A bicycle odometer (which measures distance traveled) is attached near the wbzgfy0g1z)*s z- sb2x8bj;xx heel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24 z0xb1 szyx; szbx*j-g)2gb 8f-inch wheels?

Suppose a disk rotates at constant angular veloci 6,pg+ vsw9m.td 5r ajs1i-yswty. 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 c wsjidm1tws ,a9s5rp.y v+-g6ases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value omp-ar(rafve ;9eomx +*acow6,y o9.qn6el4 gf the angular velocity $\omega$ Explain.

Can a small force ever exert azi4dxnpc. 7kzl,t66 yy/ u5js 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 diffiqs /k-/tomzbql4ow 1-z fcv,/ cult to do a sit-up with your hands behind your head than when your arms are stretched o4/tkqfzw/sobo- ,z- cl/v 1qmut in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and thuyr(f,fxu(z* ree at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket ( (yurxuf*z,f or a large front sprocket? Why?

Mammals that depend on being able to run fast have slendertyk(rl 1x3vm7 lower legs with flesh and muscle concentrated hig7rvk1xy m (l3th, 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,ia(/k: j ac/.ziy46av rm f)kq narrow beam?

If the net force on a system is oq,awg+si4b. zero, is the net torque also zero? If the net toi,gsoqa4 b+.wrque on a system is zero, is the net force zero?

Two inclines have the same m9se64.s-:h /mdgzrrg/s eqzheight but make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will tm /eg/-zhdq er4sgsmzs .6 r9:he speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) dow+yk ,k5digo)l+k ato+ n an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed yk,a)o+l+gt+okk5id there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder hafiv9jn8 dei.1(7a1diu rd 7rgve the same 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 the greater total kinetic energy at the bottom? Which has t7d 79ndfjrgidri1i1.e( 8uavhe greater rotational KE?

We claim that momentum and angular momentum are conser-6sofgpa)3+p5yxx fcy ga h 7*ved. Yet most moving or rotating objects eventually slow down and stop. Explainc-* 5pxa o yg)gf36yx7sfap+h.

If there were a great migration of people toward the Earth’s eq;:wvy)(3j l.vi o8l oi0d0/n xsj xacxuator, how would this affexvo3:d/c j xy;j0 o.lna)08(i xlv wisct the length of the day?

Can the diver of Fig. 8–29 do a somersaultttwq a3-d2o,ux8 xewh o;)*bl without having any initial rotation when she leaves tb*que2w3 a o hlt)wo,x-; xd8the board?

The moment of inertia of a rotating solid disk about an axi k ms4-p zlolnhgda; (7t1n5hk(d *.lgs through its center of s. ntp(n5z-khlk*(7lmadd;l 4o 1gg hmass is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2-kg mass in 3tzd5t o6( (y sflkdr,each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrea to6lt,r z( (5f3ksyddse, or stay the same? Explain.

Two spheres look identical and have the sl*eow2/ ihn zj:yn.5yame mass. However, one is hollow and the other is solid. Describe an experiment to .iyl5e /*wnohz2 jny: determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points west. In sfly 0 v1v-q/ 1d nlqjuy4xxmg6 v::tne-le2*what direction is the linear l-fl0 :qny*2jn l v xem quv/1exg6ts1v- dy:4velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on t ut4 c-a .iw5,qbwlr(rhe edge of a large freely rotating turntable. What happensti,wcqw( 5 -b4rur l.a if you walk toward the center?

A shortstop may leap into the air tj,- h 3(;owh69 nml,-msnfz *wapcxb wo 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. 8–36). Eznf(-w jhox*,bs ; mc,wwm69ah-3 p lnxplain.

On the basis of the law of conservatioqe 5ygsa l5p*; 4rv.z fa:n5axn of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can o ;r gxs az:f4e5n5pl5v.q* yaaperate to keep the helicopter stable.

  Express the following angles icjb gyz 830g n/q8l*fsn 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 amazing coincidence. Calculate, usingjhsp tzz4 v4 fge*6n(hrv3.7sw.q;z e the information inside the Front Cover, the angular diameters (in radians) of the Sun7t4 *evs. z nf3pv4 szqehhjgrwz( .;6 and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 kms h*gw*7t0 cc,s3dfz9 s2vgfk from Earth. The beam diverges at a fvc0gzt3fc,7k9gs swh s*d*2n 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 iw7 jlvg v4 cxt7./1cyrate of 6500 rpm. When the motor is turned off during operation, the blades sl14tgw7iv/xc jv .cl7y ow 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 floor 3.5 m to another child. m1b7xpk q.k 4v0nyp*-ag j7j uIf the ball makes 15.0 revolutions, what is its d7p4 na k pvj*j0b7qgkxmu1-y.iameter?    m

  A bicycle with tires 68 cm in diameter trav c2wb*3pjup lq1 +9gimw,om8hels 8.0 km. How many revolutions do the wheels m+8whjcq1mmp olp* i,2wb 3ug9 ake?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotatesm2 .9 lrp2iyxdoou;o8 at 2500 rpm. Calculate its angular velocityo8lu rxmoo.;pid2 y29 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 u nytnh43g7j8 revolution in 4.0 s (Fig. 8–38). (a) u7g n8y3nj t4hWhat is the linear 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+i gus8q4 z,odun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a point 4/5*9k ia*wcp whwqlaxdc0b2
(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(bg3s/64bnl rnrv0p u rotate if a particle 7.0 cm from the axis of rotatio6n3r /sbvb 4 ngu0p(lrn is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its centerm 0a- p1byc9+e7o1eivx/p svl from 130 rpm to 280 rspp iv1ex vb1+0 7m/yc9a leo-pm in 4.0 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 radius3di30ohu ubh * $R_1$ 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 aboa) pr r*i/0cv sx2ih2uqrd the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought oi i2/)2cquxvr ph*0rs f 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 tosmq,mkbm ;(57 xus)d) z9xkmo 15,000 rpm in 220 s. Through how many revolutions did it tur (xzm x;m5m7mo dqsuk)9sbk,) n in this time?    $rev$

  An automobile engine slowsqdfw 5+74ivjl1 :xuvb down from 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+qe bgokl4 ,.soy,/uw highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reacwok,oql+es by . u,g4/hing 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 uni)9f q88h xy;mxcz rg6sformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge );89sfy86 xrg xhq mczof the wheel have traveled in this time?    m

  A cooling fan is turnet0dr)m bq ccj vc1x3::d off when it is running at 850rev/min It turns 1500 revolutions befcmc0: x1vj rtqbd)3c :ore it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions a mdx8wzq1x:la.7(eyls the car reduces its speed uniformly from 95km/h to 45km/h The til( d :1lw7q.8mx axyzeres 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 reduces its speed *xbn, gv*fg ;x6c gwc3uniformly from 95km/h to 45km/h The tires have a diameter 3v;wgn6* *fc b xgxc,gof 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 per/5 r,uqk)wh gdal when climbing a hill. The pedalsuq)r/5g hr,wk rotate 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 c1zm gbj+xhyq5i-)y9of 55 N on the end of a door 74 cm wide. What is the magnitude of the torque if th yb)-icgj+5h9mzx 1q ye 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/ (bu2;hspafh lw. hr) the axle of the wheel shown in Fig. 8–39. Assume tl;h( prah ws.ub /f)h2hat a friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass ijir(v(g6x1x dw xd 1hh..)js m, are attached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is jwdv x (i.(isd1.r)hxj1 6xghheld in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine requij vm+y/r db4lorh;gy oi8y7. 4re tightening to a torque of 38 8; +r74yobi4hlv ./myg jydro$m \cdot N$ 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 the re)m6g-z4qh i ,9ec7p*ckil)hj - w xiaxis of rotation is through its centerikez7q ,jceh)6--9h*ri i mxwg plc4).    $kg \cdot m^2$

  Calculate the moment of inertia of a bicyc5 yjh1-0 elm h;2oinj qav0vs)le wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of sv2) 0joniy5h;0vl- m hq1eajthe hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the enip10(wkqx s3ns:u/ q)uj bt/hd of a thin, light rod is rotated in a horizontal circle of radius 1.2 m. Calcula(:x3pq 1/tn/0hjsw bq)ikuu ste
(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 constant angular s 2k8om)e*vss:8 e iddfpeed (Fig. 8–42). The friction force between mees8 ) *is:fdkov82dher 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 shown in Fig. 8– fc bqujao*:;143 jf*o1b a :cq;uabout
(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 oxygenm5 w0ri9bglb) n:edx 7p( :gra 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 c: 7,injeym di)ylindrical satellite spinning at the correct rate, engineers fire fouriiy)7, mj:end 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 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radiyi wl(wcp)) suai3 i22us of 8.50 cm and a mass of 0.a )i)pc 2i3 yw2ui(lsw580 kg. Calculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accel /58 iy;8t;g faztnuawerating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven merp7.(e.g8 aav7ru ;agb8yd dxmry-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. Calc8778dg.uvaya g pa(e;. rbd mxulate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is 1 rxm6(/mc- d+kgla8y gpa c(cshut off and is eventually brought uniformly to rest by a frictionalac+g (r /y amlc 6pk8g1d(c-mx 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 accelerates a 3.6-hj5rv:wz3 5jncnxh8,4 ebj e -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 throw0cn6 b 3h5zqw6iblo4e 0jozt 4n solely by the action of the forearm, which zb6hq0 ocezt3n i4w50lojb4 6rotates about 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 py3s+/ky1b+ w ndrl:,m3xg8my uuk/dlong thin rod, as shown in Fig. 8–468l3us,km:by/+dn1yrxy 3kpwm/ du+ g.
(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 d0 (+zjc5h h1uj.qx vhconsists of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest within four fu/ k rb bjb ;1a.v uoptj;1wk*rkb)c(/ill turns (revolutions) and releases it at a speed of 28br j/t;a/(v1 cbku)wp kibr *.k ;1job $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a moment of ines gwjtd(* -z-srtia 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 w3gjid,(d3o rf t z-e3cnn,a ,of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls without slipping dowhax6nvnv :v:k + i52giq2;6dud * qormn a lane qxnk6i +qvv2nvdgo ::dui m5h 2r*;a6 at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with re;b6*s :bvz;f 551ueo4ni s fmv/zpmzospect to the Sun as the sum of two terms, v1:mz/ 4bevn z is5bmu;fpo5*z6;fs o
(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 m4 pi0wm 2qc)onh;,tzmass 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 revolutionimt pc,oq 0;24wzm)hn 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 rolls withouu5xg+z igpq,5sy :)9muba: ktt spmabqg) +u,g tykzs5u xi9::5 lipping 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 ti+jy+8 mp)goxys7 -m nu wrue8 b2a7.mljq)bu-p. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? u)-rlnba+uu7.) ysm8b7po8+ 2je mm xqgwjy-[Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end of a thi-, c.;mpjy(xa2e xyhfn string in a circle of rad ( pcx.j xfe2hmyya;-,ius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum o5 a +vjqorc 7 dr(a5nmd+sggs;7p6,inpb/q7k f a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotat+qq gs75br67v +75 d/ompcki,jnnr p;agas(d ing 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 s:h50hhx f.sz-v to j:f/d1kg8vlw/ klide, on 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. 0zshtox58hd vkfk -vl:fh/lj 1:g/wation 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 otmqds7b(1d96yjo zpt9c 6ml8f inertia by a factor of about 3.o6tjqdb1z(69d p lms7t9ym8 c 5 when changing from 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 inje i,yxfj13932 zzax ccrease her spin rotation rate from an initial rate of 1.0 rev every 2.0 s to a final rate of 3zczx f1j3,aje92 3 xyi $rev/s$ 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 a fregi-kep0g e:wg y,;s; hd*3dkvquency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diam0i-;e3kgs:k ed* vygpwg d,h; eter 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 skatew1 ruq t w3+,1ft*htnir spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum of the Eaqu ;7sfhzrk8v1 x k13rou me3g03 36 llz:etpprth
(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 cfy,htsi78 k6g b 4c63zrxcs*dropped onto an identical disk rotating at angular spes3zgs4 i76h,8 rx*fckc6 tc byed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.d5h0or1c4-a 9 u,ubil-q cvjo 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 rs)qv v 1j8k9x7p- wsopotating merry-go-round platform of radius 3.0 m and momvsw)7pj98s p-vkx qo1ent 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 vaq:lpc4w7r5j(.z xiou x;ssx 4.z mt3elocity of 0.8 jxm 43ss4lrcw p;( .t.aio7zqu5x:xz$rad/s$ 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, losing aboutb8.u : k6;r c:kieadp *b(xrip 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?(round to the nearest integeria berp pki r:x (.6udc*8:;kb)$\approx$    $rad/s$ (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 120bfgbka g kyihk.xx 09r;e58+p6 z 0,aw $km/h$ 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 tnze :sfdx nr6b*zf, 2t ;*c2h$ 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, init7,nds6i,wgtsp6pn p g)0w9qau .ak n2ially at rest, that can rotate freely without friction. The moment of inertia of the person6s ,) ia9n w dg207gsp.kptpuwn,a6qn 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-si;pfn whf0px02w/h1 go-round turntable that is mounted on fricti h2if0wf/w 0sh p;1pnxonless 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 lying omvgytw/(7+*wf4t ktm wmnh**n 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 far does the spool’s center of mass mo k*g tf /ht(n+mw4* my*vtm7wwve?

  The Moon orbits the Earth such that the same side 9es 4 l-h 4vl/b+60kp igiqnm4bki ,rualways faces the Earth. Determine the4 4ibk,gq eisb9h6 -+l nk u4/vm0lrip ratio 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 accelerates fromvjm n;-bl,d;ip;vfwr 4y7t5 m 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 4doh3n + 0xwyct/ubm3 shape of a uniform cylinder of radius 0.20 m acquires a rotational rate of frobtxo0 uy+4h d3 cw/3nmm rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cyz ve,nj-h -ri0lindrical 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 calculah-j,ir n0e -zvte the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of the rear wheel tc xg2c)kznq*; +7wnu($\omega_R$) 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 great, were rot x +.fs* hxz9(mp axsiuh)6yu:ating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron9hzyuim. x+ )*u h6s:xps f(xa star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pogrvv cjt -k a:txfp: ;4zvo-21llution automobile 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 kmaozftxt v1g: ;2j4vp r:c-v- k without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates an tze,+oszd (r+ $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal surface at speed v=3.3 b. cs5wckh ;b dsnw0g2.e2)mtj/ (xpw $m/s$ 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 pivei)2gi 5a*( vgrsty t/ot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizoav5si( yt)i2 trg ge/*ntally 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 flo bvk6c4im8:s zc9q/kdor, and we want to exert a horizontal force F at its axle so that it wil :bsm 9k6/c vkzcq4d8il climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on aha0ub40of3 dfgf 3 3zk flat road is making a turn with afhf403o3 zugd af30 bk radius The 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 m and begins w+e22 b ijo)ye8ndk ldw;q-;xmzo o). vhirling 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 requio )vb;yl;no.o+-q8j zwekexm 2dd) 2ired 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 armd jkp2; pz+flrb a.zi41lpbnr+ys 45 4s 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 her body.lsd.p 5 kzra;zyf+pr 2+lip4nb 4b1j4   

  You are designing a clutch assembly which consists of two cylindricalu4a4qlq4ab 8 brz8p7 r plates, of masb8laz4 48pqu aq4rr7 bs $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 loopellzyqnvvc xoth gq98 88:2-t4d rough track of Fig. 8–58. What is the minimum value of the vertio:tlz9y -t8h8cg 2 nqqvx8v4lcal 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 d njj0slfy* ,vez1zn0jq b5.:q o not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolu/k+ xzt 1xzr8j*zgp:g tions as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each tiregx/xtg1 z r:z kp+*zj8? $\approx$    $rad/s^2$(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