A bicycle odometer (whzdcc+ vc 6pkjb7hj34t*hn 1j4ich measures distance traveled) is attached near the wheel hub and is designed for 27-inch np4+vck3cj t j4zc1jbh d 76h*wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at4jeu (gcqs9 2d constant angular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point havegeqc94(2jdsu 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 singaah()h nn(x c tp*xwna/)sq8p(nzs :*le value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Explain.yy 9cl y9 abwe5e-0qh wh,fd01

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 l f02 vt3hv)+x)dhd/hb ahu8ql+4nw b your hands behind your head than when your arms are stretched out in front of you? A diagram may hel+dlh4nb+2ufx t 0dw3b ))a lhh vhv8/qp you to answer this.

A 21-speed bicycle has seven oa-bp 1-zb10,wrlqnc sprockets at the rear wheel and three at the pedal craq1cnz ,bw0-o1 bplr -anks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able t)shvvw(x f5o+8 x qctlx/,-cko run fast have slender lower legs with flesh and muscle concentrate c 5kxlo)s-wv+t /vhx c(fq8,xd 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. 8–35m)sfh 5pg9snv,2 d8 e:rhpb9ncv83 a n) carry a long, narrow beam?

If the net force on a system is zero, is the net torque also/fp,a*z hm nd2om*gw4 zero? If the net torque on a system is zero, is the ,dmg*poa*42mhn/fz w net force zero?

Two inclines have the same height but make differentq p ucv. )co3o(uja3+keg 5i r*mq3r3a angles with the horizontal. The same steel ball is rolled down ea33p.qa3* gc)r3uome qaji(o+ 5rvckuch incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an incline. Ood p2lm7*l*ey ulj)gfd- w 8hgx3x c(*ne sphere has twice the radius and t3f 7p2cldx j*- m()guh*8 dwe*lxoylgwice 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 mass1)8z ozijay0 tv6igr.. They start from rest at the top of an incline. Which reaches the bottom first? Which hasv j)izig10r t 6.8oayz 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 lwb )boi 7sx3l-l4f2 iangular momentum are conserved. Yet most moving or rotating objects eventually slowoblb-4i7ls wxfl 32)i down and stop. Explain.

If there were a greayp+g17lwy 5u41rohj bt migration of people toward the Earth’s equator, how would this affect r y5jbgwhl1u+p4o71y the length of the day?

Can the diver of Fig. 8–29 do a somersaultnkjl x(ep0a8 ; without having any initial rotation when sheejk;x (80a nlp leaves the board?

The moment of inertia of a rotating solid disk about an axis through its centsd scssw w6abw-uah+8p7z(6 5er of mass is 5s67h sa8sbzw up+6as wc-(dw $\frac{1}{2}WR^2$ (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 holdu abufg2w f-wdpfjd73w1ijj 0*q2qk1 ,8as l; ing a 2-kg mass in each outstretched hand. If you suddenfliqqp2k d 81-0*a;gjuu,w7 f wwjj2sf1d3ably drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and havewr*q 9cjgwku 412tb0 n the same mass. However, one is hollow and the other is solid. Describe an experiment to determine whichjbwck qu n4t1*r29 g0w is which.

The angular velocity of a wheel rotating on a hok jh ajuc3*q r artph4m) d3pda6+:21srizontal 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 accelt 2r p61acp rhdjasu*:)a43hmj3dqk+ eration 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.q ong+/)a7zc;w u-utx What happens if you walk toward the centuxg7aq-t)+; uw zno/cer?

A shortstop may leap into *4unzt2 vy;qw pt-)czthe air to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will noti)4v;cp zn t2wq- zy*utce 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, d(5v 66lr1evi1eh tyykpls:82jary o *iscuss why a helicopter must have more than one royy syelt6h6p58kv r2l11: vo*ria(je tor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles8c-i ei mf(7ecbz 9ho7 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 because of an amazing coincidence. Calculate, using them p .. 5+lfiz/qlng,jk information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Elp kigj n5l/z.q,.+mfarth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,0h :a p.d: yyj:a8t 3aq; ,gbtgj.lf9ys00 km from Earth. The beam diverges at an angldjahf: g qpyay :s 3tab.j8ty :;l9,.ge $\theta$ (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. Wd* iog48d.e+ igjwrx 7hen the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as txrd+iio 4dg*w7.j e8 ghe blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the ball pkc8klfo;,f6*fn y +amakes 15.0 revolutions, what ifk6 y;k 8of,a*nf+clps its diameter?    m

  A bicycle with tires 68 cm in diametks c2h) ,c6t, abkg:cyer travels 8.0 km. How many revolutions do the wheels ,c,cg kb 6c2ya h:)ktsmake?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 250 ww9ws:n8m,3i v2wdab0 rpm. Calculate its angular velocity i,m3ww a2wbd nsw8i9: vn $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4w)fz u pl/9t)(mn)o7t 8yiyglkj d01f.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from the center8uffdz(tn) kt lj1)7m0/) g 9wiyyp ol?    $m/s$
(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 v,7l1 2/v5jzafbfttnrss h 60 orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a cfs 7r 9vz +aoytgskm e13aba 99l49i*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 particle 7.0 cm from the axe yx11v0*sun 5z*p bm.hsm6mxis of rotation is to experience an acceleratio.x mv6eb5h*xnsu zmp0*ym1 s 1n of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 1fzl9y o*9/rqkua u., b30 rpm to 280 rpm in 4.0 s. rz.qo9 al/kuybu f,*9 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 radiusezsr t u(4*ourel84 l6 $R_1$ 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 spacecra xj q6./5a,z+vkh pm2rayv cj8ft put themselvesya vk ,a 8 562p+vxrhmq.jzjc/ 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 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 22072zixiavp.m 2et9 a/ c s. Through how many revolutions did it turn in this t2.v 7ci piz2e/9tam axime?    $rev$

  An automobile engine slows 2c9vul vx.6ix down 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 flying highspeed jets in a whi2 gm;x eyr)(adrling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching;dga ( yem)xr2 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 240 rpm to 360 rpm in 6.ite+xpmx5hm0 - (p7 pm5 s. How far will a point on the edge of the wheel have traveled in this timem0tmxph p-e5 +ix(p7m?    m

  A cooling fan is turned off when it is running 1+itdyzvmw6t qyf2jk oc*3/5w/ioa2 at 850rev/min It turns 1500 revolutions before it come yi5mvk ft3j+d*/y6 2too q 1czww/2ias to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces if 0,eiz*cry2k (( m,fhnh 4boits speed uniformly from 95km/h to 45km/h The tires have,n fho4reby *(h0 imck fzi2(, 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 reduces its spvj(5),rnp1 i ddu7iyvhlv51 *xhf n:l eed uniformly from 95km/h to 45km/h The tires have a iun lfl515 v ,ihnpvd*h1yr7v()d:xj 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 he: z5b3hot3(rqkde bhs9j3h,d r weight on each pedal when climbing a hill. The pedals9:t e bbjhk3qodr5h(z3dh,s 3 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*jb;,kszwbg/ 6zd-dm on the end of a door 74 cm wide. What is the magnitude of the torqugs/ 6;b z m,dwj*b-dkze 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. 8–39. t)dm7 vyc3f3 q dg;v:uAssume that a frictimcgydv; tf7d)3:3 vuqon torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a mit:bj+a;h ;h+)vdgwdassless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction ofh;+vbiw:+dj d h gat;) the net torque on this system.

  The bolts on the cylinder head ofuc e lql)5loh9-ze 0yv;95by k an engine require tightening to a torque of 38 uvq ol5;0zl99yebc-e h5lky)$m \cdot N$ 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 sphe evab9q+aa(2 bre of radius 0.648 m when the axis of rotation isbe a9v2b(+aqa through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. The rbwu n plt*n:6(im and tire have a combined mass of 1.25 kg. The ma*6 utpbn(n: lwss of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod pi0fyck :izp5.9oz vqnx s 4nv*6)b,zis rotated in a horizontal ciifnzy9oz0s6)vpvi n kq: c5xbp,4.* z rcle 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 r5 wigt2 y;t /;vz1zla f.g nsdpxv-kwk0...qg otating at constant angular speed (Fig. 8–42). 2zv.gvwg ;kwgxsf0z5 .yk;p-lq i.d/1 t ant .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 array of point objects shown in Fig. 8–43t*ym2 i-3p j sui;oz9 :a5usol a5:p9- 2ol*am3u s uyiojzt si;bout
(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 obnvr3l.fjjbeht48 ka.c;s0 q2l9 ea( f two oxygen atoms 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 satd u(dj xm:n ox2 7,6l5/st ip9a(xqbzzellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocke(txs 6dadxm7p2 l( , z9jboi/:xu5qznt if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 cm *tjm bg*rlx96d f3vzxdw4 6g+and a mass of 0.5x9f g4vxd 6jlg +3bztdm6w*r*80 kg. Calculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelergtl3 n+2py)r xating 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-rougvfynej7(3p+; j )niind 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. 7(n+vipin;jf e)3yg j 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 rotatjr r2rt/ )a9ar2mxa e3ing at 10,300 rpm is shut off and is eventually brought uniformly to rest by a fr2mxj er/ 2ata9)ra3r rictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a 3.6*2zlfoxm /h:/b.p w qb-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 solelwh*y02x 8 :qocpy esv)y by the action of the forearm, which o:* yhcqx sw8)ve 2p0yrotates about 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 bmz ) nak(wno((wxw23ue considered a long thin rod, as shown in Fig. 8–46ux( (own3zmw2n)ka w(.
(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 consist,iq,u+a1pn0vxy0:ikmy.gq/ o c ur7 zs of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest within four full:w:k.ue le l1h turns (revolutions) and releasel:u:keh.lwe1 s 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 has a moment of iglc- 3(f7lp ii o,lzt(nertia 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 1e 6it:dy-i dj280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass w2m (p33ux7ju 3*m8szl oridw7.3 kg and radius 9.0 cm rolls without slipping down a la3 i7s38u wro( 2jmpumd *wxzl3ne at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as thegkql el8 4koirvwki7* -+:i i3 sum of twwko:e7 iq+3i4*vr ilgl 8i kk-o 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 1640 kg and 4ihjj ib/0o jzpq6 d (qnkzpu; :0u*v*a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation ra:p;iqj b* zz (ujoqvin/j00phk 4u6*dte 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 starts from rest and rolls without sp;d7dckm(v dq8y kp c6 i-7p/jlipping down a c/k j- ipy(v7d8mp c6pk7;d qd30.0 $^{\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 verticat3g.uy :azdf0.vj x*vese bxgyn;4,3 lly 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 pole just before it hits the ground? [Hint: Use conservation u4vnxggyt,:vy3x ;f*b0ade zesj ..3 of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the eb1piqj msh, 7 k8 qj5eymfl/32nd of a thin string in a circle of radius 1.e ylb8k3 ,7fpjqq2mijmhs15/ 10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentub m4*m1wjxs4 gff9w9y m of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotating at 15001w xygm9*ff44m9w s bj 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 h s9*79n1bq4uht7b7q : xojhmlpb a wom.r2n2pis side, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal posi2o. hbh:9sqajw179 bqptmxo7nnmpu2bl4r7* tion, 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 shop*zy +t n4-k xfc0fv.iwn in Fig. 8–29) can reduce her moment of inertia by a factor o0-xtp n4c.yf kz +i*vff 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 angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can 6cm9e;ya9wj ,xy *ztbincrease her spin rotation rate from an initial rate of 1.0 rev every 2.0 s to a final ryt6wmc*bezay, 9; j9xate 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 cenz2-z rc v g k(7-v4rqe)lmxs1kter 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 disk of radius 8.0 cm, onto the center of the rotating wheel. What i- msv(ze4) lz1k -kgvrq7rc2 xs the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angul r/d(ekpgrh. 7ar momentum of a figure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum of thh+*gzie /r3z0gl 0mk)tnwm. ,*g weone Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is dropped onto an identityccr*,u+r nwy e:px *2 sj8u.cal disk rotating at angular spu8e*.c x 2,t nyu:pwrrjy+s *ceed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2 aczlhlufmwh(+ a3(di44c1t( .4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg s a3:iddif)7h lctj e*:tands at the center of a rotating merry-go-round platform of radl)jf 3:i:7 itadh*ed cius 3.0 m 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 wit/r3w+ ,crgj1w.wpkfu.u-ag/qu (xms h an angular velocity of 0 wr j. mggu1/3qufwpwr(+cs,u .- xa/k.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 dwaa+k51 a5gvnna-37l 2alzgby .jmev-prf, losing about half its mass in the process, and windingg75lj2vv nkb3ye+a1lz . g a-aanp-m5 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 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 wyzmw0iy/vcg1 cdo*: 1 inds 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 gpz--ffdbadtq 61,y89ph g( x$ 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 b-1+zp0ymi ap freely without friction. The moment of i1yp0 miazbp-+ nertia 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 pers(5r;k.z6r+7l b ritlhppt9*1a/i wj qv a vf3jon stands at the edge of a 6.5-m diameter merry-go-round turntable that 5jhb 9ijq/pkzr ;l.lai1rw 6 rvp7t (fv*+ 3atis mounted on frictionless bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls o wm,u, q7q,t(jte1 c2do rup5tn 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 slipping. What length of rope unwinds from the spoo5 q(to,uw, 2 qt rj7,d1mtepucl? How far does the spool’s center of mass move?

  The Moon orbits the Earth:/p nj+3i bsiptu9/n3nwet 4f such that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to/spib /39+: itnnwjp4f nt3u e its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates frorl;x g ,5uqh 41h, :teogmek,fm rest at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shap o0n +z .encl1k pn1 m*3/tffu-dxulp9e of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleradt flpkomnfn/.-nlz1 p3c1 e9x+*u 0u tion. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindricaljjutm+7 f8s1b .aov7b disks, each 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 end of its 1.0-m-long string, if it is released from rest jbmu7b 7+sotf 1j8av..    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of the rp, tl7jzfv1t5m)4q; dyp s de1lp g(lz) 1a;fdear 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 great5r7/o w 3i9cdlymz8d2z3k .oi)bzc yl , were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that ozydk)238/lo.9di zwb7rlyc35icmz the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use energy stored indu +p ft.::k:3r7rlj4wua psc a heavy r jc:+3fw:u.dt r:lp4pkr 7ausotating 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 illustratesenoz p:4dyg3xm3w ;qc,os+ 4m 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 ho8o y45-d mzbqwrizontal 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 lengaf5zz 1l gim8z/1g f+ath 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 faffz mg5z l1z+/8ag i1orce 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 the floor, and hqk le.en 8:d/ycg579yf :xahineg,;5 xiu/owe want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig.; yn5i/iuq9c ::ey glefkxdaxo8h/h, 7 g.5en 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is d bcxieeu: ic;2 k9:7ope)3cjmaking a turn with a radius The forces acting on the cyclist and ;jc 7 92euix:i:b ko)3cdcpeecycle 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.gvcy8mu n7z -0/dy za55 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this featvdzu57g a-n0m yz y8/c, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her *m h.k knlqlnw07,w d+arms as thin rods, making reasonable estimates d.hkl w7 *n n0mk,w+lqfor the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindricav wkq/44mvrb-p7s ukws8 /c8rl plates, of masswks-v /bp4rvswr8u cm /4q87k $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 a /w5lf agg, 068if l +oxy+lx/v(ewocwlong the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h8exfx/, alovg+c(/6 0ly5ow lfgwiw+ 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 dv j w/e go(q4y)h3dr;bdnyt)8 o not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions asaqv2qo2dy: z*5 bh30rjt t1nj the car reduces its speed uniformly from 90km/h tytbh j15 do2tqjr* 2vza3q0: no 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s