A bicycle odometer (mmog7s +9 v zr-nlyl*:y pb.u2which measures distance traveled) is attached near the wheel hub and is designed for 2 lr29+mb*g.yu -vpnm 7l:ozys7-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at conkn:)bni; jc0 vv: a4nvstant 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 mn0 c)4naj;: kb: vnivvagnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single vas p2yl).r;nww lue of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Expv v8 2jm,rsf9bypx 2qpz;n;k.lain.

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 yournx w6 jovi )y.u.e hkj1:talj7d,3ma 1 head than whenjlju:)a1a .3me 7j6hkxivt,ow1. y d n your arms are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has sevenlxh/ nzb( --t ccq3n;y ;fn:ytcvn6gni74fh- sprockets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a lvy- nn;-l7n-yti:tch6 4c zf/3 ( ;chxb qnngfarge 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 a9fsr/;nin w9c1qini 9 ble to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why tw /s9i rninq9nf; ci91his distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, nanu3hkfqw:rrdd.5 a. ,rrow beam?

If the net force on a system is zero, is tdka0v +q3( lathe net torque also zero? If the net tot0 l3ad+(kaq vrque on a system is zero, is the net force zero?

Two inclines have the same he3n 5puulguo. i0k7s:s ight but make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be grea3ls unukgsou7 :50ip. ter? Explain.

Two solid spheres simultaneously start rolling (f y6+29 nf h;1:mqhs 5f5cyy6ssfuyc zf,ka/varom 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 bo fycy216k qa; hnzfas5cy, f6 9/u5:sfym hv+sttom?

A sphere and a cylinder have the same radius and the same mass. They stary5rqb t,u90+) i1sg pfhg(sent from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the g ser5,bytgif )( 1pus qg+nh90reater rotational KE?

We claim that momentum and angular momentum are consr nq+v,ams tv :udg+4sg4b *o0erved. Yet most moving or rotating objects eventually slow down and stop. Explain.r+s u+4avbos, vmng*q:0dtg4

If there were a great migration of people toward the Earth’s equator, how wgf7b+* tnp.ob5b3*uxa k:m uwould this affect the length of the 73 t.5uuobwf*bx a+*m pgnb k:day?

Can the diver of Figi a*:w;ewqsv2. 8–29 do a somersault without having any initial rotation when she leaves the :vs 2wie*w;aq board?

The moment of inertia of a rotating so o h(77rhiu wr6ekcnui0:m+ld 0wf0 p)7k tya3lid disk about an axis through its center of massh: 0n7(ryc +6uktwwihup 70ra) o3eidflkm07 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 rotat1k dq3 efj*,()i8a bswxqbn6c ing stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, wa*(sj q8,bf b6c )dx ik3eq1wnill your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identi.q .vot nx :j7cj2zol.cal and have the same mass. However, one is hollow and the othec tvl:z.j.xn.27j ooq r is solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotatingz)k v6:a *k fw 36o5*s0jylipd ojos2s 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, describdpa0)2j 6i6k*fow ojovs:y5sz *k3lse 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 rotatinhy*ze0 czb .4uu wu3i+g turntable. What happens if you walk toward the cen4ch*u y3.0 +euiuz bwzter?

A shortstop may leap into u,+0 xq/asfmxthe air to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you wil,uaxxs fm q0+/l 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 najd ,+gfly*5momentum, discuss why a helicopter must have more than one rotor (or propelle+5 fy*ganljd, r). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following ,wqqxjvge) s wfh8-70angles in radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazin m:sg0ea xnl,b33 ony+hv6 j*mg coincidence. Calculate, using the information inside the Front Cover, the angumm6bv:3osne+0gayjxl* h ,n3lar 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, fzzdj+/ t;7 ldvt-jk7tqfy +- 380,000 km from Earth. The beam diverges at an angle t7jjz--+ty /kd 7ftqlf;d+ zv$\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blenderc0 +doq 4smt+xc d6g g 83nbu0r fb,.zjqc:c)p rotate at a rate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades n6o fc xm g3bc,++ zj )ctqqu800drcsp.b:4gdslow down?    $rad/s^2$

  A child rolls a ball on 1i, mgj 21pode8oka5x63fumt a level floor 3.5 m to another child. If the ball makes 15.0 revolutions, whgux1mie2a 68k3,t ojd5mp o1 fat is its diameter?    m

  A bicycle with tires 68 cm in diameter trargell4tmqzft1.1m/ q: ra7hex.m ; r: vels 8.0 km. How many revolutions do the wheels m t1;zqrt m.q.a lr 1mer4e/ml:h7x:fgake?    $rev$

  (a) A grinding wheel 0j: a4 t)d vx4jqg6e(gi.35 m in diameter rotates at 2500 rpm. Calculate its angular velocity in6jigg dxeat :qvj4 )(4 $rad/s$ $\omega$ =    $rad/sec$
(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 (Fj)) ktj nv4su48v;p1:a iko vdig. 8–38). (a) What is the linear speed of a child seated 1pv):akj48i1 jvo vd u)sk;4nt.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 velocity of the Earth (a) in its orbit around the Sumkum l3 .ye9;mwdi gft6 *u3d83xmbeto ; 7nm/n    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speggy9j + * e9qrik.oc(c 6abce;vz *)gjed of a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.0 cm from tpj5s+tcgeu* 2 he axis of rotation is to experi pec+j5g2st u*ence an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheen3jaar 07lqq-0saw )d+r/ o qql accelerates uniformly about its center from 130 rpm to 280 rpm in 4.0 s. Determine-qa/0sq7 )odn+0a ajr3w ql rq
(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 radiukm)-75.lpwm19 jrgqim.m q ny s $R_1$ 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:eq q vzg 1((//ixws)+ysllwgd 9n3qi aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine w:lz(3selgs/d(vq qn+yq/ w g )1xi i9
(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 uni4 k2u7 h npq(c9g1cuf +q*smvrq/ae 8aformly from rest to 15,000 rpm in 220 s. Through how many revolutions did it turn hf8c up(9eq sk1a7/4n 2 mr*ugv+acqq in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s49qb d;ahf+l zar5n l+. 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 .asm s-xa;j(7/ukgavdvn -*vr; 3it d for the stresses of flying highspeed jets in a whirling “human centrifuge,” wi.-m( vsdv;adgvatjn r;*ua s 7x-/3khich takes 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 360 r xrs0j: 9-um)ip t.klvpm in 6.5 s. How far will a point on the edge of the wheel have traveled in this time? t.9l) jmv x:ik0ru-s p    m

  A cooling fan is turned off when it is running at 850rev/e9 9 mj , 76q/:cl/hkpnlvy0kpaim m2qmin It turns 1500 revolutions bm:6 h7q9/mlc0 n/l,v2qe 9pakkpj miy efore 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 car reduces 2u4yp7yholvnob u6j + 0;qf(e its speed uniformly from 95km/h tofv 4ylhpo+jb u;q2y(70 e6oun 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make i jtiesn0fje:vr wz9.;2w,c165 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diam:sj91 wz ,.f2;jtivee cr0i nweter 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 each5bif 9dl9j 0qajw( -vgtal*c, pedal when climbing a hill. The pedals rotate in99daaltvgqf5-( bi*lcw0jj, 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 7p ;f- w:sdq dq5,fvan:/jon,p4 cm wide. What is the magnitude of the torque if the force is exertn,j/d::waqfoq ,ps; n5 fd-pved
(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 torb ck9oa467of8j-e-dttz k ,egque about the axle of the wheel shown in Fig. 8–39. Assume tt b9doatc8fg e7ko, zj6 -4k-ehat 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 enw-icg e-1j 8,o cflii9ds 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 net torque on thji8fli c9iwc1-o- eg,is system.

  The bolts on the cylinder head of an eg o/a16t x5-rn e v:puq)c0udzcsp8;engine require tightening to a torque of 38 s8r1;otncpc d) p5:uxzg v qu6e-a/e0$m \cdot N$ 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 momentw vo0y /iiyk3eu:rz32 of inertia of a 10.8-kg sphere of radius 0.648 m when the axis of rotatik vei iuz: yo3yr302/won is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia(yx9 n sti6ca2)(nsiy: vqk 1f of a bicycle wheel 66.7 cm in diameter. The rim and( x2kvt i :ysnf1c(9 nq)siay6 tire have 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 of a thin, light rod is rotated in a x,o1civ *gpkck /z .xl2y(c r8horizontal cir8k vx(k1xi/*p ocrcg2lz,. yccle of radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a b8,qexprrsef,7 kw)g8 owl on a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between herf ek8e ),rqxsp,87rgw 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 ine7wxi*69j(an:rt jz :ztu c8uy rtia of the array of point objects shown in Fig. 8–43 aboutyjwn8i6:x :j r*(cza t7 ut9uz
(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 consf 8kne fib+g5 /j+177lknmc viists of two oxygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at thev2l1/1q/nhr m5 wt jrxw6 a,m.2obac h 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 m1 1o.n/ml/,6a2hwt h awjq b2rxcr5v 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 runnc+: p:7o o(0cvy znadius of 8.50 cm and a mass of 0.580 v:ycop+c:(n zo 70nnu kg. Calculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerati1 *l q;iw4bazi o9hpg6nh5uqy a99l z8ng 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-roufig c5 m9fjrv +a99o+xnd 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 dis 9i 9f5+coam9gvfxr+jk 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 rotating at 10,300 rpm is shut off and is eventually brought qx. 6jwp)*ws/ ri6o rnuniformly to rest by a frictx)6/p n.qs6iorj wrw*ional 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. u/*td dae kz6fao .s9)bo6/yq8–45 accelerates 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 is thrown solely by the action of the ff mmtsuyih .6.2j-2bvorearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated unifjm6sm2 -hv2 .bftyi .uormly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin j3i (nxm2.*zyz7s9u zxed0e q4zn; zerod, as shown in Fig. 8–46q x7 zn ;x(4e2ee93jzu dz0s.yi*mnzz .
(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 con9 a ;; 872tfqnvsthfku m2amr zbq)m96sists 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 0uz7qw1h0 e-e,1p s(o,swp;g cr wmu(nfdtz+from rest within four full turns (revolutions) and releases it au c,o us e(;mrn-w1zzw7ws pept00 d h,(+fqg1t a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a momenhm0y l5ilb1: bt 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 1 h yts/.bn0xr r,tnv)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 slippinjm od4/r)ar l,9 xdj0ig down a lane at 30rmr, ij)jx dlod9/4a .3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the sufm; tk.wd hk*vb,( 5tmm of two tehk.( * kmfd tmbwv,t5;rms,
(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 7.9qznc7k0r, 7b*,vooxq p; j;snt t)c x50 m. How much net work is required to accelerate itct,psxzk) vqo7*x qcr b9,jon;07t;n 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 mass 1.80 kg starts zri7u:r 7)w mnud4zh) from rest and rolls without slipping ddu7zm4ir n w7u:hz))rown 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 starts to fall and;0f- )rwb3 cm ,ssmco3 cdz8xl its lower end does not slip. What will be the speed of the upper end of the p sc)b3m fm-ow,0rx ls3dc;c 8zole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end of a t+b vd 6 w.yr-(hyclh+rhin string in a cd 6(vh + yr.bwy-hrlc+ircle 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-kg uniform cylindrical grinding uq0/rfejn4+ocs, 0,xaeb 6aewheel of radius 18 cm when rotating at 1500 rpm?4eb fuc,,jr+exn0saoe q/a 60    $kg \cdot m^2$
(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 is rotating;eq:jvb s5 -fm at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation de : e-mf5jsq;bvcreases 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 inerti x()fnu 7ijnh8 .:xahba by a factor of about 3.5 when changing from the straight position to the tuck p).nuh hx8bi7: xn(jfaosition. 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 rat..tu s;y,n .yv /ydexye from an initial rate of 1.0 rev every 2.0 s to a final rate of 3 yt. .yy;/x,.dvy uen s$rev/s$ 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 3cuta: ttptyvxw+0pr9 1: 4gxa 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 frequenw x+ypx 1tt 0tp3tguvrca:49 :cy of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinc35qgz v f0d.oning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum of qw2oqdh)o6fkwz, 37bq btl a() wfo)2the 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 -mbh81ey 3ifp jx yc;ekl81:mdisk of moment of inertia I is dropped onto an identical disk rotating at angular spe8kh l3e-1 e;8yf1pmcbim :x yjed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.h nd .ex9 (pnw ;isujz:m4(*g(b-rdgd4 $rev/s$ 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 platfota68yyn 3v8wlu(ph: trm of radius 3.0 t6ulp3an (wy8 :8h vytm 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-gys* l,tiabx ,umc9g2:4dvh8ho-round is rotating freely with an angular velocity of 0.8 cvhsxg ,i4 b,u a*2d:hty89ml$rad/s$ 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 colla9,x0/m/7qyiaxbw - nybu z+pqpses into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing b-i0ay /+qnux9bp 7 xz,qy/ wmradius. 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 winyby3xhx1nuw u109 9xu ds 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 m5mk- 9g0t pdz$ 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 gd2 e+zeu6z 2gwgcuc)r+( 8egfreely without friction. The moment of inertia of the person plus the )eg2w(cg6g+ueerd8+z uc2zg 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 stxor-mv/bnr2ea7 rq1, ands at the edge of a 6.5-m diameter merry-go-round turntable that is7 b1mrrxr a oq/ne-v,2 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 ground with the end of the /qpt 5hs-(d zgrope lying on the top edge of the spool. A person grabs the end of the rope and walks a distancp-5 g/dt(qs hze 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 that the same side aljd .4bs;-omsvws u(k :hppx+veb v, 48ways faces the Earth. Determine the ratio of the Moon’s spin ang (bjosk.wvsmbv44+hxdsu8- :ep ;p, vular 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 fromh8/8 fn5i s tl6gxml/s 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 shape of a un7s5 tkpsuxek+bt1zzmp 13* , tiform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angularz kz1 u3p5kb *ts,7+1etmtxps acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two su uo5k+/ 8u2(qaww)td xl.b kqolid cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and dia/qu5w)o 2 8tkuua k l(db.q+xwmeter 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 h0qw2mnmy 6i*u37c l r2mapu) 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 gg;v fa(vt-3 i3b z,axk-c9 vagreat, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a unif abxa9ca kzv-;3gtgf v3(-iv ,orm 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 user+rjy 2 g .i,p3vyo:ez energy stored in a heavy rotating flywheel. Suppose such a car has a total mr3p:y2 oe+,r .gy vjziass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrat,kx7:2rwd p nnes 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 at speed v=3. 18n1vsa /wif e7d/tau3 $m/s$ 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 Ljkps7*8d0, cc e8yazq can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21gcpzd,e8y*7cs 0k8j a q.]

A wheel of mass M has radius R. It is standing vertically on thot03h6qdrdvbx:q d:2 e floor, and we want to exert a horizontal fodq3xv t:0: 6qd2rhdo brce 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 makin4jw 6pz/cm88i t)knppg a turn with a radius The forces acting on the cyclist tc/ ) m88 k64zwnpippjand 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 and begins xm0 m5s2z:ebgwhirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to5exmm0 b:s 2zg achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skaternz) opt0 6csu .ovc0*s-tjr: n’s body as a solid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Thenus6 .ztjcv *-ron0)p0t :cs no calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which cop**ir f kt q0d 7p5;azbgy67oyhgy;m +nsists of two cylindrical plates, of mb7am+g6pyqfgk*7 y0 ptd i*;;zhy or5ass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 rad(re-o gav9; 23;+u fscqwoviv ius 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 drop if it is to reach the highest point of the loooo9 vavg;q3 e(2iwc-; v+usrfp without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, butjs1x wj yh zmkhe*)9q(+w t65t do not assume $r\ll R$ h =    (R-r)

  The tires of a car ma t ypou0y358hwke 85 revolutions as the car reduces its speed uniformly from 3 pwyy8 otuh5090km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s