A bicycle odometer (which megw ac;t)vq9( wasures distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What hap)qt awwc9g(v;pens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point o 3my34ry,c1cp h5vxjbn the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases , yp5vc 3chx43 jmr1ybuniformly, 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 sfhr)* cvza e,:ingle value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force sk 63jrpb1 f/b:yc0st? 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 w vh)g vgk;f- vbjuh0yp,wkd8m:46 rb/ ith your hands behind your head than when your arms are stretched out in front of yk:k - hh64pv0 bj ,gfwdm8vyvg;ub/)rou? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three niecz3ck/hz8ybnuz5h0 4y tf+ f79,w at the pedal cranks. In w,hn 7ffk5z3tu0yhiy+4/zwcb cn8 9 ezhich gear is it harder to 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 slender lower legs with fl m jlfw /e k)j0l2h)2(kw,jgtxeshtjl0 j/w) 2,kf)hmwe2(x kj lg and muscle 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) h1 u+cd8i bwb xehfu9)96;oexni h52vcarry a long, narrow beam?

If the net force on a system is zexpprjn.v0 r.*ro, is the net torque also zero? If the net torque on a system is zerjp. 0rvxp.r*no, is the net force zero?

Two inclines have the same height but makexb7(7rpkta z ; 5:oeyl different angles with the horizontal. The same steel ball is rolled down each inclinboa5xp;yr :t7( zke 7le. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling c,ot */bqne dok)m12-c5fhsl(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 kqnmho/ 1o,- d*5 t) blk2efccsinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. They cj5 u/d9 4lxxkstart 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 b9lc u x5x4jdk/ottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserve ap(+8o m h -yfmn )ctv4ahd7c*g3a5mbd. Yet most moving or rotating objects(c+ avn4ya d 73htm ) op5mfg*cab8h-m eventually slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equator,+l 29rndv.r6ost fb9b how would this affe9d6f+ ltobb.2rs9 v nrct the length of the day?

Can the diver of Fig. j6uwd)*ifbz w3 v/ym78–29 do a somersault without having any initial rotation when she leaves th b ufw*7iy wdj)6mv3z/e board?

The moment of inertia of a *1 k;)bytkbgvg0v49tsr 3w h drotating solid disk about an axis through its center of masv g)sdtwk;gtbh1 v b439k*y 0rs 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 each ou2 a839scm 8fxkbw3 v6(lerxzttstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay xxsar vc8 6z(fml3 b2tew398kthe same? Explain.

Two spheres look identical and have the same mass. Howexs gz*i +-txh7db8zs- evx2fai:1j8 k ver, one is hollow and the other is solid. Describe an experiment to determine which is which. xfvb ki+x-:8*t7-si h1xzs28 z gjeda

The angular velocity of a wheel rotatin -;o/wiufo/ whg 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 t u-ioh/;wf ow/he top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating turntable. Wy*uhmhq p/n g,h2r7fphhw.(bd+ . m8z/(mcwe hat happensm.chb2zye7gm*h+dphrf(m(h / whn. /pwq 8u, if you walk toward the center?

A shortstop may leap into the air to catch a ball and throw it opy5v:0 u7 jjdquickly. As he throws the ball, the op:u7j50dyjv 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). Explain.

On the basis of the law of conservation of angular momentum, discuss whb7dfqrbb bvnjd2*kr3t*ft.c,fp 3, 8y a helicopter must have more than one rotor (or propeller). Di.n3dkr37c fv b jp*qd, f*b b2r,8btftscuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following d(g p3ic5v qnq xtn7-4angles in 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 Earth b g7sm5iki01q)ut4 k voecause of an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on vm 7u k0sg5 tk4oi1qi)Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The b*k4 (mpdz cj*jeam diverges at an c jd4zmk*(*pjangle $\theta$ (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.7umjb -gm ob*w6 jibp(- +fof of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is th.6p jo7um m(ibb- w*- fojfb+ge angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball oniu2p636.wbei p6b / 2tchkall a level floor 3.5 m to another child. If the ball makes 15.0 revolutions, what is its diameter?hpl /a ec2 6wi6k2bib6t3.up l    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many rev5m3xisn gz -4eekoy+e-nn +o8olutions do the wheels makieegeozkoxnsm53n-+ n8+4 y -e?    $rev$

  (a) A grinding wheel 0.35 m in dqx9, e w:izhx0iameter rotates at 2500 rpm. Calculate its angular velocity i x0zq wixh,:9en $rad/s$ $\omega$ =    $rad/sec$
(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 co2( bq.*cmjrh vmplete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child hbrc.(m j*v2 qseated 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 tx0 h-- xz+dgww2mb :c:cv1+pox jemkk +w)0xuhe Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed o2 d1ucteu *e /kra3l+af 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 irl/ 9, tf)lrbtf a particle 7.0 cm from the axis of rotation is to experience an acceleration of 100,00f/)lb r9,lttr0 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rt(be6ed;,mkq(dctv3y l 32grpm to 280 rpm in 4.0 s. Determi23eybvredq,tc6(g3 lk (mt;dne
(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 (reylpvfou .7(x mu/i+l. g y9$R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboard the Apollo spacecraft put themselvesy9 q5 5ceunwq1 into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameteewu 951cnyq q5r 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 acceleratey0mnk c2ss.dg z+/fe 4s uniformly from rest to 15,000 rpm in 220 s. Through howsyfg +/. medzn04sk2c many revolutions did it turn in this time?    $rev$

  An automobile engine slows dowcpis q8o9l8a;j1 pqg/ n from 4500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying4 lffd( h8 +g- hpoxmb(mw5w;s highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutiomfsm f- 5pwo(hx;w+8 hg l(db4ns 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 diamete/ p7vvdtg3gjwikujj:- ; -rn1 r accelerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a po-j3u gvj1:r v7t -wn/ g;pjkdiint on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/misz og.fvd66;k n It turns 1500 revolutionzv.s k odf6;6gs before it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 61 5ngaep6ha ttn/2/29yj. lz* gjcdjm 5 revolutions as the car reduces its speed uniformly from azt.ya/jcn5dg *g/hp6nt1l9j2mej2 95km/h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car -7jqyw ql 9,svpa8dw4 reduces its speed uniformly from 95km/h to 45k8vq -w4pqyd 97jw a,lsm/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 climbi5k shdjo8hl: sv;)e q o7ac+;cng a hill. The pedals rotate in a circle of radius 17 cm.)+7 ohs;lvh kqs5;jca c:e8do
(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 ofxpf8hbcy lc,prs-8 jb ;7ov2 6 a door 74 cm wide. What is the magnitude of the t7v8 cy6hj8,2s plr -;pcfo bbxorque 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 Fig. 8–3k3b/v c/jpbr9 9. Assume that a friction 9vpkb//r3jbc torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass qi; x ffjih3+ciw-vh1, :(tut m, are attached to the ends of a massless rod which pivots as shown qtfh; +j(i:if,tu x wc13-v ihin Fig. 8–40. Initially 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 tightenij:f pe3r,f 2bsng to a torque of 38pe2s rf,:f3bj $m \cdot N$ 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 0.648 co*( p2udz y 27-1dasdcevv*fm when the axis of rotavoep - ddyd1z7vcs2*(*fauc 2tion is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diamd2z czzz(b*3d4o3 qn qi*1jacs 7(i jqeter. The rim andj71 b(2qzsiaj3*qz*(n czzoc3ddiq 4 tire have a combined mass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod *jmyie m d i+isq*;f5-is rotated in a horizontal circle*5*q iijf;e misdym+- of radius 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 rotatingmx5vn6 57p 3;xp unc/s itwp2i at constant angular speed (Fig. 8–42). The fr3p7pi n v6n/mi us5;5 cx2xtwpiction 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* -n d2 emzfvzet s9,w/a*/cxi of inertia of the array of point objects shown in Fig. 8–43 a-9s*/cxw,dzv/n t mef2* ze aibout
(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 w 67up oyj4upc9jj,wg,hose 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 ce (b vx.i9qhkf:k dk8,orrect 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 kikb8fe h: k(xd9 .,qvis the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder frcai,rl6dd;cg(a g h76 y8a8/loqd -with a radius of 8.50 cm and a mass of 0.580 kg cay/fdo d6ql gag6r-l8 c,; ahd87(ir. 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, accele-g f5hur x16iurating 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 and iij /2j 3wq lfilp2(cj2s1.u ues 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 edgej13ul2 (sj eu2pc.il fqwj2/i. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is eventually bc54p or3li,/0 ummhcirought uniform 4imocrp,uch 0/i3l5 mly 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 acceleratesxshi)2 )dkccnld. ((n a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the action of the forearm,bfreba7w1e :* : p2ary which rotates a *we2e raaf:b1pbyr:7bout the elbow joint 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 shown ;( i txr.dd ;;g-zzteepb*6a)td yvx-zdl) o. in Fig. 8–4i)-y ttd;povd.z e-rda.(; z*g zb dxl;)t6ex6.
(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 )iifty+:hf,4 . ezz scconsists of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hamx:9 )oj u wydn,b9xc pm6(c r7oxl-(sfmer from rest within four full turns (revolutions) and releases it at a speed of 2 :w9(,- nyb xd)(oc rc 6lxujmf9sx7op8 $m/s$ 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 , .kswy1em*pcmoment 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 devei6j46 ln(con. ; giikjlops 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 radius 9.0 cm rolls wit1 lk8ozz:jqg4loyzk4;x 0 (bxhout slipping down a lane ax8 q0zo okjzxg(k 14 ;:lzlb4yt 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy os)6cz y*u jh yza(*atf /m)0rzf the Earth with respect to the Sun as the sum of two tey0 /t)cjah *za) zsmyr(z6f *urms,
(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 sr 1m(6xst6mc radius of 7.50 m. How much net work is required to accelerate m6tcssx( 1m r6it from rest 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.e p*c,d u 0 ca/15;9xsxdbapai80 kg starts from rest and rolls without slipping down a 3/xcpsd ;1i,abaexd90p*u 5ca 0.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

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

  A 2.30-m-long pole isgyktz/7p6h2g balanced vertically on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pol/g 2t7 phykgz6e 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 .g yyri:ig*d2 the end of a thin strini: r*dig2 gyy.g 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 of a 2.8-kg uniform cylindrical grinding whee2 0:nnek0fswq l of radius 2 0q:ewskfnn0 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 platfo8 /i)wso36 yxbwzchy0 rm that is rotating at a rate of 1.3rev/s If he rawzys6wi)b c 8y/hox30ises 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 reduce her moment of inerti)19tviihi d*ynep5 w2,jg,qea by a factor of about 3.5 when changing from the straight position to the tuck position. If sh 5)h*,qvjp2,i1gw d 9ey tneiie 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 ro1 q444fhvaut ptation rate from an initial rate of 1.0 rev eveq fav4ph4t u14ry 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 thr f5.)pg+ ehmf4sqjtb6ough its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat +5tpf46) jgm.bhs eqfdisk 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 spi +cmd /kcsyk(nq 9n6,ynning 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 Eqr.)m 401ng5kicm-4zkvf rd y arth
(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 mom3ii;l :ompe0.fs2 fuggegh6 .ent of inertia I is dropped onto an identica3;ef ugf hsmgei .2p.ogl:6i0l 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 2.4vyfs 8a,0huyv) o 4iw; $rev/s$ 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 rotatidv (2ezs tym5-ng merry-go-round platform of radius 3.0 m and momenye 5v(szmt d-2t 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 frem 9a5)vm xnz)uely with an angular velocity of 0.8manmxu )9vz5) $rad/s$ 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 white51leq *y -9dnkh wltf1*cm vo7 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 rotation rate be?(round to the nmo* lhcw 1-1*d 7yenlqtf59kvearest 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 excess of 10wq ,oyvqti40 pyw,n-20 $km/h$ 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 l vy ,;cuc .-m/k:vvii$ 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 rotatetn9hy a7 leq+a2q24p ,g;9jqosx( k kz freely without friction. The moment of inertia of the person plus the platfora l(7phsk kaeyqzn2jtqg+92q9, o; 4x m 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 stapkp7l avyuo 98(lvtyb57t;q1 nds at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia of lovk8yuq (59l;apbypt t77v 1 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 rollymq) c8po6 f5 (spi.hhs on 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 distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slippihfhp5 m(s 8.q6yi )cpong. 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. Dete;llvdta8o.)kf 3pi) rb 6z.nirmine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbiti)zo8dil)pn 36 bt;lv.k.fr aal angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a ratela6 9f pzr0pf t5x4c-e 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 0.6o*if/(wi7ry7 kc uh s20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Cal7ui/o k c*fihy(67wsr culate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, ea+nv7p83 ihhw6rl)j bv ch of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the envb +r783i 6hvjh)nl pwd 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 o69i( i icsr4 g6shkyfxmaj9. yufx v:5;fz )(kf the 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 mh( f)os9v;nmdass 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 spemnd h)f(s;o9v ed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to:l(ae+ syox ;epq7ku2)ou8 tc use energy stored in a heavy rotatipuy: k;cl us7exa)oeo8 q2 t(+ng flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates a tx o2d6cy)-kln $H_2O$ 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 hpkcv l()e1 mxxm)y6 o-;f :ig8q z8icaorizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., we ignore fri8auaj3 wy(hlk;-0sa lction) about a ha -0su aa(8jlky;w3hlinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on thr6*bk: a dx*a6uar x3u ey0ev3e floor, and we want to exert a horizontal force F at its3x0u:66*ea * ry vbdke3rxuaa axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is makin q.:e x/ k-ckjmrb-hx*g a turn with a radius The forces acting on the cyclist anbxq:jm -xhe-/ r.k ck*d 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 slinbp -7tl6uzupfn- y2 :bg of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his u bltb2y-6:p7pnf-u zhead, accelerating it from 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 body as a soo8 q,o,-oxnyd lid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of t oqn8o ,yx-,odhe angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consi,yf r4-q2c*j ioqq0swsts of two cylindrical plates, of ma- 4yijfcqqrw2qo*0s, ss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 the looped rough2k-c6xctj1 w w track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the hijxc-wwkt 162cghest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assumejhl . ;8x(am :i6by8hzs)unv/ kw jfa( $r\ll R$ h =    (R-r)

  The tires of a car make 85 revoluti-w :2*nqs najb* . sthw;jh9agons as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of eachj*a-h wt .2;bsj aqg: *ns9nwh 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