A bicycle odometer (which measures distance traveled) is attached near the wxe/d 1jxc-isxs* x3d 3heel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-i3d/1dx ej-isc *x xs3xnch wheels?

Suppose a disk rotates at constant angular velo)flx)r6ykaf exk 1v::city. 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 accelerafxv):kfy61er : ka)x ltion? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value o.3dm78ii bh : x7yhjlgf the angular velocity $\omega$ Explain.

Can a small force ever exertd bfy kb, 64:fracsr:0 a 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 hands beh r3y p(/vrifk2/zqjr +ind your head than when your arms are stretched out in front of you? A diagram may help you ti/(fvpz rqk+j /yr32ro answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at thf4.4att so/09fp soodd8n u9te pedal cranks. In which gear is it harder to pedal, a small rear sprocke dtt fopo du4/o.08sfna 9s9t4t or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slen,9k i,aik4gc vder lower legs with flesh and muscle concentrated high, close to the body (Fig. 8ka 9ckg,,ii v4–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walker qzs00 ,mbf qrcj (ce5e,h6ti.s (Fig. 8–35) carry a long, narrow beam?

If the net force on a system iz (da9v6on/jfi )zbwki x2 /i1s zero, is the net torque also zero? If the net torque on ai1nf2 bj()6a9 /w kz/ivxi ozd system is zero, is the net force zero?

Two inclines have the same height but make different anglewftt1zy fs+elzklr0421o7 ni,-n h p 1rc96yzs with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed ofzlfnwyzn1o ,+fki rhteztr4y l9 10 spc1-67 2 the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling t.9o3pldha d+ (from rest) down an incline. One sphere has twice the radius and twice h.3tdp9oda l+ 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 bottom?

A sphere and a cylinder have the same radius and the samo1 t4*3-,(l3medy hedqem ryrb/4kohu jt+ j0 e mass. They start from rest at the top of an incline. Which reacht/,r eyuootmk *-hm 1(rh4dq4e3b+e0y3j lj des 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 momentuuix+ 9toy* 0lymxnlj: 5c+/ szm and angular momentum are conserved. Yet most moving or rotating objects eventually slow down andy+ c 0o usy5tm*l:z9ilxnjx /+ stop. Explain.

If there were a greavlh n m5 uw0s,+o/qd8ot migration of people toward the Earth’s equator, how would this affectv/doo, 8n+u0m 5hlqsw the length of the day?

Can the diver of Fig. 8–29 do a somersault without having any initiaruu391v(zw q hl rotation whev u wu9r31z(qhn she leaves the board?

The moment of inertia of a rotating solid disk ;av ur uzbj9n /x(:;cdabout an axis through its center of mass (9 /;aj nbcx v;udrz:uis $\frac{1}{2}WR^2$ (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 sto+*cn n4b-eio eol holding a 2-kg mass in each outstretched hand. If you +* 4enne-bc iosuddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is hollowaz 6ri/wd .xr. and the other is solid. Describe an experiment to determinei d/6..rrzxwa which is which.

The angular velocity of a wheel rotating n+cz)gt3b1e4 )mjdev9m *emy on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear accelm c zvm3mgd+ee)n1ebtj 9*)4yeration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating -nc3p-kxxy;plh l:z yruqk: vq+n+ 7/turntable. What happens if you walk toward the n y lk+p;: ycxk+q7vzq xl:nr/ --hpu3center?

A shortstop may leap into the air to catch a ball and throw it quickly. As he 8w s3hv l9gsr/l*0xk ythrows the ball, the upper part of his body rotates. If you look quickly *9xgvs8h ly 3r0ls/kwyou 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 momentdzi)pt tb-u7/e8t1h4ph:il n um, discuss why a helicopter must have more than one rotor-dpu7ipb/t:)1enth8h i zl 4 t (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in r:oh1owk75umj6kdt,h adians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincidence. Calculate, usingmfw3 l)2it.-- spawup the information inside the Front Cover, the angular diameters (in ww paut-l.3f-2)mps iradians) 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 divejos m. ,cnv*gnie2aim r;c;7,w2rqz5 rges a,a;v5.m nwsgqnr7r ;moc2i2 jizc *e, 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 xndm o,nvfmx/a-u27 3 tog /,kturned off during operation, the blades slow to rest in 3.0 s. Wfokov,/xg 7ta mdun-x,m/32nhat 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 the balid u2smwt7+)qxyg zc7* *tq7ggq,+dcl makes 15.0 revolutions, what tdqu,z77 x+ci+w )2g7ygc s*tqg*qdm is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. Hb zbd fv00*0cnow many revolutions do the0d *bvb0f0cnz wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 .cckzax5qlbg ys ,tuu : :((29p y(dbe5js)mprpm. Calculate its angular g:a( ju, (qbyltu5c) k.9dc s espy:pb2z5(mx velocity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makejrkl 1f h;7z8xs one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linearkfh;1 xz l78rj speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orbit around the S: eu4 /l0xmol jefap0:un    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear spes1ocp1 zx1jg .h* g+cjed 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 the axisspze/, 5wj g ,1)t)8g hqkxvnb of rotation is to experience an accelerationp1q,xg5 ze),8/bsvw gj )nt kh of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheelo.m2r k1 si r b7i6m8m*nry0st accelerates uniformly about its center from 130 rpm to 280 rpm in 47b.k ro8is mtmmn6is1 0r*y2r.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 radius3nu r2ruy 6ykx04qgzi+;p 3na $R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboard the Apollo spacecraft put thems/9p ib ywr3b4l (txp4celves 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 i 4 94r pb3blycxw(p/tinterval. 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 unvt o ;q6v+ox5ziformly from rest to 15,000 rpm in 220 s. Through how many revolutiox;5 zq+otvov6 ns did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s.yf 7mbsm.,1 3+npq evszo1 rh -wsvk74 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 .q(y)m r oed/t5sh;w2 8hpv2dh ir0ze the stresses of flying highspeed jets in a whirling “human centrifuge,” which tam 2hehsho t5vdq;8ir p )( e.z/r0yw2dkes 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 uniforml. s (gk0fl5gnhx6xx0l,ydqa5ipi5 q3u d d 12py from 240 rpm to 360 rpm in 6.5 s. How f p3fq hx1l qd06ld52igd5xx,0.py (na sgi5ukar will a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 1500 repcz9jom(9qh :i y(t5a5pibw .volutions before it comes to a stocwb5 ij.t9(iomqp h 9pz5a y(:p.
(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 its2u:t8 wcol o * s;ab lhqixtbq7;0b70e speed uniformly from 95kil;7qo*t b;h8ow2qubae7:sc 0b0lx t m/h 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

  The tires of a car make 65 revolutions as t g-) yyyw*vi eryg08v.he car reduces its speed uniformly from 95km/h to 45km/h The tires hav0 )yvr.wvy-yigg* e8y e 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 v uky .otpx2 85nup0e7her weight on each pedal when climbing a hill. The pedals 8eu u pnokp7 ty2vx0.5rotate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cm wide. Whatz/w -c6aa/q)w54gbbs yrec*zl j4w0,xa igz . is the magnitude g)ylw wwzq/a 4.s0b*c-bgz 4ca6a5r, zj ie/xof 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 shown ijx*8mdcu x5,, w zr4srn Fig. 8–39. Assume that a fridu4x,*rjmc 5wr,xz 8sction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass rbxs xc9. (g+bm, 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 pob+sc9rx bx (g.sition and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylin;0) flnajo e+ smb r.pv7w)/m2pznpa(der head of an engine require tightening to a torque of 38pmrj)vo0 (zfen7 )p2s+ab.w m/l p;na $m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of radius 0.648 m when t-i oqnbs 7,rw33fk mq,he axis of rotaqnm7f,i, brkoqs3-w3tion is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bfx3se wnv5 1;ug(* ;:)9qwc gqo*c-orifhrtricycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of thsfg1g9 wo-hc(;ce3u*x qro: nvtfr*) ri ;5qwe hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thinjmih02*zl7pg o0 .f+bbc:x iu , light rod is rotated in a horizontal circle of radil* ou . gi0x :2hbmcbz+0ij7pfus 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 bowv ki hk *9jqpo( ly9-xb.l,hz8v* l-ezl on a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her h*hjll yl(9 xv o8h k.q 9i-z,vz*-bpekands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point objects sj0rx- wn m7c9ahown in Fig. 8–43 abou cjn09x7m-arw 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 consistwyu lv9 83o*s aa+x+bxs 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 satellit3o* 9uja qsax(e 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 each rocket if9xsja ua(3oq * the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinderhr4/e- ydd.o- k9 ymwhd3g4ov with a radius of 8.50 cm and a mass of 0.580 kg. Calcyook h/. 3wev4dm- 4-dy9hdr gulate
(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 ; +i4yxn )pa(gqw7t9i2rl q xw1:mqys 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 ta1 equ ztoz);*te58bqqy( mh2fngentially on a small hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, ymq ;e8)(qouehz f5zbt2q*t 1 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 evy,dz7o + 7,-mydj doqventually brought unifor-7v dqm,ozd y,+j7 oydmly to rest by a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 lpwmw;bb n. i94jvi(sirid3) 2*/ bxh 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 forearm, whib*q .dva k kh+sa9wk/-ch rotates abou*akk b9sdv /+ ak.w-qht the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, as shown in Fihoje 00m5w lx) c oxn2m73t3rpg. 8–46.l mx33ro phec2)w 57t0jm0oxn
(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 massob6d l/;f s9w48nae ines, $m_1$ 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 hammerwoywglq;:u6 ) hm75zv from rest within four full turns (revolutions) and releases it at a speed hz;gw7:6 o m5lwv quy)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 momenti*fvk3s r:0ru 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 ddvy x(+ / r:oc lqoeo/v+yhfk d8(ld 7b5pgj27evelops 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.i tqadam)pj8 2/ 3ni:z0 cm rolls without slipping down a laneamq ida )z/tnj2:pi83 at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy oovz9yxn vw i2u8edk;u33 z2x 5f the Earth with respect to the Sun as the sum of two terms2xu;vd kwv38ynezxz5o2i39 u ,
(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 mt3kkv ei)+a icw 6p-k*ass of 1640 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 k-tkp+iiwe )kv*c3a 6s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm a f 0eohnk6;m;r,8djeand mass 1.80 kg starts from rest and rolls without slippim j;0;oafer,8h ekd n6ng 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 vertically on its tip. It7k. 60yjt l(z0v/qk,n g nco3jr g,kgz starts to fall and its l.lkvy(c6qg3n7kn zorj z0,g 0,t/jg kower 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.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating o-hbao z /7g*zocm4dn, n the end of a thin string in a circlecd/n,zo zo4- mg*b a7h 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 wjm81 cw)j*589r piz-tew cez npq (:iqheel of radius 18 cm when rotating at 1500 rpm? 8r )izeminqt*1 ze5qc-p: (w98 jwjcp    $kg \cdot m^2$
(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 rwytzjn*nz4l 31; c. 0ljfxe)is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the spe wl ztl)fj0c nrxjn.*13 yz;4eed 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 F- iq-w-aes.6i2el7b el4j dezig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to tl-e-j7i2eb6 dq sz 4ae ewli.-he 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 fj2qqg .,q.(o o lp gf*gbc*deq/u 4ql bq,0)xher spin rotation rate from an initial rate of 1.0 rev every 2.0 s to a final rate of .,pq/g4oqlfe q2l gq*quj0,. c(bqg*fdo)bx3 $rev/s$ 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 cf /t isj-h)g0a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0s h)/-0g tfcji 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 spin r0w/ rktg5z,.hhq cx.-3grboning 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 mome b6te0wp *xaf)ntum of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia (wyf mw2)cwl/I is dropped onto an identical disk rot2cw) (wlfw y/mating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 6wo 1aif/ah*r+ a-joxjcy 33 h2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg s- rs d6itr sz(huo* 9q2m:j5rxtands at the center of a rotating merry-go-round platform of radius 3.0 m and moment oor*hx i5qssud:jz m692r rt- (f 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 szb*lt zlf1mt.bqhdx+5 ,ysx / ;o c70of 0.ofb*,xs h+zct5 s;q/l b.ldt1 0zyx7m8 $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, losiaq 2yr c*fg-1 89gwcnxng about half its mass in the process, andanr1f9gwqygc* c2x 8- 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 )xjx2yo2st+l/m+ h vr winds 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 jn,w:i4gl - iw$ 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 w-ekx5g 5meu /cgqgv -)ithout friction. The moment of inertia o5) xgeuk-cgv5 em-g/qf the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diameter merry-go-round turc6-9.v kyly knan9x0bntable thatn .y9-b v9c06kyknxla is mounted on frictionless bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of the rope lying7vm) 0jhf:h 6relg wx4g xu:-j on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How faxvmx6uhg4:j r :0wgflj-)h e 7r does the spool’s center of mass move?

  The Moon orbits the Eartq5peo*y t-+q-2 wnz ,aqb6 ialh such that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter casaobqlnzwq5i yet-- 2p*6a,+q e, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest:gf3dn,k ia)2flao. dra; /i e 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 tzlowy0/0mv;2 vk o ld2he shape 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 acceleration. Calculate the torque delivered by the motor0kv d2/yl 2wo;l 0ozvm.    $m \cdot N$

  (a) A yo-yo is made of two solid cyfjrbx/oyxep s,lf q/ )s0;b3th5ir/ )lindrical 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 diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it,o/eb)p)jxf lxs biyfr h;3/tr/q50s is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular sd)y/c r/uh (iepeed 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 hd9:xy :j o/mi:b+*dns .ami n;)pohw great, 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 i;wmho : :xi+n)9m yods/.ndp:a*bhjthe 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 aut3; c-w)-m5iydr l2he ctxr,db 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 and mass 240 kg, and should be abll x,wch-3di ce -t2;r)ydr5bme 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 a+ki3z9d vs s*nn $H_2O$ 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 hori ozhct5.f -(-hnf)u j,dbvb:kzontal 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 ignore frictionc/n4aj- f;,b2m jkbdl) 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 centeb4l;2a-d k bj, f/jmcnr of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It isrx-tx t9gh 4km z93m)dm3lf r5 standing vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, wmxld 5tgm 9 t439 zrh3r-x)kmfhere h

  A bicyclist traveling with speed v=4.2m/s on f: b pz gb-f;ygx)ip:6a flat road is making a turn with a radius The forces acting on the cyclist and gg;:ib 6 b:xzp-)fyfpcycle 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 roc buabxf88d l-+k into a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm f8 xblu+adb-8after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her armz3b u:d9/ nxs w.f.vhks 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 as .nz93du /f:b hxk.wvrms held tightly against her body.   

  You are designing a clutch assembly which consists of two cyl*0ihbj )xlls4indrical plates, of mb *)hj l0xis4lass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 rolyafz161jwvi19 zb.1(xg rukv1 8jz auls along the looped rough track of Fig. 8–58. What is the minik ji1uzg(a6f8j 1bu x9 r.1w1zv1yavzmum value of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assum.kk qz;lh8e8 p 77qrreb4h y/;amzv7he $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed unifot2 5hnp: gb 1qfmpx:6nip*8 bbrmly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What wpp*1t8 pxq5:bnh2mnbf g:i b6as 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