A bicycle odometer (which measures distance traveled) is attache2fv 2l4 d4ylxzd near the wheel hub and is f z4xy22vlld4 designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at cons2hy t th .bg65a,4bdlxtant 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 have radial and/or tangential acceleration? For which cases would the magnia5tb4 h lhbgy62x.d,ttude of either component of linear acceleration change?

Could a nonrigid body be described )9 c. gnhch54+qpqsbxby a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque 7gjpd.5w.ud r qv6 h0gthan a larger force? Explain.

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

Why is it more difficult to do a sit-up with your hands behind your heaxxcmm savn *u/him.50v1:3hv d than when your arms are strenv m0/mch*x m a1hv3v us:5.ixtched 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 t*yuwa/0ftio;h65gqa wy0w b, hree 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 it harder to pedal, a small fr ay5 tq60h0o wy/wgu;a*wb,fiont sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slenxfp* h.dyo/e3der lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why .3d peoy*fh/x this distribution of mass is advantageous.

Why do tightrope walkers (Figq z2cx0 h.v0ylzi(a g9. 8–35) carry a long, narrow beam?

If the net force on a system is z ),26cmrpaaph ero, is the net torque also zero? If the net torqu cmh,)r apap62e on a system is zero, is the net force zero?

Two inclines have the same height but make different angles with the horizonthsdh( b;tv4;wal. The same steel ball is rolled down each incline. On which incline willt4(;h vsdwb; h the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously startdfe ) 86orqjk) rolling (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 the brqde8)j )k6o fottom?

A sphere and a cylinder have the sax* nf p1r+a8rjme radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Wrpfnjr *81+a xhich has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angularr/9;v)zu;f(v mxi nwrfi. 4ua momentum are conserved. Yet most moving or rotating objects eventually v)xf;u/zrfrai(m;uvwn i9. 4 slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equator, how oa7 lzf2:* vgpwould this affect the leo*plavgz7:f 2ngth of the day?

Can the diver of Fig. 8–29 do s)4t7,.ca5,oiha 5q op, eollx 0 g6b;wbei dia somersault without having any initial rotation wheno xh6s,gbiibe.i ;e a4cw,dt5lo),q0ol 57ap she leaves the board?

The moment of inertia of a rotating solid1gmn e/in8v7 z disk about an axis through its center of m i1/7ngve8mzn ass is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2-kg mass in each outxy,o8qmivikj: cv8 2 0stretched hand. If you suddenly drop the mj q8o8:x,vc2mi0yk viasses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. Howeverf9opwt.+ 5mo xxt)fh+ , one is hollow and the other is solid. Describe an experiment to determine which isxto9h 5tofmp+xw .)f+ which.

The angular velocity of a wheegywyw r, ga+4 me:lzg)v,z7:zl rotating on a horizontal axle points west. In what direction ig e)7wg 4w,+ v:gy,:z alzzryms 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 or decreasing?

Suppose you are standing on the edge of a large freely rotating turntable.iisq1vqnh u wz5/;sq,cwk7yk./ 91 o o What happens if kc/h1k7oq s 1n;,5i/s wq.i 9quzy ovwyou walk toward the center?

A shortstop may leap into the air to catch a ball and throw it quickly. zr1fv lh8,5oe As he th hv5lozf8r1e, rows the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, disktry3 e+4,w. 7bt ,*ejj wjuopcuss why a helicopter must have more thaj*3 w jru.tpkboy+jwee ,t7,4n 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: tr4s*0pv4zvc (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 tdvfp2 :m vt+rmpx)5 a gf6n*ta-h( 0jbhe information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth. f(+:j*tva mdm a x2)h60gf5vprp-tnb
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Ea b1qwd l:*qlkmlev931 rth. The beam diverges at an angle:qq1llv m1k*9d ble3w $\theta$ (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 6jpa:x zvf//ned off during operati6va /j/p: xfznon, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on zm5;;8qhc a8rby t q/ga level floor 3.5 m to another child. If the ball makes 15.0 revolutions, what is its diamhbmtg r8q aqz/58cy;;eter?    m

  A bicycle with tires 68 cm in diameterxn6bq boy32z: travels 8.0 km. How many revolutions do the wheels maknoqbyzxb: 6 32e?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calcul4hp2v iz.8a -u.js z 3o.fiiclate its angular ve.j3 .2sz lvi-4puch iiof8 za.locity 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 completev 1m-mth4wc;gwz/8hmcu80w t revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a chwh8; wc g1twvt/0 mhmcm u8z-4ild 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 vel:p exeg2 j*gxm ;87obygr /d:pocity 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 of a pk6w iwkqxxji 2(0)1g koint
(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 c2dmawhr 9f 1(lhu1+kk m from the axis of rotation is to experience an aa2 w k1k1d(hlfrm9u+hcceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel avgvap v6zs+ .3okp;nw5n)yt3 ccelerates uniformly about its center from 130 rpm to 280 rpm in 4.0 s. Dp3v3 vtwgn.zo p k+ na6v)5;syetermine
(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 m3 pry nslm2i6c-(m0j $R_1$ 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 spacecrafto7/. :scsvq gt 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-min time interval. The spacetg/ 7q.vs:ocs craft 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 ib/ 7 fhojtu8 t/:1bvbin 220 s. Through how many revolutions didh8 71:t/j b/ftb biuov it turn in this time?    $rev$

  An automobile engine slows dwr /bm+y:ng2 bown 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 whirlingrxprg l,70hs ) “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its 7glsh rpx0),r 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 accelerat r)2v nw5g7s+9s:gfm nspe2x c0a.z cfes uniformly from 240 rpm to 360 rpm in 6.5 scs 29pcrz+ae2fss0gmv7gn x:fw .n 5) . How far will a point 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 1500 revon(u pab0o+bpb;*u te, ods41q lutiopbs;e4unt p ( *1doauq0bb,+ons before it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolu6pr5r. bfaqj, tions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diam pfj5rbr6q.a,eter 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 e309ohh wcao3t/8au,xo4,a4c9l goju ixvv1the car reduces its speed uniformly from 95km/h to 45km/h The thcx ouvg1493, o,iah/0aexjcv39lu o w ota84ires 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 al)u zlzohc3 -0:na yf*)5p h,tl0p xsdtl her weight on each pedal when climbing a hill. The pedals rotate in a circle l)atf5p n *ocs,dhh lpxt:3)zz-y0u0of 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 tou6 uu8zy4wvg v6ix;/ he end of a door 74 cm wide. What is the magniz iu6v8wuo46g;/ yuvxtude of the torque 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 sho5sw .a;qxrl k7wn in Fig. 8–39. Assume that a friction torque of 0.4 r 75w. xqsk;al$m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached t 7 i1gr2ks/hd,aof/if 1y3vvko the ends 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 direction of the neikf7/1rvdhoy 1f,g2a v3ki /s t torque on this system.

  The bolts on the cylinder head of an engineqw 0mgc*oz1g(m5qy,c 3x)u ua require tightening to a torque of 38 x,mq0u1c)ayqwg3*5ozc(ug m$m \cdot N$ 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 oftpnai9xvt kx7 pc;-48 inertia of a 10.8-kg sphere of radius 0.648 m when the axis of rotation is through its centeakcin -pv x;x7t894pt r.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicy2*cck/w622ybv ucn iq cle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub6bwk*v/iqnyc2cc2u 2 can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated /nadz:leaf3 m vy( 0x0in a horizontal circle of radius 1.2 mafyn 0vz/l a( 3xemd0:. 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 sp(7r0 gg5iy/wqij 7ijk f9fhl5 kxg 2)geed (Fig. 8–42). The friction fwfx/ )i5i2g 9jkyq7j5(khgi f gr07glorce 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 invg6ojrb5n 7-tllp :+ hw55wkj Fig. 8–43 aboujrgwoj5wt75ln vl:h kb-p56+ t
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxyg c*xo-e w71nmzen 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 spinnig;,6: vpqhg;u5ivcot r a0 gm:ng 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 rocket if ,m0o qcp:ghi;u ; tavgr6 :v5gthe 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 ojmb*,1k, ihyu od8x4rf 8.50 cm and a mass of 0.580 kg. Calculat roid4b 1,,*xkuyh8mje
(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 rzmo5gh,) vh z2dve7 u2est 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 tangenti*j r:eo6gk6ufally 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. j6ko6*u :frg eAssume 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 torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor roa/s;hb .ji;d6 cc tii8pe**j itating at 10,300 rpm is shut off and is eventually brought uniformly to rest bj *h aiii;;dpjc/bc 8.es*t 6iy a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a pgd , 1. qpql*ynle:d(3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown so+pu)0fbcoc )7j e5 ljzlely by the action of the forearm, which rotates about the elcje0ou)jz7fl bc )5 +pbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be co. 0 g4hqecqnkbh w;3.ujke40 znsidered a long thin rod, as shown in Fig. 8–46.eqgb.hkjecw qu0; 40z 43nh.k
(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 consis4 3 n9e ziam2 k:rk0mvw;(jghbwp1*ivts 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 r;uyz s/ v*k0mnkx1u b,est within four full turns (revolutions) and releases it at a spe,mun; 0ux/*1b zys vkked 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 h9ytq4kq7 p sm/as a moment of inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque of 280 j+krir+nmij/ jp 7w j /fg)j*.$m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls wi9zladh13:b 8la/nhbsmo9fh 3thout slipping down a lane/zsal 1hhh am3l39fbo:dn 89b at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy ofle: f5fkc 6pfqhk3bwj-g94 0u the Earth with respect to the Sun as the sum of twou eh 0bp w4-9kf:f6qkflgcj5 3 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 a radius of0.:yugu99ij/uuz io v 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 ij9.guv9 uzi/:iuo0 uyt is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.l -kmxw1z, p-ss8ygr- p5+qvo 80 kg starts from rest and rolls without slipping down a 30sr,x-lkz v s+mg1y85pp- -qow.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 bflz tj2m3)0 kmalanced vertically on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energ tk)fmjl 02z3my.]    $m/s$

  What is the angular momentum of a 0,k5 + z :g4edlc;v(r7ltlmkwr/3 airs.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at a:csmgk ;+ea 5,r4lvwt rk(l3l 7z r/din angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grinding whee1f8zml7:l b youea13 ml of radius 18 13l f:18m zyb7laueomcm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, handsk3noe q: 4hij2 +nt0jr at his side, on a platform that is rotating at a rate of 1.3rev/s Itejro+2 njq ni 0:3k4hf he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce her moment of inertia by6rkjncpt(q 9- 9wpb :fcetn6- 2rp0eg a factor of about 3.5 when changing from eqw-9gc-tf6p:tn k( pe2j crnpr90b6 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 rotation rate from an initaza o ,y4oo :25qx 0q*k+s/lo0ha bknkial rate of 1.0 ok q0:n0abl *+k5oq/os aho 24a z,kxyrev every 2.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis through its center at a frew;p/n7e 5imooquency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter ;o 7oe5pn/ miw0.40 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 angulauzz:m* 7ee2 r3ewwh3 fr 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 momej,emt)xe 6z2kntum 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 dmk/klt6 / ;epxisk of moment of inertia I is dropped onto an identical disk rotating at angular speektxmek6; /l/p d $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2xbw fmk4q (x/)s+ g)dq.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 stands at the center of a rotating merrygq e:a7y6mj2:vwx f4zf -w8p5diqcez 1y m1e +-go-round platform of radius 3.0 m andq4:1vy6j2ce 1mpefzfgm5w:yied7- za xw8q + 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 zxvn:z67bvwal .:q- u rotating freely with an angular velocity of 0.7xv :wv zb6l.n:-ua zq8 $rad/s$ 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, losi8 7tfwb.ry)i+yc 1ncl ng about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to then7ycc)8lbr t .wi1y f+ 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 winds in exce a,l/if1-wbewss 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 6yh9o s+7yg6eslvw9x*r rt2h $ 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 freely wno,kk i: bkrkk*wj*)z4 1aav8ithout friction. The moment of inertia of the person plkzarkv )8 j a*in:bk,*kk4w1ous 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 i xfl-l i3rnlirt/np+ / f+br3;n8r4iof a 6.5-m diameter merry-go-round turn ixi+rni f/+ lfn4rr3; -li38bln/rpttable that is 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 on the groundfk3zql:.2u3 ow+ kn ht with the end of the rope lying on the top edge oz.w3:u+h qkol tfnk23f 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 spool’s center of mass move?

  The Moon orbits the Earth such f4bmlo fkzay,.41 /rdpz *ly1oocn 1: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. (Ino1czlm r:*lbf4p1o4 oyf 1ynkz.da/, the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rat/kt)fm 88 bhoqe 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 ol(xir6 998vg9xb,t2( aa)ow zo yagbbf radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interv9(i9 y)6tbgv abx8la2oz brw9oxa(g, al at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cyli8 .u* nmdo5syf :to+anndrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 5 u+ooty .nd:mn*a8f s0.0050 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.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of the(svx4 v9i9lh r 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 gre0vavp2 kc:a6c3fg7eh at, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapgve23v7fcakahc0p6: se 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 energy ,0ii k k0pw6z 5c3rdaqon 1w7pstored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m ,0a w6ip1p3k5k znqcroi d7w 0and 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 illustrajxohtgm/g01g 8.j.l f tes 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 sur wf4(b*an5- biwo;bi)wt 5otpface 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., zt +x,c x,3hbcwe ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horcx c3,t,+z hxbizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on the floor, and wemnicb6r h xpy 7kz1k 166;0yvj want to exert a horizontal force F at its axle so that it will climb a step against which ityc1k mzxp0knvr6 6; 1 6y7jibh rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with y0 apan;v x,y.m6ngj5speed v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist aa 6pax;y.0 m,jvyng5 nnd 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 rockg .*h::ofkywi6vb+a y into a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating bhyi.g : o+f:yk 6a*vwit from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thin rods, mam4y+jv8r)r 6, oet vsm vm,a8lking reasonable estimates for the dimensions. Then calculate the ratio of the any,slmva )metvov ,+8r6mr j4 8gular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consis6ph ;s3qa(c9wg/tm+bt frtv hx:, +ryts of two cylindrical plates, of map sgqh;+m +fxt6y9t3wrvc(r/, hb ta :ss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and rx c.b /em y+fgxu9jr+/adius r rolls along the looped rough track of Fig. 8–58. What is the minimum value of the vertigbcxjy9 r+uem +/ x/f.cal height h that the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not3re2h/f4rn /0 9 3xu 7saupqd7 yeskek assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed va)y3ck2 uen0 uniformly from 90km/h to 60km/h The tires have a dk2vn0 c) 3yaueiameter 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