A bicycle odometer (which measures distance traveled) is3jb,37dfrg y3e*dur9f(pg 1d; ru rd y attached near the wheel hub and is designed for 27-inch wheels. What happe; brruf3r ddd71p*3, f9eyrjd gug(y 3ns if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a pb+x (v7s9j ry5zpa /duoint on the rim have radial and/or tangential acceleration? If the disk’s ang sj/v bxuad(p57+9z ryular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single 3:xhph.b,,wy1v p uf xt j2g:/x/icdfvalue of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Explai9 v7 jbe)rr dwkr3 wiqobz374+grb(*yn.

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 youjpo99 :h ng1d27c-ho 4t m/divs9l kosr head than whe9ijk4odp -97 hos2omtd/cvgl9 1h ns: n your arms are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle h,g+pitbx -tl8 as seven 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 large rear sprocket? Why? In which gear is it harder to pedal, + 8ltpit-bg,xa small front sprocket or a large front sprocket? Why?

Mammals that depend on being a llf6nify:2wr e ,/ha8kg .b;able to run fast have slender lower legs with flesh andfn/ah 8rk,illbfye6a2 .g:w ; muscle concentrated 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 (Fig. 8–35) carry a long, narrow ,ut4gwwf ro2+beam?

If the net force on a system is zero, i c t onwj56xv y/h4.lv/:0kkkss the net torque also zero? If the net torque on a system is zero, ishx 6kw/4j5onk:/k vv y0l.cts the net force zero?

Two inclines have the same height but make different*4i*m q:yduyrljt ;2w angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed ofi j4:l tqr2;yw*m*yud the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an incline. Og1u *x,j,mn1z-jozh g ne sphere has twice the radiugg,,*o-j x 1 uzz1mnhjs 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 bottom?

A sphere and a cylinder have the saci1p*, joh 3n uf*g;kfme radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has;unpof*ih,1 jkg c f3* the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved.7u:ls--z5vz 0cy cx*m m:(ai jof fy,t6;zfuv Yet most moving or rotating objects e,x v::(-v ;z 6*smuo ty0fy lcf7cfaz5zjium-ventually slow down and stop. Explain.

If there were a great migration of people toward the Earth’s errww2*yj v91h quator, how would this affect the len jh*wr9wy12rv gth of the day?

Can the diver of Fig. 8–29 do a somersault without havl ,yceehu0hpzzvjl4 ;pg-8 88ing any initial rotation when she ue l-zpgzpc;e 4v 0jh88, ylh8leaves the board?

The moment of inertia of a rotating solid disk about an axis through its ceni 3zugn3 ix0260ug bmnter of mass is 03iizn2bn x0m6g 3uug$\frac{1}{2}WR^2$ (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 in9m6*d c:y .x3tprk zxn each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay3.nc p6kxt9z: * dxrym the same? Explain.

Two spheres look identical and have the same mass. However, one is ho 1egfr,us1 5brllow and the other is solid. Describ r,ru1gf1s eb5e an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal a gdbwu2s/x7-i xle 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 acceleration of this point ats bxi /udwg7-2 the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a l. i at-q jexr(.zuw:z;arge freely rotating turntable. What happens if you walk toward the ce:et .qj.(awx-uz;i zrnter?

A shortstop may leap into thzz7 (ir)6fsns(hj9 pnvd;j qpy4gm/8e 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 will notice that his hips and legs rotate in the opposite direction (Fig. 8– s nzv4n)j9;z6y7(pfi (msrgjd q/ph836). Explain.

On the basis of the law of conserva4wxn( zc 1nnb ;hc(z0qtion of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss (wnxz4(;n0 1b cczqn hone or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (a) 3 p9 eijzc9p t;(/izp7l0 $^{\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. Calculbs)x;9hrzr)v(s8b ly ate, using the information inside the Front Cover, the angulazh9 )bxv r sbrl;8s)y(r diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 ) qi5rk-1nur l;4ffozkm from Earth. The beam diverges at an anglqio;nf zku1 4 )5rr-fle $\theta$ (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 um+. ;vvgel(fzxp +l3 the motor is turned off during operation, the blades vmz .pf (+exv3 gl+ul;slow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floovxmg0numuhi 5 nmn7(58 jg ;a:r 3.5 m to another child. If the ball makes 15.0m uungn78a5nv:mm0 j igx5( h; revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutions dob0s 9a2qc8qsrw 8s9+s-add x f8wwe .m the wheels make? 8q q+ dasw9fw e02rs.w8dssab9 8-xmc   $rev$

  (a) A grinding wheel 0.35 m in diameter rotat 66ein4s7ydfi+ cj)hkes at 2500 rpm. Calculate its angular vel h d4csij66f inyk7+)eocity 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 . h8:a4xyp.t-9 ru3 iyrhwj)ivooe- s(Fig. 8–38). (a) What is the line.y . jo: ihr3tvw98 eyus4)r--piaxhoar 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 angulasgy- kf 9*bkks-a9z/ar velocity of the Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a po5 ajvx7. le64pho t(m n:obg 2()dqjfzint
(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 ifb 8zk1p:loiz6av7c1 d a particle 7.0 cm from the axis of rotation is to experience an acceler:o8p1716 zacb zikldvation of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly ab7,du (e s61j/cp6rrpf ohnlx 0out its center from 130 rpm to 280 rpm in 4hls,c7jed p6p1 frro0/xn(u6 .0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radius r8i jg ( tmh;qt8cm yxh6)x6;j$R_1$ 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 space)f d9l*u( co5cp +aah-obbymh4 .hd9scraft put themselves into a slow rotation to distribute the Sun’s energy evy(h *d l-c.d9smu5aph f)oca+b 4h 9obenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpm in 220 s. Thr9xi 9(zpbc en4ough howb4e 9cxn(ipz9 many revolutions did it turn in this time?    $rev$

  An automobile engine slows down fro fheb9g+ xk-lvcxx93h p,4 w-jm 4500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying hig3rdhngwtjr:f,y 95vm *fn (d 8hspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete 5,yff 3*gv9 whrn(jt m:8dndr 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 acce2xs-xa t1qadtp58i )olerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far xa1i25tq ad s8p-xo)twill 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 revol*+gkpu;lye4wi+v -x 4)ebek21;vh b/syo ti autions before +oi2/e-yt ewk1eb p ;+g;*l4a)vi 4vbskyhxuit 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 tv+ il5*0f jrp hcq4aee0z7 l9ehe car reduces its speed uniformly from 95km/h to 45km/h The tiresijf9 recp +47eq50l*0zlv eha 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 the car reducesy 0k*vcmfw.;nj :oi g6 its speed uniformly from 95km/h to 45km/h The tires have a diameter0 ynfmwio:; vj*c. gk6 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 ridin g27tq /au/tmnj4hci u9rf7 t(g a bike puts all her weight on each pedal when climbing a hill. The pedals rotate in a circle of radi (74/jt c9 ttq7ufh2ri a/gunmus 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 wwk r5al*.,zx;j4tem e3hl+bp ide. What is the magnitude of the torque if the force ih mklw eb.+,e*;ra l3jz txp45s 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 sho;fajq)vweniu a nkc88--eu7cm9e sif 6p3(v5wn in Fig. 8–39. Assume that a friction torque of89qv8aee cw(ukin cpmfv) 57- n3ufs-a6j;i e 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a massless rod whqc:t; tkp q+hg0cafj(uvo5+ 0ich pivots as shown in Fig. 8–40. Initially the roq(p ;++50h:g juk0cto tq afcvd is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylomb8fo) x(zd6 inder head of an engine require tightening to a torque of 38b xz)d68oo f(m $m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg r:vo8nw;z,h*n63h4 ifg opxxsphere of radius 0.648 m when the axis of vx68z , ghoh3w* fx4inro n;p:rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia 4p(gv,u y:6 y7- fl e1*oa5idpthl y0anq8ert of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignore 5ynqa7ope-y:f0g uthe *8ti16pl( l 4ar d,yvd (why?).    $kg \cdot m^2$

  A small 650-gram ball on th)m n:g xg-8k.qg0nauue end of a thin, light rod is rotated in a horizontal circle of radius 1.2 m. Camg uxgn .agu:q0-nk8 )lculate
(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 os 9s5f vdtdcu,toyg,(,mt3( rotating at constant angular speed (Fig. 8–42). The friction fou,s5,f9(s,ttvtogdody 3mc ( rce between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point objects show1.i8 n5 rwy*qw yua-jun in Fig. 8–43 a58iuryw1.ya wju * qn-bout
(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 atoms whosyo*d yqj4ghk*le7w zs+:eyl 1s8c;;l e 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 correct rateq iwe 8q:zqa2o2kn4; 3q zioo :-pkid-, engineers fire four tangential rockets as showq w-eio:2onqi3zq4 ;k :2-idok8 qazp n 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 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a sel,b),lny ;,pzn 6 af g7+h 9mfwqk9wradius of 8.50 cm and a mass of 0.580 kg. Calculhak,fn+y;,eww )9 ll,f6nz gqbps9m7ate
(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 to 6 yxwd3q*(gk lmy1vz83 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven merry-go-round and is ablzt (;cx7tjk* 20*ars;ggbeua e 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 accelekx*e (;* urgsc z2a t0agbj7t;ration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating af( bq./kd3pkm t 10,300 rpm is shut off and is eventually brought un/q3k dfm pk(b.iformly 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 accelera k 7tv9+qmi-0)xxrt yotes 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 actios amlz/86y7+chi a i gb*t)/jln of the forearm, which rotates about the elbow joint under the action of the tim)a/ iyhgc8 za+s l *b7l6/tjriceps 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 consideredl lbu0n 5/lta l: erhl*za2:l* a long thin rod, as shown in Fig. 8–46l u:a*/lr5ae :hll2t 0nllz*b.
(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 ofn 2h mdstt)(d0 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 fr( 4vy ompls 1u.c80htsom rest within four full turns (revolutions) and releases it at a speed h10s8om (c .ytlvspu4of 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 e yo,;.jzv ro 1zh 6xl;z81fn-o/qtpwmoment 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 3qu 7wbcr6rt) 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 / meo)v.v(zsw and radius 9.0 cm rolls without slipping down a lameovw/z(.s v) ne at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect u:,bf ggn t,r+to the Sun as the sum of two term, +gbufr,: tgns,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius of 7.50 md4 e() -aq a0cns:lzn 0ew8urn. How much net work is required to accelerate it from rest to a rotati0de -:4 8wen qnsrl0c)anuaz( on 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 from rest and v,y vte m-fyss8/;o*7 jbvpl9hmw0 -crolls without slipping down a 30- 9-wof vctps/me7 m*y;vs,bh 8ly 0jv.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 i64 li.yoj :wtsts lower end does not slip. What will be the speed of the upper end of the wjl4o 6yi:.ts 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 on the end of a v zo /*,ymnse+thin string in a circle of radius 1.10 m es* omy+/,z vnat an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrib bu wk5q8c+5f2dwjbo8v-/h s cal grinding wheel of radius 18 cm when rot8ckbw- w/b5juvh8ob5ds +qf 2ating 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, o-hq j+,)iw54pm o bnl7pd1ura n 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 rotation decreas alp)4b,nw+u 17ordhm ij-qp5es to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one show;.,vj rv vbl4wn in Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.v .4v,jw;lbvr5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial rate d27 j+.pj5d.q yag-0fjjxm/b5z sx 3bi dt9qoof 1.0 rev eveq xtj xmj9f.5bq 2jz +- d/d5y7a3dg0bso .jipry 2.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotat*rzvys s-5vz 1l;raii1y9; *eulqt 3ging around a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of masserq zzru**v y1isv3-stylg1 ; li5 9;a 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 n5/9w-i 3 apu lspa2bdspinning 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 6urz3u t- 8bl3c7lvcv the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$

(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is dropped oy0mau r/ mkwi5t lv51k:m yz35nto an identical disk rotating at angul 5:mk31ywr0 5 5uiavl/kymtmzar speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns ah,) wybn/t(dml frxa9b/w-a d9+dl: ft 2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rm bp p8z (a ,f(+qntlp1x2cw-pq vkn11otating merry-go-round platform of radius 3.0 m and momeb np1l-tm,px w c(vp1k8+zq1 p2af (qnnt of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an angular velocity of 5ip :3/l/n2f awcb ioez.jkj0fxj.:t 0.8pf 3z. twlx/j :k/.c:j0e 2ijin5bfao $rad/s$ 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, lo* nxwzo.0 6ovfw fmwd f5t((6nsing about half its mass in the process, and winding up with a radius 1.0% of its existing r z(wfm 6 0*5fvwnnodx f.(otw6adius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in gcniyr6/:k/d nc v22 gexcess 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 5o*k m5oocj /,epf1bf; oi/fe $ 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 z,3hzkcp,ux: dgrr kf7/7c+bithout friction. The f kphc7d,+,/c :u xz37bkrzg rmoment of inertia of 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*opas: l.yb l. gmaow9mlz;(1-go-round turntable that is mounted on frictionless bearings and has a momen*ms;w.yal1 gm(o.b:apoz 9llt 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 th pr*tx: hb/5sje ground with 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. The spool rolls behindh5jr/: pbxts * 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 Earthlq8j 2jmbt-ig l1) e:f such that the same side always faces the Earth. Determine the ratlj1g)-b f2itmel j:q8 io of the Moon’s spin angular 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 acceleratesmyb,jo2z,5g8x *ya bi from 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 uniform cylindd8g7a zwzv(m ( ce0v3/wnv*xcer of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular accelzzg0*cv(m dxnv vww a 8c/(3e7eration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.ug,i y6iktj0b+/ vgsa-nrif xpb6/. 9050 kg and diameter 0.075 m, joined by a (concentric) thin solid cyay6ki6brbgtf/s,9. -un/g0 jv+i pixlindrical 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 is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular splput 1; q4b9bdv) zyvi2et:fkz3r+c/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 8.0dn0hl/ ,wht. m times as 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 the proceshdht / m0nw.l,s, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is f0pnmeu*dl8 )0)x (rjwb8g ea kor it to use energy stored in a heavy rotating flywheel. Suppose such a car wk)enrg*lp0 ad(8x80em) jbu 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 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 aj 164f6fdxpka r99ett5k 2w qqn $H_2O$ 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 surfaea 90n-5mltqvz hv8x ,ce 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 lengtj-z gow74prm -bj+2ts h L can pivot freely (i.e., we ignore friction) about a hinge attached to j 7bow2-zt4+pj-gmrs 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. 8–21g.]

A wheel of mass M has radius R. It is standing verticall5+ta/mqr tii6y on the floor, and we wai i/tr+tqm 5a6nt 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, where h

  A bicyclist traveling with speed v=4 f/wd,pq(f ha93kml /c.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist and cycl,q3a/fm/wdc k9p ( hfle 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 xzneaara q u11 2ey+avy( f(.81.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the to+ qa1f ea(unexy(.8zy1v2 raa rque come from?    $m \cdot N$

  Model a figure skater’s body as a sol5 6gnsa-yq -eci *s3zoid cylinder and her arms as thin rods, making reasonable estimates for the dimi ez3qosn5a6-*c -ygs ensions. Then 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 cr( b6xr; kutvc4y;pp 5d9gr )which consists of two cylindrical plates, of maskc(x u4;gdp69cy v r 5rp)brt;s $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 along the looped rough fd8q eupojr,,kn3qq.i (0f9 strack of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reaci8fq u0( sfn,j kqd9r .3p,eoqh 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 as*ch,t8.hf 7qi lpo*)wbna-dlhon zr9j8 ei ;;sume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its .ixv6* ozkfj-speed uniformly from 90km/h to 60km/h The tires have a diamete*i xfkvo.jz6-r 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