A bicycle odometer (whicd(0) 8x9h1y5tf(6w h vl)3che.hpqngi naf gch measures distance traveled) is attached near the wheel hub and is designecgyxh e6t) q 8wni (.5(09vl13ghchhpn)fafdd for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant ao 8 fhl(uic0;yw/eej-d ,iq;s ngular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s u(j0;ilw-seh/ed of q8; iyc,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 velo) j4fjjdo)/.ah asf b:city $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Exp74o)oh-ezx b ro,l l9 xq(ib9llain.

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 behic;s (3jnf*wp8e 0doj cnd your head than when your arms are stretched out in front of you? A diagram mao *jcp f8; dejw3s0n(cy help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel a37lu, i ucvh,na2+xrand 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 it harder to pedal, a small front sprocket lu3vcaah,7ixur+ 2n , or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lowerd279kul (oyau legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotat 9o(u 7ualyd2kional dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. vm- zw2t2d2lrxam0 jy xns uns+7b3(,8–35) carry a long, narrow beam?

If the net force on a system is zer;3cl;1jeb a(llqo /nf o, is the net torque also zero? If the net torque on a ; jl(3;oqb /1lnlace fsystem is zero, is the net force zero?

Two inclines have the same heighte ((jg,tzf+q6ayvpf./o wwmrjw :;7e cj7q3 d but make different angles with the horizontal. The same steel ball is rolled down each incline. On whiq(vfr; qwo p afey :7mwz(d7j/.gc+e3jt,jw6ch incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from 9ugw,zwgt x7 0rest) 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 great g7z0wgx9 w,uter speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder hai1yy 08rj hj lcdur7sv15(jr zsz/-f+ve the same radius and the same mass. They start from rest at the top of an incline. Which reaches the bo l7rf/susjjr8rz- 0 15+1 jyvhizdc (yttom first? Which 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 conserved. Yet most moving6w)- 878ts lx uontr/dngm 0zc8aja k/ or rotating objects eventually slow down and stzt8konw6mn/8r)0ts / u cg -8 lda7ajxop. Explain.

If there were a great migration of peop7:d3h nrj2 wk1p9nzjrle toward the Earth’s equator, how would this affect the length of31j:r j7h9z2 npdnrw k the day?

Can the diver of Fig. 8–29 do a somersault withou y/)+/psq5toayns p /ot having any initial rotation when she leaves the board?o )atyy +posn//sp/q 5

The moment of inertia of a rotating solid disk about an axis thro9eys.- 052gb7z eqz es4jcd6/yileuough its center of i4 2 eoce5s/97.edy0u glezqszjy- b6mass 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 eau o2m gv)pt(7xch outstretched ha uopv2m(g) 7xtnd. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical ad cd5n3-j ccs4ohc) l7nd have the same mass. However, one is hollow and thcchdn5c3)4dj s c-7 ole other is solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a hi07fc7 qy .2s9ybhrwzorizontal 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 tangentials iy7zbhwf 2y.097cr q 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 rottnex:*6 itdyui78im 8 ating turntable. What happens if y7en*6 dy 8xtii8:u mtiou walk toward the center?

A shortstop may leap into the air to catch a b x;r zv;xkq:h*()vnu:ucf xqy 4g7uj9all and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly 4nyu;xju7uxq;)gv :*k( 9 :xz cvhqfryou 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, dcob,gbrcwx*l 9l p;;;iscuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stablegxw9 * cb,l; ;opcrlb;.

  Express the following angles in radp,4.i1.qf*1zllhe)m ug f dvw ians: (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 Earthuomvestfhkt) ejkr f, 3)9: /) because of an amazing coincidence. Calculate, using the informatirs:9) fe3),kuvk tj /)fohe mton inside the Front 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 beam p.wvzge , b1ap4xxz1 -diverges a1vxzzpa x14eb., w-pg t 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 of 6500 rpm. When the motor is turned o,zbq2- v4fr*l qp.xqhff during operation, the blades slow to rest in p*r l x-vf2z.4bqq qh,3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If thep p 1**q(shu,cemc* oj ball makes 15.0 re* cq*1uj,*s oc(ppe mhvolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutions)oe97 syhmdqf,etb h4x043o4bivv , y do the whb304q fxs)d,,eh4vy9h 4oivt e m boy7eels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm.nm6,yer 3(xe0 wa q8x n-g7l eolxp6n; Calculate its angular lexxnl 3nwp7xy g8 6o m aen,;(6eq-r0velocity 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 ovx9e3 t o,ga 1olu(-qjne complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from the centq9g e(xvoa ujt3l-,o1er?    $m/s$
(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 orbir(ueljd3h dlcy +28m2t around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a p9k 9wue;/-tunff ./llyl(;qod/zhdi oint
(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 c+kxlel dou ho8k: 96e 2.dalx.m from the axis of rotation is to experience an 8le2 a 6xxlkeu .ok:lddh9+o.acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its cente2hjmr:n 5 zm6wr from 130 rpm to 280 rpm in 2m z :rwn6j5hm4.0 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 gaw-n q-tk 8rxpu8y;( ic 1gr1p:rz3d $R_1$ 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, as8)x.p5 q+i6anccp5h zsx1s o7n2 eyxg tronauts aboard the Apollo spacecraft put themselves into a slow rotz7oy+x1c s26p5g ) ai5.snxcpxhne q8ation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly 7uoxjl *wd:,w *f sr7sfrom rest to 15,000 rpm in 220 s. Through how many revo :uox*rj7,ww7*sfsl dlutions did it turn in this time?    $rev$

  An automobile engine slows down from olqp3q xu x*2(lsqa( 44500 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 highsiq;kgo/:2;g7q ubx il peed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutigqx ku;2 l7 ib;gq/i:oons 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 diameteuq:88 nez- 5qb dw0cb,3kku4 nxgah-k fki30 hr accelerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a q-nk83uk3ca ,hdq 0bzwkn :4i 8u5ek -b 0hgfxpoint on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is rue1 86dyu-hb knnning at 850rev/min It turns 1500 revolutions u81h-yde 6 nbkbefore 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 revolutions as the b, n qo+p*qjno88op3 jcar reduces its speed uniformly from 95km/h to 45km/h The tires have a dinp8j8 pq3o q,obo*n+j ameter 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 revolutionme*y,kfw5 0cmp4w x 1rs as the car reduces its speed uniformly from 95km/h to 45km/h The tirw4fp5 w,ym*r c0k xme1es have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weigiofpk+ rbw+ 0 t0g)mz,ht on each pedal when climbing a hill. The pedals rotate in a circle of radius 17 cmw kp,0bm+fg zr t0)i+o.
(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:sp69 rc(ouht cm wide. What is the magnitude of the torque if the fhc69p : usro(torce 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.by) q jp4 n,v/u ox0c5bh0pg5j Assume thqpgyv0jh55x/jb0,bc ) pon4u at a friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a massless rod o zmu9xm7m5 q5mg 1w+hm8, .ylglkhgh*t /k:qwhich pivots as shown in Fig. 8–40. Initially the rod is held in the qkggzmo m 9+hg5 ltl5m1q. mx8w :hk,*/umyh7 horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening to a torqu)q1,ccntl jp2*i 0zq xb./2w+ emna* o xjhef6e of 38 )c , xa bf2h.cj+ pnio2lwx*eqm0 6qe n*1/jzt$m \cdot N$ 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-k*d q4hcxmpph2l;1xg3 g sphere of radius 0.648 m when the axis of rotation isx m;p2*31 q4hxhcplgd through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicyclqe,w ytfc 5 /flo4-e4le wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignored (why?t4ow ,lceelfq45-yf/).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in a horizontvo(2o zsrdov 3xeb/i9w v/s(2al circle wx2soveb/9 rsd (voi(o/23zvof radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potcm38 t 1c7xyekter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and the cy7t3xk8c 1meclay 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 ob;cj : ezoeus3*jects shown in Fig. 8–43 abouzjeo:;3cu se*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 oxy vo j,4(jwr1s4+u hbivgen 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 cylindrica,1b(.-5cm. )xhrqvmpukjr svryom(.l satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of p5kv jsv )u,(hr rbx1q .rom(m.c y-m.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 8.50z4q g;revd r,) cm and a mass of 0.580 kg.de 4 v)r;zrg,q Calculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, cqm 9),yp fx b-v)mni43)rbfw 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 tangentially on a small hand-driven merry-go-round and isq :ti 0stfn. 5;lgoy 4esq-xf9 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 prodqltex;:. sg-i f4f t5 s0oqyn9uce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotatvzs fb/ 52xr3xing at 10,300 rpm is shut off and is eventually brought uniformly to rest by a frictional trv 2x 3x/bfsz5orque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a 3.6-kg ball at ca+ xgqa:-ktv( 4xs,fuq86zz 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 t7 jkye/ ,o3taihrown 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 accj ot7 /3yie,akelerated 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 consideredf r4 o6ncny1fz1eign1-;1u.ef8sdi k a long thin rod, as shown in Fig. 8–46feif1 es4fdk.1rcy8 z;n on1i-u 6g n1.
(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 masses, 5oo wgjbit7g86v snk** gy ;p:$m_1$ 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 fors)0 7yr3u5 avvn1,ay rn. lhfur full turns (revolutions) and releases it at a spnysnhy37rr1a0,.raul)f vv 5 eed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a moment of inertiay kspj)u6gtg( 3wf 6(yc1n9tw 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 devihl yj7wwo*p o6(3la/ elops a torque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls without slippr :h,h z mp(:m,,mbhzding down a lane at z,:,mdzhrpm h ,:(bhm3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect q5tf8kjx)n f1 u2g lvak/no 2/to the Sun as the sum of two terfjq5gn1 /) l/unvo k2 kat82fxms,
(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 of 7.50 m. How much net work wiyyj 2b2/gd0 k.:dat j-ny34ai)vun is required to ac3 kijy-v4ud2bng 0)ay.dyj/n wta :i2celerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and-fp12be7*y nu - cvbtw24iuwv mass 1.80 kg starts from rest and rolls without slippinv2vt*7-24yn cbi pu eww-uf1bg 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 ve8zb nnbb k*fj7 l 1i(l*: hkog;,vab8frtically 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 energy.l8 l abffkzj i(vb7,;gbbhno*: 1*8kn]    $m/s$

  What is the angular momentum of a 0.210-kg ball tcpedk 3cwu 4u+lki (8h u+k3(rotating on the end of a thin stk+(e3w+uput l (kcu8i4kd ch3 ring in a circle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-k(-5x njot5ox jg uniform cylindrical grinding wheel of radius 18 cm when rotating at 1500 rp(oxj-5x jn 5otm?    $kg \cdot m^2$
(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(dd 6u.ad;xmpz9z)wn is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speap;zx 9n). d(dwm zdu6ed 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 mom r,n1w79c shofent of inertia by a factor of about 3.5 when changing from theonr19,7 swf ch 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 rat5p+ymep .nk9 y+9niq)yw ev8ye from an initial rate of 1.0 rev every 2.0 s to9eyy k.e8+y5 +9 yqppvmnwni) 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 vertiv uxzfwm8b6opnv 2e0y,1 c.2q cal axis 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 as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the freo6pm8b21yvqnz x .cw, vf2 ue0quency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular9exl 9fs luu+) 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 momentu,dcuatz*+mtix -3 qi-m 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 isnmzfq764d xm0 4- rhmy dropped onto an identical disk rotating at angulaqn 6mmyf0 rxdh m4-4z7r speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turnsf10r1essesqi. -pxu -ug/c+ s at 2.4 $rev/s$ 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 s4th (qpf haj ;hp9c:0kg stands at the center of a rotating merry-go-round platform of radift9 0hh4c(sqppj a;: hus 3.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an angular velocity l,+,t3a/z-fm2 aw*y t nlga9mgnk+ nv 9pb0 bdof 0.8 k t b l/+3npza9an,20fntg-mw m+agyv9* ,dbl$rad/s$ 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 dwa dt vof:*9al.:)vtv kh rr.g.krf, losing about half its mass in the proce .9ov: .) var:tdv*kkft.lrgh ss, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the 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 8q p1 wodzuf7b9 4gjp9excess 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 q4m raob fw+ 062z9ntm$ 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 withou af)b*, g me6x9cz:ptbt friction. The moment of inertia of the pers) ,bzmc9et abf*6pxg:on 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-m diameterfn-* 0l e0mzuh merry-go-round turntable that is mounted one -0lu mfz*h0n 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 ground with the egbtm4. 4(tkc4 dhjq,rlh89/ hun, s pfnd of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding 9t4hqk8.bjup ( r ,hgf/m, s4ndhc4tlonto 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 that the same side always faces the Earwt7zxyd .td57th. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angwy7d5z. dxt7t ular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates fr 0poa;x/t*nbv g9qec1om rest at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the sn ez2 8x ;r)q8 fzv6zuwt0dm:vdq y(z3hape of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleratio) y3fzr:z2 u;8 tdvvm(zx nw8dq qze06n. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, eye f n,+xal y8/fdltx5,id/:lach of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long strinxt/,ffl 5iyy8n/d ed:,lx +a lg, 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 thzm8uuc +9dy2k e 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 siz) e,0g04lnxeff 9 lizvv/uzz 9e of our Sun, but 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 neutro9 )zl/f gvx,90nz4uz efei 0vln 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 aut uaggpd61 x:*t paw6t;omobile 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 diameter 1.50 m andp6ta6 d gtua*:;g 1wxp 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 an 5,hoo-iy6y kui.xc-m $H_2O$ 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 a qt1a;6fpc)y*mz g.o on a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length e0k0gft1i /bl L can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in F0gt 1keb0ifl/ig. 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 ra)( f4 h an)zxdn 47czrlosoa+7dius R. It is standing vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step agains)s(n7fo7zdx ao n4c4rl)a+hz t 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 i *.8gwub6ky fes making a turn with a radius The f6u.8g by*kweforces 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 sling of length 1.5 m dk:/*vyk ajcb2mn, vxcc;*7 s and begins whirling the rock in7dcvcy: kv2/,jba* n;ksmxc * a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s bodeul0**ay8xncpopkq t 3;t3s5( qhk)by as a solid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the rncs3 a)xt k(;5*k8*o quq0etbplphy3atio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch aswom qv5gy7ja4:6kh 0x 3oss7zsembly which consists of two cylindrical plates, of mamyhsw 7o45o kx6s7v0 3:gqazjss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 r vmsh k t:.dzx7:bgf(/s ok. ep()rts0adius r rolls along the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drf sb ot)/0e: td7h(.rg(svkskxm:z p. op 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 not assum asy7rl +w1muofz1 06ecnj.6ne $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uyspnk(n:ls3+2v1 g crh6an- n t0xj1yniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleratiognn+1pja6lhs s1yn y-(3v x:0nr k2tc n 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