A bicycle odometer (which measures distance traveled) is attached nearw*; +2ykiliep dhc t .p6ls*6fke +wy1 the wheel hub and is designed for 27-inch wheels. What h+lkwiecp 1*yl6 +ep*hyt.i skw d62f; appens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at co,ajkc d-buyol+7y * b0nstant angular velocity. Does a point on the rim have radial and/or tangential accel-dabcj, y+l*70oky uberation? 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 describediyvpxu /399 kz by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque t9jcjb4k* z ccu gl10-whan a larger force? Explain.

If a force $\vec{F}$ acts on an object such that its lever arm is zero, does it have any effect on the object’s motion? Explain.

Why is it more difficult to do a sit-up with your hands behind ydc* y3me f8xe7our head than when 7xced8 fme* 3yyour arms are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the pedal h,;9sr5 x *m)62zmqyt vijj2yqj ip0acranks. In which gear is it harder to pedal, a small rear sprocket or arvaj9 q2 ,iy2 p* 5 );zxqiyt0sjhmmj6 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 witqfa5 4 lj5u(m0 vw:savh flesh and muscle concentrated 05ujaf5avq4vwls:m ( 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) caryjw7)vh7/my0v +v wwu r.1pax ry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero? I.hb+ jz f lcww;g09i:3g sgm6if the net torque on a system is zero, is the ne gwh6+fi. gsljmw z0b;g39c:it force zero?

Two inclines have the same heizgbj x un+t/p,8- j4jfv n 22(oxvzi6tght but make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline wivnxjjb,6 fxu np+g2 z-(4/otjv t28zill the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rollingyl zm.2xnp50yf22dja 6 v ;eiy (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bot;v. imljp2y2zay2e05fy6 xd ntom 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 wbm61p5 v4m dmwfk74hsame radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Whik51 vpmm 6bwdw44hf 7mch has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are cononr(f 3 95ss6 nrpg g-(kjnjf3served. Yet most moving or rotating objects eventually slow down and stj3nsfgonn69f-rrg 5s jp(( 3 kop. Explain.

If there were a great migration of people toward the Earth’s cim7g nsp.3p.7vgi k) 5z e;itequator, how would this affect the le)c35 pnimg is t.z7pig7 ;ev.kngth of the day?

Can the diver of Fig. 8–29 do a somersault wit) i:s6bay)/me dk sv8shout having any initial rotation when she leaves thea) /b8:6v)k yssemd is board?

The moment of inertia of a rotating solid disk about j3 k276 tzimkvh:r,cvo pz/d-an axis through its center of 27h,6rk cj3kmv:viz-zodp / tmass 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 sittinp-g8bd it n ysj:h:6j:g on a rotating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angubnjs -:8:d tig 6yjhp:lar velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have tgzw6r9u xk.ie,u+v( jhe same mass. However, one is hollow and thuk 6 re uig,(+vw.jzx9e other is solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotaep-wd 9s .sgk awj)xiov:r1 sf;e 87t+ting on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing ot- ewsgjviws:1sf9a.x k+ ;8od p)er7r decreasing?

Suppose you are standing on the edg8x*/.beuhpv+oa ptbo6gi; - be of a large freely rotating turntable. What happens if you w8g. a eoxvi+pp*;u -hbbbto6/ alk toward the center?

A shortstop may leap into the air to catch a ball and throw it quickly.a 9*yh sh2 s7s+e h152eimuprw As he throws the ball, the upper part of his body rotates. If you look quickly you will notice th5sru+ hh29s 1 mipeey 7as*2hwat his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservatibqwa;:;r tq0 l+irv 1yon of angular momentum, discuss why a helicopter must have more tha+lr :qiwta;0rqy v1b ;n one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (a) w pjppof5ee62*6yv( b30 $^{\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 coinc20z u8 tkpbq)v- 0q,ey(z drsnz7a(yaidence. Calculate, using the information inside the Fru8e(,(y-v dz )k0zbyaq taz0n2qs7 pront Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beqjw1kk, h;gb4 am diverges atkh ,1kg4bqwj; an angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate -jikq 0 26ct7s sl6jxmof 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What jmq6 -jt ki cxsl2076sis the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3 u ijj*4(s 8ak3)*qzkauy k5nk.5 m to another child. If the ball makes 15.0 revolutions,zjnqkas3 *ku8jk5()*aiy4 u k what is its diameter?    m

  A bicycle with tires 68 cm in diamete:04b tqdoknw5oh8+ c9c3rxm6m u2y zyr travels 8.0 km. How many revolutions do the wheels make?2qy34chz 6k:m 9cyo 8nxrtw 0u5b m+do    $rev$

  (a) A grinding wheel 0.35 m in diameter rota xxx*;ho4ghd qbbik u7ir 1;13tes at 2500 rpm. Calculate its angular;gx*xbhu3q1iho;xkb i7r 4 1d velocity 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. 8–3xkj8*xplq i) 6td4aw* 8). (a) What is the linear speed of a child seated 1.2 m from xj *)w4dx8p*ka 6liqtthe center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orbit around the Sun 6*84y/ l-sxvwn/5gqi vxf oa-d: lurj   $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of yk c,1zg qwdp3ro(/+7 sagxv9a 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 ib3kkb0 bjh. ) z4:xw vjs9fm0af a particle 7.0 cm from the axis of rotationk zv:kbs)jbh 03w.amx94j f0 b is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformll.n7a)u zcl y*kwi-/hy about its center from 130 rpm to 280 rpm inul-zl/7w)h* nk.cy i a 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 radiusy/o-xw2zt0 tggv3 w l,y0ul(f $R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboard the Apollo spacecraft put themselfv0ols ;0js)xves 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 duri0jl)f0 s;sxo vng 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 accelerap;a rsg97 uu+ntes uniformly from rest to 15,000 rpm in 220 s. Through how many revolutions did it turn in 7asu; p+rgnu9this time?    $rev$

  An automobile engine slows down from *f-10sbj:9 (tnhncc fhak + lx4500 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 whirling “h4x4a dlg0 aik,o6(o3pbu.x3) te1 yqu wqpz2 buman centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching zxt o4 g3dybbku 1p w6ueqq30pa)a2(,l4o.i xits 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 acceleratesfe m niug(p22; uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled in this u22e;impf gn(time?    m

  A cooling fan is turned off when it is running at 850rev/min It tv4 y;vzcfl 0zt1h-9lo c iqc:;urns 1500 revolutions before it com;ol-0q :v9vct zf1h ;c 4lyizces 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 its spe-jh51 djxan l,irg :f9ed uniformly from 95km/h to 45km/h The tires have a diamjr n dahif,95x1-:lgjeter 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 itlfm *grr6 y j3.9t*p,y jpx /tvm9y8tc c+be2ds speed uniformly from 95km/h to 45km/h The tires have a d938jp2ry9,*y.gf x ctty6jr bm e/dlmc p*+vt iameter 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 climbing a hpq q.g2+ qaru:ill. The pedal+. gapru2:qq qs rotate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cm wide. Whxat0cnz+, c1 xat is the magnitude of the torque if t0x 1nt+xca c,zhe 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. Assume;0zuq +;hb uxb that a friction torque u;zqb 0 hb;x+uof 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 endqp:2u7k u b1bns of a massless 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 dkb2np1:u bqu7 irection of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening t;gcrm3cmc4w4pxw3(7 kg a. mco a torque of 384m 3 (xcm7cw;mw.4k rc ca3pgg $m \cdot N$ 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 zx:xzib cpbo+: 0(z.g of radius 0.648 m when the axis of o.x0+p:gb cxbzi :z( zrotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. The s (49c f,upa7i l4cyuyrim and tire have 9ac,fui4c ly u4 pys7(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 on(oq:w-oqvyl c:;x d1f a thin, light rod is rotated in a horizontal circle of radius 1.2 m.;: qyq 1n-wxcovlo: d( 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 potterk2 tx r/hbzm8t((lg. m’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and the tz(.brg 8(/l2th mkmxclay 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 thw+vmqag: 9b9hmd :3r z ji3g-ae array of point objects shown in Fig. 8–43 a3 a9vwmiz-9a:g:rh 3m +dqjgbbout
(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 tq d(*crlrf m.2wo 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 satellite spinning at the correct rsup)atqe5 w5/g:t p+sate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg anw stes)5+tpa5guq/p: d 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 :(sa p6 xqxsk6of 8.50 cm and a mass of 0.580 kg. 6 axps(q6s: xkCalculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerating it from rest to 3xnr g y(*5iix35yih rss+2)ye $rev/s$ 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 mewhue x0,7a q8grry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torquug0wa8 e7 h,qxe. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut i 2o p1ehh(xry2)p*ghl -imu7off and is eventually brought uniformly to rest by a frictionam7 g2 1you e*x2)h(lihrp-ihpl 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 2)y6fgkc( r wqd:*hh*dj odyh b5k0v.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 b) gb0ggve1 +-po4 9q+cjimbs is thrown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from resc0 1g9vosem)pb+b- b +gijg4qt 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 ls- ,/bd7 7etblga65dsmd nm0b*y:am nong thin rod, as shown in Fig. 8–466mtml,5 m7 dga0b7 n/d*bey-b s :sand.
(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 6/n +dp5x.dh(;w iqy52 sixy.nhckhjconsists 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 accelerate/v*o/ b)h0xua eg*bwws the hammer from rest within four full turns (revolutions) and releases it at a speed of 2aw/ hev xbbugow 0)**/8 $m/s$ 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 momenthrm;6s go8.ar 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 torjcn9a1xbo,jp:pwkz /,- xp5 oque 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+k ;,sk svfvrmh* *p;y cm rolls without slipping down a lane at 3.3+k*;prm*v;y sf sv hk, $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energ ig9236tthkdy-c*j w ey of the Earth with respect to the Sun as the sum of two ter*g2j3ke dy h -wcit9t6ms,
(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 43fojitj m-0v*xags .3f cjb(of 1640 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.jc .4j0(og*vbi t3sxfmf a3j -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 w;gubuwf mdo 3h3tf 8+1ithout slipping owf8;dtu ub+mh31f 3gdown 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 tip. It startsu so(egny;j*fnb2, b, to fall and its lower end does not slipnyfe;ng2(b *suj,ob ,. 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 of a 0.210-kg4yd( lh;ntww t+atl *+ ball rotating on the end of a thin string in a circle of radius 1.10 m at an angular speed of 10.4 wt+wld+4t h;y(ntla* $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.3l nwjsg6kk,s 0f9svdt l s66 bkq)g(.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotating atd.f6wq 9)ks,lgg 3(n6 tks 0ssl6 jkvb 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that i)y z(5wv6h5dv+onqqs2q sxy e-5gmk 2s rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–4q m-kz+vy wd6onvs hy q5xq2es()255g8, 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 momentei;py28uph8. vz-l sg of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes . el;-2higyu pzvs8p8 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 rotation ratqjn0.d ddosbylq ,- dn)o1y. 4e from an initial rate of 1.0 rev every 2.0 s to a fi4oq- )yl0bds..1d,dqnjo ydnnal rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis through its center at an t4bvu4;d403o feswm frequency of 1.5rev/s The wheel can be considered a uniform 0uen4db4m4fws ;o v3 tdisk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinning at e q:q+fu 8rq4z3.5 $rev/s$ 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 mome) hp;wmmgoz/ .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 drth gil +/i62udk *dzp5opped onto an identical disk rotating + d i2pugz5d/ihktl6*at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.42seld ny ii)8)0(m nxt $rev/s$ 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 rotating merry-go-round mxhfo :b3 -j f yfe6zn/n 3ty5.loea +8ul*:ihplatform of radius 3.0 m and moment of inertia 920 o l/+xiehumt- fney *fzon::fj863l a h .5b3y$kg \cdot m^2$ 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 rot/ -tsz*gnvf5lvy8 op:ating freely with an angular velocity of 0. lz*5to -8/:f nygsvvp8 $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, losing a6r,l z8 jmng4vk. pi .oedz3v02*gawnbout 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 rotatioen dn v.a rlj,g*438z2 6iom .gv0zpwkn 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 win;golx v042 en-n+tp8 br,bg8ct zous3ds 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 n3jrn 2r.z4fl3k fh 6g*eqyy+m4qzf2$ 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, ind4t4u oen(0wsitially at rest, that can rotate freely without friction. The moment of inertia of the ow(u44dstne0 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 persoj , jai;hw,pqsra +7h,n stands at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a mhjra;w,is7hq, +p,ja oment 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 let*xbmt( +i;h ying 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*; mb xi+tht(e 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 fa+2.vti3fn *h:zmrfbb ep;h t4 ces the Earth. Determ3 ifzt:;rtb. p*+hve 4b 2hfmnine 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.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 1 m8sd)r;vp3gbf -s nx9x /$s^2$ 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 uniforpb - tn6z+dp5um cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6zpupn td+b5-6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, eai;bn)(kpu1t.jzdanx f z -kp;ms4 13uch 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-longsamtkj np1zn 4fk;zxup(-;13.ud b i) string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is fxp7j3v7d5i k x.l4kqthe 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, but with mass 8.0 times as great, were rotay54lugbv( *qsting at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational 4 b5y*gqs(uv lcollapse 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 carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use enenx7;35l j7 *xcflj rb.ho5iyg rgy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheelrjhfnob glyil5 .537* 7j;xxc 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 illustratesu cmk mg3hc;p 0rpqb;uo/7(9l an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal surface cedu qn*7n5 t:i9iq(nhjd(:j 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 i4yz ;e3qh4b efgnore 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 (b4;q eheyb3 4zf) 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 standie e ne51tndvd+7*j o)kng vertically on the floor, and we wa*) 7onv t+5edeekndj 1nt to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is m )tjl90rv;a avu.ihmi,a)q e:6n8wb raking a turn with a radius The forces acting on the cyclist and ci6w;qv),thlrmnaru8 : 9i )abj ea v.0ycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.5 m and bega8gv c k-v+g1+zf7 gudins 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. Whv g +fa+c7g1uv8 -gdkzat 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 solid cylinder l(/j n:/;wp*coy fvk zand her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skkfy; vjp*z/lo/(w: cnater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists ,0cfhovrf) h7 of two cylindrical plates, of ma)h h,7 crffo0vss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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) lr6bzr/suq, rolls 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 is to reach the highest poiq/)s,lu 6rb rznt of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not ass)5damj34m ,j8gv xpcrume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolu d 1,kgs9bl)cfu*7-plp (b luctions as the car reduces its speed uniformly from 90km/h to 60km/h The tires hav,dpgp k c* -c(719sbbu f)llule 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