A bicycle odometer (which measures distance traveled) ita7t(;2rjb *qpdiio .rh n2bk.1w7pq s attached near the wheel hub and is designed for 27-inch*; t1ra w(pi 72jh oqpi.tb.72 bknqrd wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point . urr;/qhzne.io m85y on the rim have radial and/or tangentia.;.5 ry/eiqnzomur h8 l acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the angular vgy+-+h) cd/i ndqgdvz0+b8nkelocity $\omega$ Explain.

Can a small force ev5j 0i38mz yvzwer exert a greater 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 do4 gon (6qv1nzhvv/2im a sit-up with your hands behind your head than when your arms are stretched out in front iv n1(6gqovh m2/4nzvof you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear -x 58bs hh3k9 ygh+wy(g0jvd*:kj jeewheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is jxv3hykhg8gdb+ye 9j e0(*s-k :w5j hit harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on bei6cq(r towy;g; c;1iv eng able to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34)t;o(;iyq 1gv6 ; ewccr. On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow3 did3+hn8 o,clb3gqj beam?

If the net force on a system is zero, is the net torque also q.xp9q r1l27f2f3qz.s ad k xfzero? If the net torque on a system isx.37r2q.p2axslff9 dk q qf1z zero, is the net force zero?

Two inclines have the same height but ma i/ q30gsxd,mnke different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will t/gqsnix3 ,d 0mhe speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rollinzy k:o. l)+ocpg (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at to.: +zkcol)py he bottom?

A sphere and a cylinder have the same radius and the same mass. They s;i)f eow6kkq*/ 6j s vms,itc+tart 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 greater ro+it/qj wkes6 6 )vi cmkso;,*ftational KE?

We claim that momentum and an3iyih5bvo71s7 bo3 ghgular momentum are conserved. Yet most moving or rotating objects eventually slow dogi5y 3h7bh si3b1o7 ovwn and stop. Explain.

If there were a great migration of people toward the Earth’s equauz/7g014ksec j dpko z0 +zrc0tor, how would tgzrc4czuo00k +1kejd7 0 zp/s his affect the length of the day?

Can the diver of Fig. 8–29 do a somersault without havingp2ry )k osb24 wr:(furj0i(do any initial rotation when she leave(r4 :of)0bi(yd po wr2skru2js the board?

The moment of inertia of a rotating solid disk about an axis through its cenmp:9dev cs8sy ,m klny)9x93w ter of 9 ecdvm wx3,9k)syys 98npm:lmass 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-kg9r zsz3bary 4c m(63eg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or staszcgyrm b3r3e(z46 9 ay the same? Explain.

Two spheres look identical and have the same mass 38 qng:k)m+(ynq)l/qcyegg msx gz;. . However, one is hollow and the other is sn;+l gqg s/. eg:qknm8)qx cy()3 gyzmolid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a h 1jf,vu5, m/ha5o8s0e vdodj morizontal 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 t,d50mfeo1 u8 jav5/ohs , vjdmhe tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standi+zvk- ha6( m5 + 4oz6mbq6og k5nieeying on the edge of a large freely rotating turntable. What happens if you walk toward 4(+ey b+vq iekza o6h6 mgz-mo6n5k5ithe center?

A shortstop may leap into the air to catch a ball and throjz ya x:6 ;qg7kj4io)rw it quickly. As he throws the ball, the upper part of his body rotates. If you look quick:y;jq)6k47 axr ozjgily you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of consesmoay0tj2( 2o rvation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operam 0t(2 joo2asyte to keep the helicopter stable.

  Express the followings92k;xo) llih/vb(1*x jmz q a angles 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 the fth ;tmhefa) jo iejtwe+; epm*n2/17/ (e-nd information inside the Front Cover, the angular diameters (in radians) of the Sun and the Mot e*ahtn impnt2e o-/h;(dwje7f/m; +ej1)e fon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 ,ex mj61 z8plbkm from Earth. The beam diverges at 1,xpmlb6j e z8an 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 motor is turn:p2x3a 5li wx/ s. e, w;h8tn2ozbr xnr)gm)rled off during operation, the blades slow to rest in 3.0 s. What is the angular accele;wt n25) eix2pb r/.,xrml rolh)xn8 g w:3sazration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another chil61tz o,l+jl2kc+p-xcc ecl z6 d. If the ball makes 15.0 revolutions, what ilzco l+ ,zp6126 -etkcclcj+x s its diameter?    m

  A bicycle with tires 68ycfc bjt4: 71i cm in diameter travels 8.0 km. How many revolutions do the i7bc:cj 4ty f1wheels make?    $rev$

  (a) A grinding wheel 0.35 m in dnw;fm c,kp k*w dp+)caa3;7m ziameter rotates at 2500 rpm. Calculate its angular velocity m czmw3kck ;pwa7 ,+n)pdf*;a 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 revolution in 4.0 s (Fig. 8–38). (yrg13r82f7ohkym0o: )di bzsa) What is the linear spib d7r1 fry:0 8o )yo3hzm2kgseed of a child 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 vegpe cx(-ys.3sv 6v5ou rq0qo3tj f/ 6llocity of the 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 ofsy rp; * s*u9m1tb x4s +*ejqir66nrpj a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if zgrn /x 7x92nme g-pp fh1l0q1a particle 7.0 cm from the axis of rotation is to experience an acceleration of 100e 9 07grxmxpnq2 h-/fpn11lgz,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center i)l1.l09b6lw ucg5+dljm p mhfrom 130 rpm to 280 rpmp gl)ll+9c1l.dujmm h6b 5iw0 in 4.0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radius s-xaykx7/a ( x$R_1$ 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 aboarv -brx,t x.f) pjw,yl9d 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 during a 12-mwp x .br )ty,l-,f9vxjin 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 15,1 fhe9.bk/ s7y sv*o wi96zymy000 rpm in 220 s. Through how many y9mhb9iyoz.k*/7 ey f6s sw1 vrevolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calusjh7 hhh-x,8enk0)slvc ki28fw ;z,culate
(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 pcj+hu w)3 i1bflying highspeed jets in a whirling “hump3+bu iw1j ch)an centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 360 rpm in 6.02c xy,tkplcts 6g p+05 s. How far will a p20y t0+xtp,c 6p slgkcoint on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 116jup s5 t4glpvm* ,vb500 revolutions before it comes to a stop.vp*jmvgb6 lp ut5s 41,
(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 makb8ocrbn7u np 7snm7+2e 65 revolutions as the car reduces its speed uniformly from 95kmnnb2upo7s8mnb c+7r7 /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 mr -g g4kq h(md usv4ea8o,6s7xake 65 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h Th og xush-e48,da46(q srkv7mge 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 a:(2rfk;p,l; otwexqjzsn 0x 2ll her weight on each pedal when climbing a hill. rxxnp (j ,2;fqz;ose0t: 2klw 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 forc xwqro aih5akn)o-,(wzb:42 j e of 55 N on the end of a door 74 cm wide. What is the magnitude of the torque if the or-,w)w(2xan i:a ozhkjq4b 5force 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-- jcaer.x9t9 9bhuhbj6mj z; the wheel shown in Fig. 8–39. Assume that a friction torque of 099jrje .-9hj zuc bt-axhb m6;.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a massless rod which pivessz9 ykt,)eyqdwn tz2:1r53o-+hprots as shown in Fig. 8–40. Initially the rodnrs o2 e,+:y9r q)hd 13wztepyks 5t-z 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 cylo7uq5ec)skrg-0*, yco 33 kunwjtaw:7w cy s8inder head of an engine require tightening to a torque of 38 ycujwko7sc 87:5y3*) tnwc wr qe-, gus3 0kao$m \cdot N$ 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 m when tn09bob- +hqq cx7bd0cl, x(pkhe axis o,q+d0qn lh -cxcp79ob0(kbxb f rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertiah)k 2r- rci )bl/g8dsc of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mas)rhsicd8 r lb)-/c2kgs of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in a hey0p ehr16nl dyns 3)6z8 b h-2a*kieqorizontal circle of radiu2ky068s l*pne r 6dz1 3ieyha-hne qb)s 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 rotating at constant angular+sq/raco.4iwvt vlj- m) kxxkmp 23+ ( speed (Fig. 8–4k aiclq /k2 m3pmxj+o. +v )rs4v(txw-2). 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 point2a; j mru615uy-eplutr w oh,) objects shown in Fig. 8–43 ab),r- ;6 wuyt2mup lera5 h1juoout
(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 t.fsj9-xh8igh.ko1 c5l6j d6 +c aspn uwo 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 satelljrjiep/ 99u7iite spinning at the correct rate, engineers fire four tangential 9e7ii rp j9/ujrockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket 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 an9ze +(6k,wgjkay;epd jl +b3nd a mass of 0.580 kg. Calculatdekg yw6 ,ej+(9jlna+k;bzp 3 e
(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 swzj 5 y6h 0d+huv )cvoo9f*ovi2ings a bat, accelerating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentz *oq:(emod xkitc7s ;h1ti g740m 4ayially on a small hand-driven 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 accelerati( 1t;o:tm7ayc*x m d4khso i4eqg z70ion, 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 a .ehm9w89j9.h are2 x pbrkj5brought uniformly to rest by a frictional torque of 1k9pa 9 m8 xebh9j2wj5.rh.rae.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-kg1r9amq5ue). k wus) gh 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 thr d)vsw/5 mneqo .nxm3/, decx3own solely by the action of the forearm, which rotates about the elbow joint undwd)en5/ /d m. qnx,3 scv3eomxer 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 t35y9:miicncl/mt tr gr5b. bq6z m,msxl-*j9hin rod, as shown in Fig. 8–4bz,5.nl icm qm99:mrctil6gm3 sxy- j/rb5t*6.
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine consists of two masse lwr8a-gsbt2,s, $m_1$ 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 turnm xtu:nl2f+t k(h+9kx(d:g du oe3im1s (revolutions) and releases it aumx:n+31 li(o t:e2hd x 9(m+gfdkktut 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 moment70l6qj xei+ u)bqx gu tkq6w4( 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 t 8uotj5 zw3e3ehmsy /:p(vmkco e/, m/orque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and t0n1 h8q9dqjl)wul e+7njxe 4 radius 9.0 cm rolls without slipping down a lane dt0hlq +lunx 178neq je4)wj9 at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect s1nzxqed19 2k75fchck d7: nc9tvpc ;to the Sun as the sum of twc 1sqhpcvk797ndxn5 ; f ce9:z1c2tdko 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 k qe 5xj2,fgwicek:.8fg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it ix :qkg.c8 i2 5fw,eejfs a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and sbgu5. +dj ,pzrolls without slippip sjd,z+u5g b.ng 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 balanced vertically on its tix;nac/ acqic:q1 6k*kon5 bl/p. It starts to fall and its lower end does not slip 5ik *k1acx6c:lonnq; c/q/ba. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum h- :psbcm/uf3zx txhldm 65l uv7 )o2q+4 gy*pof a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10g z73 -5mlph*l64+c2)dbxq ou:myp fsvtxu/ h 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 grindik; dhypn-j.avyal m)q,c0o 7io7)w: vng wheel of radius 18 cm when rotating at 1500 rajo. ,v7);ol:c 0wkd-)y 7 iynqv apmhpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, handsv d3 d18y(bl8rphn,4 :f* qyjah xqiv: at his side, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases dv:1nh xdjb8iy*qqp(h,a l4r8f3 v:y 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 ol(y+5 .klgp(lkghe*mdp; yv+u ,gg ). 8–29) can reduce her moment of inertia by a factor of about 3.luk+pyl5*kd+y,.hme(g)g( g pv ;g ol5 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 increase her spin rotationdal37u4 7jk zq rate from an initial rate of 1.0 rev everz3kq4 ju77da ly 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 qy qwdx1),s :eaxis through 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 ax1d,q ):wyq ess 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 s8)p o+,vb e* kg.pft5dm s8 nyemae,2lv-6 slopinning 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 momedezlr0g8v),dfon9, klca d,(ntum of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is dropped oqo6y3us9e9 j9a jg;ntnto an identical disk rq;93 tsa9ju 6ge9y nojotating 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.si ,mwijs 1()d4 $rev/s$ 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 attbrnyrbp hayvu67)kk-73q 1o f2m 7w- the center of a rotating merry-go-round platform of radius 36 pwnrk) fyu rb k1vm--y o7h7t237bqa.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 rotatin*eltx g9( mx*tg freely with an angular velocity of 0.8 x*x e9* tt(gml$rad/s$ 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 int,ofj pb*oejw9e 68l4pcwp* 7i o a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries awoo8p *4p6p7ibwjj ec 9fe *,lway 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 involv9zb-15y dr a a :i,v/jv*bmgipe winds 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 qiqybuy 9 3(ze/ bv)y-$ 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, mloc a0vw /xc34r8z e:initially at rest, that can rotate freely without friction. The moment of x4ez3am cvo/rc0:l8w inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5- ntaur-nww2 c:ke9 ) qt75)zxbm diameter merry-go-round turntable that is mounted on frictionless bearings and has a momen 5qwn79t)zn -)xbuwce:atk2 r t of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of the rope lying ,n hab rx7l3e;j)zjp(on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the sx( 3nzh)a bej7; lpjr,pool’s center of mass move?

  The Moon orbits the Earth succq*z9 4o+*eez :jt ,pjtt j7qowrr+a. h that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.9,or4z jt7epqt j*tcw *rz.jo +e: aq+)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate 2-rxabbb n2 td0)y9fpy 4 gjm*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 0.20 m acquixdkj dp0ijm(9i+8 u3qres a rotatxj + d(p9j38kuqmdi0 iional rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two -aej*bcfd5. qmz42 a-m -rsqrsolid cylindrical 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 the4md. -sjqb* fma-- rrc 2aqez5 yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, hozy r3z:wi89o yi(9ellw is the angular speed of 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, bucoab v v)cw; r+6i+6hxgd++ wst with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass x6a+so+ch6b+ v c +wwvg)rd ;icarries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution a 9/ou), qi qgg el.q,xuxmsf6:pvl18s utomobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of dvf 9xgu:lp qiq,l1 ) su/o.mxqgs86, eiameter 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 :y-jb,o; 0io.a8j p+by3p 5oepchr nwan $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal sub6h,wm2m2v mcfn3ye* rface 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 frictioni/in-p mki+p *(qkv/q ) about a hin qi mkpq v+n-ik*/(/pige 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 t y:1iyn5h()dpb0i1kt elm1 mvertically on the floor, and we want to ey(tm5hbi 1deti kn l1:0)pym 1xert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with sn kv 7am)5n1rdpeed v=4.2m/s on a flat road is making a turn with a 5knnra7dv 1)m radius 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 slinvpzw 3lhzu0a oe6c4 5(xx*: dlnn(my ;g of length 1.5 m and begins whirling the rock in a nearly h 0pwv4:a;menc (3 l(doz xy6*hu5zlnxorizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skaterpvm9 aw2r27g y’s body as a solid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater wi2wv y9g am2r7pth outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindrical ij0sq e7,x5pi plates, of massxq p5ei7j 0,si $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 rx,1n,03wl o rm cjpuja v3+2mvolls along the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is0r1jol3vw v,px, njum+2mc3 a 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 aw4r 65v l ;6wue :m ka+wdyr3h;vlyp5ossume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolut)y* uyx8)0) rt)ntdjnicjvso0a1uo9 ions as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a)0vti8yj)u)1n *oj ct)dya xsuro )09n 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