A bicycle odometer (which measures distance tragg8pj+5pp ij3jc*t+ b v.tl x)veled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on) 5iv3 gtxclj*pj.ppb t8gj++ a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on the rn/-w 9ycq 0l 24b0 vbqdjoque6im have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point holbw9c-uqdv0 24q/j 60b yqe nave 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 aphx0l( afq+.s single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greo0xsor09c87id o 2pp rater 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 1:rbb4h17c:1vco tza hqri6 athan when your arms are stretched out in front of you? A diagram may help youh ha1z1:tr 6:vb4 barc1ocq7i to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the pedal 4y.7*,2ydozq ohtvm n cranks. In which gear is it harder t,qtohn *yvm47 z.2dyo o pedal, 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 able to run fast have slenxt7qda(o 968vptezjt -tq(n 7der lower legs with flesh and muscle concentrated high, close to the body (Fivo6(d8(ttxte9t -aq q7j n 7zpg. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) 6 * wa*3.wo6qc/8lqftg/ykqpp8 nsp scarry a long, narrow beam?

If the net force on a system is zero,j6*)ee +vfxwu ot26yz is the net torque also zero? If the net torque on a system is zejew6*yx +et2 z6u)of vro, is the net force zero?

Two inclines have the same height but make different angles with th-2(ioj cd+p*93qg. gifzj e1pa doi x4e horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greatere dpdg -p2gic*. fi4jqj91ixoo3 z(+a? Explain.

Two solid spheres simultu6pl-h un y6dd26vh y.aneously start rolling (from rest) down an incline. One spher66vpuy -y2n6u.lh dhde has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. They ske87m3)s* qox5 gygz6v0 ejsttart 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 total kinetic energy at the bottom? Which has the great*gg vj7)ks5yx8zqeem6o03 ster rotational KE?

We claim that momentum and angular momentum ic 6ijlbv vz-/c6cx// are conserved. Yet most moving or rotatin /j//vbcc6i-zv xic6lg objects eventually slow down and stop. Explain.

If there were a great migration of people towardc*cq :lo;i e 5re f;+hr20mk1zutauar:7nbt 5 the Earth’s equator, how would this affect the length of tz21 h; 0*ruc+r5t k:riab;:c7al5eoq e unftmhe day?

Can the diver of Fig. 8–29 do a somersault without having any inip-dce 4: uoc(otial rotation when she leaves the board? :do( c-e 4pcou

The moment of inertia of a rotating solid disk about an axis through itsw3)n hxsi-16snh6w ra center onh) xs -rws1nwah6 6i3f 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 holding a wx oc,ejt6 zdfa5+0y62-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or 60+exwj d,czya6 f5to stay the same? Explain.

Two spheres look identical and have the same mass. However, one is * iw pea)knbv ,c;(ou 1nt0ny/hollow and the othinaoc/epkt *;,wb) n0 (uyn 1ver is solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotatingamvajh1o ;oh*i0up5yj p4c3h z5 /q:q on a horizontal axle points west. In what direction is the linear velocity of a point on the top of thuz : j0yh oph*qh;3jao1miav 5qp54c/e 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 turntable. What zhvy;2 hjf54 ehappens f5 2ev4yhhj z;if you walk toward the center?

A shortstop may leap into the air ttn, c/r gza+co5gv6 d1o 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 his hips and legs rotate in the opposite direction (Fig. 8–36). Expg+1ztg5o6 r acc /vn,dlain.

On the basis of the law of conservation of angular f:bd 1pd:/d v lgl0*m5wdis4d momentum, discuss why a helicopterbpgd5m1sd /wi0v d*l:f4:ldd must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles i,)rnc1 ojtj2 s,4cy )kcy+/+w ud;z5rfuiwcmn radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earthq4ckg0pg0t qe m.; tdz968iaoqf(5 oz because of an amazing coincidence. Calculate, using the information inside the Fronzfcq 4; dtgki 0e5zg p.8q0tm(qao6 o9t 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, 380,000ok.bt40 q3b mq km from Earth. The beam diverges at mbk.04 bot q3qan 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. When the mh llmnb6:2d9c l3hj*x 4k kld. l;r(hsotor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular 4*rh6(3;h9ckdxlld k: n 2bl. jmslhl acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the ball make- t-r7tooa,c qhb6g2ks 15.0 revolutio7t2-htq6g -,rko cbaons, what is its diameter?    m

  A bicycle with tires vsfct8 zk.0h.w6ka;zd) v q,l68 cm in diameter travels 8.0 km. How many revolutions do the wheels m.6 a.fdz,zhsv k8);w ltkc v0qake?    $rev$

  (a) A grinding wheel 0.35 m in diamete:xna byb)fp-o: w. g9.xhndj;r rotates at 2500 rpm. Calculate its angular veloc ;. n9.jyfn:gda b-xw :)xbpohity 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 revoluti* ihk 5exiafs/h gc,/5on in 4.0 s (Fig. 8–38). (a) What is the linear speed of a a*hf55i sxi/,ckh g/echild 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 (ns 4(bnq xo*.dypw8j-xwlu o67e cw4orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a pok+6 v o5gptv1k)t 3gy8de dw.(o ;ashjint
(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 rotg8t f(nqd:-kkgd 1g6 po*r bctig4w4; ate if a particle 7.0 cm from the axis of rotation is to experience an ac;4*-gfktkcg4i 1t: dodp6nq b wgr(8gceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from u0djywqo- hr 5wf)jo7 c l3*x*130 rpm to 280 rpm in 4.0 s. rqjdoc0*ljx yoh -uw)753*f wDetermine
(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 9mvpvyg:ywf3*;kkf)lqa9 c) $R_1$ 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 ab3s8q dkroo.g5 hx)b*v oard 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 revolution every minute dubhrkdq8 sg* .o )x5v3oring 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 unifobq0f rhi 9een(6lh6)ormly from rest to 15,000 rpm in 220 s. Through how many rer)qi9fb( oh66e0 n elhvolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.54s 7kwffy wg:) **t lz*bp7jqj s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeed jr1 705htabsr fets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutior7at0 bh5rfs 1ns before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 24c-d.q,dlm7 bz 0;qvx l zba(/qm1 z1zv0 rpm to 360 rpm in 6.5 s. How far wizl;m11z z7,qbq0m/ .xqd(-c zvlv da bll a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is runningl)jp-pl u -n sp7an4p0 at 850rev/min It turns 1500 revolutions before it comes to a stop. p lp- p-nn7u 4lps)aj0
(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 spr5hu fw, 6/9 j;kocvkneed uniformly from 95km/h to 45km/h The tires have a diamek c,u9o6kn fhjvw;/r5 ter 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 reduces its speed uniforml bx 20 tslk*7r3j),,dzjeiw:arp * tny:dv/kwy from 95km/h to 45km/h The tires have a diameter of ,)7kj*sx y/l02a dw, 3ij dtpn: rt:w*bkrezv0.80 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 pe9 :bm jdb j8 1lf7lmb;x8wbk7jdal when climbing a hilllkfx1 :bj8jbld9b8j wm7b;7m. The pedals 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 Nkv32cenw :j )2,d7 c(vrrnxte on the end of a door 74 cm wide. What is the magnitude of the torque if the force is exe:) xwtdek2rcrcn,37vnj (2 verted
(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 torqecij/w ;qj k1f1m8ev*g6tk +j ue about the axle of the wheel shown in Fig. 8–39. Assume that a fr1*8 cijweg +;vqkk6t1jm/e jfiction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attaj4 jb.-y53l-kgq c7s fs*.l.pfeknbu;d t *teched to the ends of a massless rod which pivots as shown in Fig. 8–40. yj.-sf.*jfd*pne7b.4; l qkl5t3k-gst bec uInitially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening to a torque of 8 cp*jyvpeg 8u -;g+ *cu(aacr38 gvc yaa8puj-e8 (r*cup;*c+ g$m \cdot N$ 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 0.64uhk /-)er 3iyv-wqb2e )z +bci8 m when the axis of rotation is through its centerue r vwkh 3 -2/)+i)ecyqbibz-.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle w9puxm:dk8 6:zk3wma lheel 66.7 cm in diameter. The rim and tire have a comka lm:zmd6u3k:9 xwp8bined mass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on th c wd z764hj*w)z(kyqse end of a thin, light rod is rotated in a horizontal circle of radius 1.2 m. Cal(h k6zs4w)* zc7jydwq culate
(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 whcu,.u p+ ps3h)m:k ebleel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and c,mkep bu s)u+:3ph .lthe 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 inertiak,q1.dx5c sg w of the array of point objects shown in Fig. 8–43xkqdwg,c s51. about
(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 :axw(qf j) 9y+an7cwpmass 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 a+6ki nr qwvk.(ne/6uht 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 n kh(6q+reik/ wnv.u6 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 cylinbk6 g le.i/o:v4r6 yhyder with a radius of 8.50 cm and a mass of 0.580g.e4y ori yl 6bvh/6:k 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, acceleratingh;hcv6 v;g(ih ,jpm9h 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-drive0ehzt 4b vzr5 z*r/o52ko;woan merry-go-round and 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 produce the acceleration, neglecting frictional tot zavo/okwzohz2 54re0 ;r5 *brque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 1*)zex usegis l(rs3450,300 rpm is shut off and is eventually brought uniformly to rest by a frictig)sx3ur4es*z 5 e(ilsonal 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. 8–4arwc 6fzdv4 - 8+vl6vvlv 4b1p5 accelerates a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely (9jl2hxf(.7 f : bpej8xkif mkzg*0rpby the action of the forearm, which rotates about the elbow joigjk rff7 z( .l(pi2 kp:m*9jfxhbe80xnt under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, as sdw07a jttqb8u y282 ihhown in Fig. 8–46. jhaw8 8y720qi bd2tut
(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 consis3p l 7qsecboj9vsdj* 5qzto9*2/2uih ts 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 trqahborur,, p/8;w5h he hammer from rest within four full turns (revolutions) andrwb8quarhh;r5/ o, p, releases it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor hap n**u/dy) dh36 ns4 tas8uncns a moment 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 torque of 2800qu3 owx ftal*hmv1 0fg3uojsbc) 529ak/ 5f u $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls without s,mj8cuglj9x)9r 3;a h3me z aolipping down a lan;mc3oa)8juhj9a9 g3ze,r lx me at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth withpn .umixv5ra1t.vi 1 j3 +kqma3 wqz/6 respect to the Sun as the sum of two termsv6jap.aq3xmmn 15zv i13rt+w qi/k u. ,
(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 e,);qqa0gudr+wxzbip 4 ij:; of 1640 kg and a radius of 7.50 m. How much net work is required to accelerate it from reqiue qp)4j r; i bd;zw0x:ag,+st to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg startsl j oy+2waoa33 from rest and rolls without slipping3w3jo2al y+ oa down a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

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

  A 2.30-m-long pole is balano l;qrjy* 3r:zced vertically on its tip. It starts to fall and its lower end does not slip. What will be the speed oqr;or:jyz l *3f the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end of i3ww1gikwg4 4fqb*d8s52tiu a thin strinkwqig tif*gw824 i3 w4s 51udbg in a 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 o*i.;cq6jk mp df a 2.8-kg uniform cylindrical grinding wheel of ra q.di*jp;mk6cdius 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 sidz(hog pw34qox /a3b )ve, on a platform that is rotating at a rate of 1(gxhv/a 3o4 ozb3wq)p.3rev/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 shownj*g:u9vfvxgf 5 -s3ll 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 when in the tuck position, what is her angu 9* f3v5vx glsg-lfu:jlar speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initi*l i7w/y msff2al rate of 1.0 rev every 2.0 s to a27lmfy/w*i f s 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 w.8w9c6 .is+bf/wzk0s-xii jc9gwx u around a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kki xw +j9ci/ sisw9- 08..wxbgfzwu c6g 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 spinninkenaq do tq*;q--wt4f t. (t6bg 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 Earlgn*2ejeh* vjusx9pr l bxa+360j;ve eh8p6 +th
(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 drv:b3t-9b p8ywyhsc0q g8nb- vopped onto an identical disk rotating at angh- b3tvy:n89c08q-ysbg p wvbular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.oxh,2aa8ziu;fx.v t/ai: e2c4 $rev/s$ 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 rotatin77u rqu a2+hs l6tfqzm 5)fd7a/z8w wag merry-go-round platform of radius 3.0 m and moment of u7lmhz7af)7u q5wzra8ft+ q/a2w6sdinertia 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 vnyfqm8 tb 5 bf,l8b h;er1)v7ye:rer.elocity of 0.,n7v581)r. bqr;emr:bh 8fflteyy eb 8 $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, losing about half itsow ko, )i;y g-i6k2img 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 rotation rate be?(round to the nearest inte yiow ,ogi-k; )k62migger)$\approx$    $rad/s$ (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 3 9fu*w2jd mny/vt7r kz6ix; dwinds 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 9-rwapfwn gf* ic02xk9+tk j,$ 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 atod27kp/xzb+u ic+ 7t v rest, that can rotate freely without friction. The moment of inertia of the person plus the+tp ckiodv z7 7+2u/xb 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 / vf/7 dy:afqtthe edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia ofyvft d ://a7qf 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 lying o xuiu688dypp vegzv3 g)- qub:o35:xpn the top edge of the spool. A person grabs the end of thuuv e36gbz3qv5y x:pup:o) px -8dgi8e 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 center of mass move?

  The Moon orbits the Earth such that the same side always faces the Earth. Deterfkey6ooxb3t 8vh 9l( fs.-1x cmine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentumo tkffvx xcbysl3eo-89(.6 1h. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate*sg0ly 1 ; vabek m(7+v6rnujh 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 osg3ax-oo/1u u f radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the 3uog1 -au /xostorque 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 d2el0:3 io+u 2rsfrujwiameter 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 f: 2jr2orlu30+us e iwthe 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 dyegygku ).3qq(h -o7r5*tnsthe rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times as greatn n l9.58 p4p j: ci7,qlbviwqwt;wys1, 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 bp5 j n:,9yn s.l;lqvqbcwt p17iwiw4 8e? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to u dcv,w9-dgq+ nse energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”,v+-ng9d qd cw
(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 00gphziqg 4-n an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizo p;lavu*7dx/mn5yg1v ntal 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 andb1 nd -5uqn k wyigm2c5,4yxe- 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 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, as4g15c,ywnqkn5ed2bu- xymi- shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. :z(letbt wr*e ym93/v It is standing vertically on the floor, and we want to exert a horizontal force F at its axl*ewtbrv (3:l m e9/ztye 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,:g 3 p/bxsyjvke7/yh=4.2m/s on a flat road is making a turn with a radius x/ s ygb3, jp/khv:ye7 The forces acting on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.5 m and bgnadjh.-uw c z-v(x29egins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest t z2-gna( 9x. vdh-uwcjo 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 body as a s s*ishbu/uhxld-wa5fnd;)z 2um2. /w olid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skawd .f ibuu-hd /)mnsu/ h*52l wzax;s2ter with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of twsfp21 d dlm)kk40e)o eo cylindrical plates, of mep)e 0mod2ksldk1 f)4ass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 ro 667ofm/liey mlls along the looped rough track of Fig. 8–58. What io 7e6/6 ifmmlys the minimum value of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assa h;2g vznfvo kb6(h1)ume $r\ll R$ h =    (R-r)

  The tires of a car mk2qoeg +b6 :28v x-v6a: ezvjlrq7wyjake 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 tqvewjr8a22:lbez:+ 7q ovvjyg 6-6 xkhe 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