A bicycle odometer (which measures distance traveled) is attached nearlwqc hd9s 38:e the wheel hub and ih:8ls e9wq3dc s designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular vo+/cv/i hr+lw jy:1agelocity. 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 magnitude +wc /+ i1g :yhavljo/rof either component of linear acceleration change?

Could a nonrigid body be described by a qs3fjrp *r 7c5rqxr08 single value of the angular velocity $\omega$ Explain.

Can a small force ever exert 0 4bfp ze cr)emi,4hjh3r1y*pa greater torque than a larger force? Explain.

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

Why is it more difficult to do a sit-up with your jd sg1 f59q8mghands behind your head than when your arms are stretched os m 91gjf85qgdut in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and thmyvn0x75owldds .a*h yx zpj*yt5)47ree 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 s4)n dm5sazo y*yhwj50x7 x7*lyd. pvtmall front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender loweq, fws1 n-te0ibph13fr legs with flesh and muscle conce10nitp s,bhfe -f1q w3ntrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers dh/o 0kb3 rrn4(Fig. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the net torw0rql4eb3o *m 1:b fexque also zero? If the net torque on a4*ofrbm ble x031ew:q system is zero, is the net force zero?

Two inclines have the same height but make different anglrm0vh9gyq a/gd7 s0*y es with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed* gq90vdryy sh g0am/7 of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an incline.5-.dou q.8 b2a tetkac/z0aq q One sphere has twice the radius and twice tt cukdq 2.e5- a80zb/taoq.aqhe 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 bottom?

A sphere and a cylindmp l1)avfn,*mt8pb+d er have the same radius and the same mass. They start from rest at the top d+n a) ,p* l18bfmpvmtof an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentumko9w vml t3(h. and angular momentum are conserved. Yet most moving or rotating objects eventu 3o9k.lv wmht(ally slow down and stop. Explain.

If there were a great migration of people toward the g1xh n f9dsx,v.kq pj89ymf.44uu+vr Earth’s equator, how would this affect the leng8.r y ngq +99h4mpdxxf uv14fj,v.sk uth of the day?

Can the diver of Fig. 8–29 do a somersault without having +2ipwl;-4- (vfopcbo xoc 0jbany initial rotation wheo4co0ip- v+x -lf wp o2cb(;bjn she leaves the board?

The moment of inertia of a rotating solid disk about an ax z3-c g(hc2n emg1tpm1is through its center o p2hc1m-g(ztgm1enc3 f mass 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 inhtm92h+f6801 mqf.vzyf . zw1g lwl j5.wsjcm each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the.1l mjs .8mgfh6l mhqzww+v9y2fwtzc 0 f51j . same? Explain.

Two spheres look identical and have the same mass. However, one is ho x15xjr vcn9h6laqm(ky u58:dllow and the oth8jl 6:dkn xqxvahu r cm15y95(er is solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points west. In8qdfrzwy68dv **qvkh1 *;s rn what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acqd6s*rnvk zw ;1hd8*vf q*r y8celeration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of awxjv /(m7 ;5o ;a1oum pb(mv8fpjd/2f lc )gvs large freely rotating turntable. What happens if you walk toward m)g5 f bm;x1mcpw;(vj/d vosf /7up o2jav(8lthe center?

A shortstop may leap int(g qydrhb+gw*q--,xx n dr6qgr-( *ishqe,+ ho the air to catch a ball and throw it quickly. As he throws the ball, the upper part of his b ,qhxgy qh re -, g*qsbdgrdwih -q-rx++*6n((ody 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, dxcgx .9ezuia*lv fu8wt6:x3w48tz j i h8x4a.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 helic 8 6wi.ug4au8itx a9x*38 ec jwth4zl:xxvf.zopter stable.

  Express the following angles inqgv*fasf3r9: e h2 3o y2nog9k 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 amal(3sbusp:x-6zw-r u ;hme(psjk 6 v:izing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radis3sehwpsm ; bu(r p:i6 uljx-:-vk 6(zans) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at tp3u hxh4h 3+dphe Moon, 380,000 km from Earth. The beam diverges at an xuhp 4 3h+hp3dangle $\theta$ (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 .2 h/-t m5caz: hr4fc( nsixstzq:3iz turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades sl 5:m-i/ ifhxz(4s:qcz2. a 3t nzhsrctow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5rd /; 9p.k4hwfc-tjxpl)yc;p m to another child. If the ball makes 15.0 revolutiok/r j)y-fpdcl9tppch . ;xw4; ns, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolution/h c x( l1wx1teil7br2s do the wheels make2 xhi tcrlb1x e1/wl(7?    $rev$

  (a) A grinding wheel 0.35 m in di mwj-04 vos83jgl1braameter rotates at 2500 rpm. Calculate its angular velocit0j 8-jgvwl4o1sa mb3ry in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.0 s (Fig. 8–38q 79 ,iesc;oy ;luoe.byzdbjk 2 j3h8*). (a) What is the linear speed of a child seated 1.2 m from the cenc 7.os*ey83zbojl2 ; ;q9eihkbju,ydter?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of ztc+ v4j0x*dg6mnl 6f 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 aqcke fr(6.ku 4 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 rotat s8/xs4,6covenc-e xyx6)k vje if a particle 7.0 cm from the axis of rotation is to experience an acceleration of 100,x-v jko6ex cx 48ye6nsvcs/, )000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformlyms74w,a lq-/s zb:anrk +u nh0afzy+ 6 about its center from 130 rpm to 280 rpm in 4.0 s. Det+umbzy:a z0sfkq-sw4 na +r/n7,ah6lermine
(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 radiusxp*web0); sl o $R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboard the Apollo spacecraft punn; rc2-.xh skt themselves into a slow rotation to distribute the Sun’s energy evenly. At the stahr k-snx 2;nc.rt 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 from rest to 15,000 rpm in 220 s. Throughs 8 4em,8k y/-bcu zbuktt8ly8 how many revolutions did it turn in m tkcy8 bekb ,uy4u-88lstz/8this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in lrp.*+ braxj , i mha;2 xe:y8/w+ a7vgk1vlbd2.5 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 highsc jo *g0m)a0c*z n18obo nnci.peed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutionsoo )a1zn *c8bm.cj*ni 0gn oc0 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 unifl7 :)) f(rf3tb*hvx j 1x2hzxx2lvc zformly from 240 rpm to 360 rpm in 6.5 s.fx z( )tx)f r :*2vjlhchxv2x1 73bflz How far will a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned offthbf2s ;;w5k25f; r domkn7ygb rf;l9 when it is running at 850rev/min It turns 1500 revolutions before it comf5nk f2sfwd9tg 57ymr r l2;bko;b ;;hes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its sbi:d*:il)3.jlbu ,lud)f u azjj h +*npeed uniformly from 95km/h to 45km/h The tires have a diame)ln+uf)i3idu a.l:*,j b : *lbzd hujjter 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 ) cr91sz,i gh tx-4(svd(bs mdthe car reduces its speed uniformly from 95km/d1-(g ids)btszm,9v (4x rhs ch to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each p)3rcom.qd kqbw x :6daho+((xedal when climbing a hill. The pedals rotate in a circle of radius 1oac d6()+q(w o xbhkdq:.m 3rx7 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cv f/*, y rcfy*s7q tfh-pl qj/c*9q.kum wide. What is the magnitu vqf./*y, ty-c *c/p*lj 9hqffs 7qrkude 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 shown9k9r fxiyuk)) snz 3uk-i)qw8 in Fig. 8–39. Assume that a fr8xq f skri) -)yukuz3)nkiw99iction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m l+3 i(w blb0lnqlk)u,, are attached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal plli,+3 0k l(ubwbn)l qosition and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of a +b*8k ts4y9fmgh67esxfq, e 3tprq/ln engine require tightening to a torque of 38 9 6+, /ltsym7fqk*eefghp3qrtx8b4s $m \cdot N$ 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 izs x( zs/e*qn 3z a-d:)ku/*;v kntzqrnertia of a 10.8-kg sphere of radius 0.648 m when the axis of rotation is through its center.ak3 snq szne *rx)/:-/uqtzk(;vd*zz    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycbix0h 8-inoxk9kb 6r2le wheel 66.7 cm in diameter. The rim a8- 2konikhxr bi6 xb09nd 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 rf;;vo:jac kp6z 9 ueg(mr7pc0 3 nc3ofotated in a horizontal circle of radius 1.2 mgvz3k9;f 67(rpj: ecm ;uafconoc 0p3. 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 constantcah b8tln 7n*k5l/0oo angular speed (Fig. 8–42). The friction force betwenthol5l 8/n0a* kobc7 en 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 oflpb151ap 1 vtj inertia of the array of point objects shown in Fig. 8–43 lpvajt1 b11p5about
(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 oxygen 01io (5 zthawmatoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the co0p38v;7 8 bremzfn1;ngmt ynrrrect 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 the satellite is to reach 32v8zm3 b;1r gfe8 ;0yrnn7pmtn rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with j2gs*(d r( n3cj rf:tca radius of 8.50 cm and a mass of 0.580 kg. Calculatc:js (3rfn2* crd(gtje
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerating it from rest tq6o8g prelts. 1w1x( lo 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 sz3cui;l-.xvi z an m1k2hqr* 7mall hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has az3a1.i;vn7ilx 2-zm rqhu kc* 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 ro +ayn :e-3onlrudev 5it(h3/utating at 10,300 rpm is shut off and is eventually brought uniformly to rest by a frictiona3r-y ulo5v :ndta3+u ihn/ee (l 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 acceleray8 shdg6+gzya 7za.xc2f q4d/tes 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 thv45: wekie 6enf0tnn4e forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10w4n k n54t6:ei0 evfen $m/s$ 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 rod, as shown in Fitagdm l/; 8v+mg. 8–46 d agv+8/l;mmt.
(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 mt e 8zsq2*-cyiasses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest within four ful:f;h xk.hjgd 3le0udg 2v,6 tl3yk d;ml turns (revolutions) aufxehv6gyk ,g0mj:k ld2tdh3dl .;; 3nd releases it at 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 af-0mw -gf9o. oxgla*j83jqhy7 xz z3s 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 ajk ta).(.m.mac z u.. pzs9d ;yf7uvxs 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 rollsz5:/q kxqn5 plmef5: s without slipping down a lane at 3.q kl 55mz f:e/p5sq:nx3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the 4x4ty sfesbu(m-s5lz+j5l+g sum of two terf u5sg+-4s bz4y5 j(t+ sxlemlms,
(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 osou(4 4 v3hadfz:d p l-tq*sx9f 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a -*u l xs 9do:fpa4h3t(vds4qzsolid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from r+ p7 zp1w8hmxv sc6c-qest and rolls without slipping down a 30.0 6zvc-8ch+wp 7sqm p1x$^{\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 b qq0gf)eq-8sxalanced 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 pqex qq s0f-8g)ole 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 : npz 6gjv)j,zxw n,h(a thin str)zw nx,,n(6hj p z:jgving 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-kg uniform cylindrical grindingg akj tly55ric2):*m5h )5tentipd ,o wheel of radius 18 c5cj5o )h: )yti n*rid 5g kpl,at5t2emm 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, hands at his side, ob0 (u*p2 neun(8e.*, fq3jed pnfyipnn a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rot*. nn ednif *8u023qe(y(,puppefbjn ation 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 lx:o;wyv:c 3ot-gi 9y inertia by a factor of about 3.5 when changing o;itylo y::39 -vwx gcfrom 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 heyhk9mzv6, ; ysr spin rotation rate from an initial rate of 1.0 rev every 2.0 s to a finakyz6symh,9;v l 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 c p+pc .d-oqn:denter at a frequency of 1.5rev/s The wheel can be considered a np +d-d.qcop: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 frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinning at 3 q4*/jub4c.z;y dhgpqm cw0tobk7j95 .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 momenij8r:fans;s4 g 8kpv 1tum 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 cylindrica26 f9raa5/ce- rm lti,5h yinol disk of moment of inertia I is dropped onto an identical dis neoi6 af 55al9hm- c,yirr/2tk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4)*zsm5qm ;th f $rev/s$ 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 au:6jls*xh 5ih +d *tygt the center of a rotating merry-go-round platform: sjh5+*lg*y6hx itdu of radius 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 angue 22l (/p)tcspib pysk4ymwed z 8a2+:lar velocity of 0.8pl :4zke mec2bd ty)psyw2+sa(8i2/ p $rad/s$ 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 up zw/:va1m3 )kgdou(dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(rou(z dg omwu)3u1/kpv:and 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 excess of 120ht1twf 8sc* e9 $km/h$ 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 vcmq+(odpmp5a/i 3 , u/g d;xi$ 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 witho h4 lpqeq c25hyx-oe //r6b8geut friction. The moment of inertia of the person plus the plhorh48-b q2/ xqc/e5ee6l gpyatform 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 persv0gm9*l aa3 :4qwv-u-wv s rsyon stands at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on-g wvs v*-vra40s3ml w9 ayu:q 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 wznfbo*1w 5bu*ith the end 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 onto it, Fig. 8–50.zu5o 1b nbwf** 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 tj*6la3cq6rhg zfj8x 1he Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum.jqgl1fxz6 chj6*r 8 a3 (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from 1 f 3t 7*bj/:oi ;xljj lk1mb+v dujm/ac.i7qkrest 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 uniform cylinder of radius j3cl x0 yf( f**bovf9: lnuqk;0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by theu fjfo*:3n0* (blxq;l9cyv f k motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, 1b*pzv nt8rbx5x k/ 7eeach 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 v 1rp*bnx 7k8b5xzte/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 sp2 jt:bkrwm00k kmb84+ mu klm+eed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times as gvxo4lkch7vx/l -)) nve yw nag)-e 73lreat, 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 vlwx37 )/hnn l7oe)x ak-cevl -4v)gybe? 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 automoox7i0 du2f8 cfbile 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 and mass 240 kg, and should be able to travel 350 km withououdci f7 20xf8t 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 fgw-d/; ai-gl2u n-nc:h7u z z;vf5uuan $H_2O$ 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 horiz nre8i 1l: zr:akkj.-mg mt-s:ontal 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 L can pivot freely (i.e., we igi1rz(4xo khtpk8(dz: nore friction) about a hinge attached to a wall, as in Fig. 8–54. The ro k oitrxd:p184h z((kzd is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on the*u lj;+ bsav-b floor, and we want to exert a horizontal force F at its axle so that it will climb a step agvabsbj ;*lu -+ainst 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 mak7g k ;:oqdfp8king a turn with a radius Th8qf ok7dgkp;: e forces acting on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.5 mbpr uw(v)z/.c:k+tnzwy: **y nittt. and begins whirling the rock in a nearly horizontal ciytt*::p( +)ni / uywwnz tzvt. rc*bk.rcle 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 skatery lgor9c4yp el46mmd)dy3s 0 3’s body as a solid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against herps9)my4gr3 y6d0y3mcll 4 doe body.   

  You are designing a pcm;41uygkd qyiu z+36u .o*dclutch assembly which consists of two cylindrical plates, of mas6m1kicz quuy ;4p.d d+3yug*os $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls alongwh-jynw-;2h bp2* d ee the looped rough track of Fig. 8–58. What is the minimum value of the verticad-hj 2hwybe; en w2-*pl 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, butxgv bb:tas0 hr+z:qn1 ermm2s03p* 1h do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revoluti0e jt6 wxxn nu 6r/0xc +x5vhq-nq5gmn5j-wm2ons as the car reduces its speed uniformly from 90km/h to 60km/h The tires mnntx626/jjvxn ww+c n5x-0 gr0 -e 5q uxq5hmhave 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