylinder with a radius of 8.50 cm and a mass of 0.580 kgg mrk;y.m3 jv4. 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 rest wa4vgp:h) b o (utz5bkc9mo 59to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven merry-go-round and is ablev6rl :w :tm-oko*n.im 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 o -:ln i*omv ort6:mk.wther 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 n+qhoe;yu;g4 and is eventually brought uniformly to rest by a frictional torque o;q4y gu ;o+henf 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 atnq zmuu:vz k(q+e;9 8b 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 apj.hs8f jw76 unw nqe 4up965qction of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The balnhuq6qus e f49p7.w65w8jjnp l is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, as shown inlnrr2n7woi ,*y)rmp 6 6uukn9 Fig. 8–46.i u uy2,nn*k)rrpr67ow l m69n
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine consists oe+6as2+ ta1l ht i)fe uhqpg;:f 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 accelera qp(5b45rzgf ptes the hammer from rest within four full turns (revolutions) and releases it at a speed of 28rgp z(q4p 55fb $m/s$ 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 momenfl hys bz vsvr8 w/+;:-4fzb)et 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 develops a t5upv2 i*go5fc orque 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 gfs wt( +/z/ z5wc-gltwithout slipping down a lane at 3.(+lw tgg/zct wz/ s5f-3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respem)fa kvb.mu,- ct to the Sun as the sum of two ,vb ).afkmu-m 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 16 f4 inuvw-o7)x40 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Aso4ui wx-)nf7 vsume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from r .b)flv0.:8ypxzpe rrest and rolls without slipping down ap.b: e r)xr0zy .vpfl8 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

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

  A 2.30-m-long pole is balanced vertically on its tif i d34jn3z7olp. It starts to fall and its lower end does not slip. Whaofz74l3j i d3nt will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum oli 9+:o onj- bm,uc;akf a 0.210-kg ball rotating on the end of a thin string in a circle of nc ib jkom:,+uo -;a9lradius 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 cylindric*usy78 rcgym9tpec0x 5chi2h6yu5 h4o* ab7wal grinding wheel of rad cwr xguhc mhyc ih2b7 *679oay4ts805pey*u5ius 18 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotating ateof *3x /7uvqj a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48jeu7q/x 3vo *f, 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 o8g1qp vo)y-d3lak*a 2b hpoa8ne shown in Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s a qopa)2 1v-3ophl*gd y8 kb8awhen 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 roo1 z2 lxsk7apya ev:10tation rate from an initial rate of 1.0 rev every l1ase v: z2a0po1xk7y 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 center at a fndknc)xa/-f 8 requency of 1.5rev/s The wheel can be considered a uniform k)/ -ndc 8xfandisk 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 momentum of a figure skater 6+pra5h chz4.g bkyu ;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 o r ))qo-a;ngy7mbmau:f 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 izev 5 6ta.hzi1nertia I is dropped onto an identica. t 6va1ih5zezl disk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at hli(z nr5ja. n8l5vf(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 9o6or zfv u u* qou(w19,cim4tkg stands at the center of a rotating merry-go-round platform of radius 3.0 m96v(oz,r ic u14qo wmtfu *o9u 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 velocity b .i-o cpb/r kt3.-ea,/c vgnzof 0.8onri-3 -eabtzb ./,c vk./p cg $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, losing about hal3 c 4wja90i* ;asx4qir9bwmo sf its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lo o4ws0c;ri9 a43xbq9jim*w asst mass carries away no angular momentum, 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 winds in excer o4w32ldr)n 4zcg2zb ss 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 xu(9 u*s)zutvu ec0tndq)e . *$ 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 f6l-0aerphn ,)* ts3pqbsc6(xad5ilv reely without friction. The moment of inertia rcp)p-htnva6 iss5 0 (da*xl3qe,b 6l 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 diameter mh:8lno s abg59erry-go-round turntable that is mounted on frictionless bearings and has a mom a:hlgno b859sent 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 onbhp x grn) wf,0m8p j6r2xe0ucm8wo7* the ground 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 dis6b8cmxmej u,rppn*)rxhg 20o8w7f 0 wtance 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 center of mass move?

  The Moon orbits the Earth such that the same side0slxad 3:3j wj always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particlaj sd 03:jxwl3e orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from m -g zr96)vxd18x +u4izhyclmrest 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 0fyqkst1 wz((,5xk vb,.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. C1 skt(,5kb xfyz,qv(w alculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 kg and d0k5 a-hl;x rw:ik a:obiameter 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 w0kih5; br-:ax:loka 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 reo - :cufeu5wcc:)b0kyg1hb9 uar wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times as great, ; 9 b;fc4fdh 7g k3mtgvbnl-0xwere 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, losi4ctgkf ;9fbd70m g3n ;hlvbx -ng three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for rydul 9p4-po )a low-pollution automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has o p-9r4 l)dypua 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 0pq9sh-n8 -63 nbwlkjl fu6tx 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 *1wxp 31lt)s plc efr(rs s1+espeed 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 (i91icn m q1ddn h1gtz51(3+a /ykw umvb3 bfz.s.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. [yb15 ubgd91i1 dhwm 3 .sa(k+fz/nczmv 1 q3tnHint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on the floor, and we rv3-dd,e)ecftq+nw96 3ch jz wav) fe9-6hn zcdt3we+3 ,qjcdr nt to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling witrj*km lrg.dy/2si0(nh speed v=4.2m/s on a flat road is making a turn with a radius The forces acting onnmg(2s r j*0.ki/ ydlr the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.5 m and begins whirohmli j1, *q i jy)9),lqwqbc:ling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, ani bwy,q )9qmlh jqlo1)ci* j:,d where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a syh1nz lo 05k.ox*.bkhu*yua(olid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then lok0o.nk1u*.(z*hyb5h u ax ycalculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindrical plateseb5f1n*85 sa 2c hrrm y2p3xjp, of mass 12 *ybfe38pjmnprh5 25sx car $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 th)(e el*khr ekyf90x bl2k- (hae 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 rke r-(9(eheay l h)*20kklxbfeach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, butoiw+i4(j( ern do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed unifosxro8q)l w 7ucp(d ;h7rmly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the(rd7qculs7;x8 h )opw 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