A bicycle odometer (whcqfybdf5q 5 )a fhk*;fg)1z-b cjl 4i1ich measures distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicq5f-f )1cj 5;ch b)bfqliazdkyf 1 g*4ycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on yx 9azv awqk1* :+j- uw0jdiz.the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the pointja1q.yvj*z9 duwi x :0 zkw-a+ 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 value of the angular velocitymnb9ii- iy:dg-es(+z:.g u zx $\omega$ Explain.

Can a small force ev 3sapkd)isy m pxn57xyk8f)qx/ u0r74er exert 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 difficu jpe b,fy7)2zwlt to do a sit-up with your hands behind your head than when your arms are stretched out in front of you? A d2ybjz)7 wfp,eiagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the pedalm sp i l73(yqz jurb*;q17zxm3 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 or a lzs ql*j3ib y qm1(mu3; 77zxrparge front sprocket? Why?

Mammals that depend on being able to run fast have slender lower legs with fles )a,42fdj dmoph and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distributi,af)p o2jm d4don of mass is advantageous.

Why do tightrope walkers (Fz0c5ypf)l o+fig. 8–35) carry a long, narrow beam?

If the net force on a sye umhfsx*+;,rvgq8 sm;a)o3 (q gf :m4r(mffnstem is zero, is the net torque also zero? If the net torque on a system is zero, ,f gqsr( ;3mv4x *r:hmuagof)m;8(sfme qfn+is the net force zero?

Two inclines have the s/ k(rr-hz7*syvc0r (d,1vx fas2 mwk zame height but make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the b(r2 1s-f7,*krkrdvhyw(z /z0 vascmx all at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down am0a;hdi 5rci6n incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Whic05i ; ircdamh6h 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 same mass. They start fa qvso6+*cb zc2b68u arom rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has t*vc +szub66 2bcoaa q8he greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentugam3iwe64 z/g u*uoi sq:8 p7bm are conserved. Yet most moving or rotating objects eventually slow down and stope3 suw4i6qgip/: m8*zau 7gbo. Explain.

If there were a great migrati r)wfr)k v-.xxon of people toward the Earth’s equator, how would this affect the xwk) rv-)f. rxlength of the day?

Can the diver of Fig. 8–2 i.xbktw/ uva-w,m8 4xj/iv/ t9 do a somersault without having any initial rotation when she leavesv vtw8b-jx/kitm wu,./x4ia / the board?

The moment of inertia of a rotating soldlp9(6b(72 3b ; nyeigoerwvrid disk about an axis through its center of masslgrbyve2 (o driw3n(6eb;p9 7 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 holkt,7.oon26 .l1 p5yqf d/2ns d3pafexx2u l mfding a 2-kg mass in each outstretched hand. If yd2pqn51l6 muaskp .ff d,/o7yex2fnx.2l to3 ou suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical arf0zaq 7:/yfx5 qziloz)k h9 1nd have the same mass. However, one is hollow and the other is solid. Describe aq 1 l:ho k5fyf9/z zax0i7)qrzn experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle pointuo4dy /ks7mo y :75plss west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceos7ky :m4up so dly/57leration 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 the edge of a large freely roj/ r1 e-nq:uyhp nrvr0v ,9lq8tating turntable. What happens if you h /-8:vrn 9j1,peq ulvqr0rynwalk toward the center?

A shortstop may leap into the air to catch a ball and throw it qy*9lghw8ch.f/r 3eu8of z u5l1jqsa3uickly. 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 directi 5zja8rh .w1oy/9f guqhf3e38 lucls* on (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum,c nt(7q-6cot3 2l ggdm discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter slcg3qmd 6o (g27tn-t ctable.

  Express the following angles in rlsbrx v:ir-34le; c 6badians: (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 coinciden qe c(8th(i9vcx4dd6nce. Calculate, using the information inside the Front Cover, the angular diameters (id6(e9dqxhcvt c48 n(in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at thol.u9 fhvh9m+cby ,t y8*f98p-y spjue Moon, 380,000 km from Earth. The beam diverges at an ang89. f*fpc9 sph9y ym8j+hu,lt uo -vyble $\theta$ (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 o(jb73h vb2f3 ; 3qtxg(vqmywy f 6500 rpm. When the motor is turned off during operatx3y37vg;fyj t(( b3bq2qwhv mion, the blades 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 floor 3.5 m to another child. If the ball makes ihw hp0 +81d n2+sz(fnh9wfon 15.0 revolutions, fihhzon(w1+28 phnd fn9 +s0wwhat is its diameter?    m

  A bicycle with tires f(eyfspe8/8et lhiex8b 5scb zj-0 +668 cm in diameter travels 8.0 km. How many revolutions do the wheels make? -5eyj sz8 x0(8l8s +i 6/efptcbbeehf    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calc,;7wq sc)fc ltulate its angular velocity istc w)qf 7c;,ln $rad/s$ $\omega$ =    $rad/sec$
(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 (Fiz20dlq1)q slk gy1o54wtm d2og. 8–38). (a) What is the linear speeds2g zwlyqt d4o2m1q50l1 )odk 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 the13nj eyzl x-m8 2*koqr 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 poi;s4s f*9 0hletqpjtg;l bqquxd61 -k4nt
(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 nptu 87 a dc*e:bl+2pha particle 7.0 cm from the axis of rotation is to experience an acceleration of 100,000 8: +tl*a ned2p7buhc p$g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its y o6ad.im;i +icenter from 130 rpm to 280 rpm in 4.0 s. Determinia +.idom 6yi;e
(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 t k07yuc v/t*y6dua7 3p geem,$R_1$ 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 pup ll 2(r gwrnmrzmhz :jzi1 :9z7k1y::t 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 interv h91:kj:(lwl7ri mm1r: npzz gz2:yz ral. 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. Throug: g 4+w3vffsdzwj, s-)ah 2ayih how many revos+y 4wiajs)zdvg fw 2-, ha3f:lutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 :3zb p(ngbb1 m.z,xxt6vr.zn 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 stash; bb 9tpr/1resses of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reacb/rt9; hba p1shing 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 uniformlfy3+3c 9h sdod oq0nu*+xk.u o/qxd0ry from 240 rpm to 360 rpm in 6.5 s. How far will r+0d.qu oq+ do3xnydchx/0usk39 o*f a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when57h0ttol*0 oivmgbk6 z(xb m* it is running at 850rev/min It turns 1500 revolut0zmm*(hix bo v7tl5*bt 6o0kgions before it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed uniforp,c2t,*jol.:bk/a;l pf 7+n ol qlbsy mly from 95km/h to 45kly;lbq, lof2,o lk nbppsc:*at 7+j/.m/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 revolution/2i)f:vbd 3yyzuq l j-s as the car reduces its speed uniformly from 95km)q:ufzj -/d vy l3yib2/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

  A 55-kg person riding a bike puts all her weight onpxw q, 33 ym1)l+mkx;er pgn(f each pedal when climbing a hill. The pedals rota mfn+(,xp1l3g)xp y wqm;er3k te 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. Wph qb cld830 pwerx. 9wb;.)rshat is the magnitude of the torque if the forl9 0 q)pwh3r.rxc spbw.8;bed ce 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 thee/a6 lay*v+ug axle of the wheel shown in Fig. 8–39. Assume that a friction torque of e*/ a 6vua+lgy0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attacnt-6x u6f( mxlzb9mz5*v7m ubhed to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then r l uvm5m6btz6*-um(n9z b 7fxxeleased. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening to a torqulmo:)t b6ch4 icc7 :qo ,vb8xue of 38 b46blm 7o:oq)vc:,ht8c c i xu$m \cdot N$ 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 1lj6:rr3o(87n 8q )rsuxc mbf u0.8-kg sphere of radius 0.648 m when the a6jfx(lour8 mb:8 r37 q crns)uxis of rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 6d/;nvmyvx0 x)6.7 cm in diameter. The rim and tire have a combined m d0;vn)yv/ xmxass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball oneg qzk5v+2ri0oaw5d4 the end of a thin, light rod is rotated in a horizontal circle of radius 1.2 m. Calculatr izao e4gd +502vkwq5e
(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 wheec ,70g;6wm bc8xfiu 2 bdolk(v3 ffy4)+toichl rotating at constant angular speed (Fig. 8–42). The friction force betweffm +47v3 o6ch ,(u )0ctlfkbiw2cbigd ; yxo8en 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 gq9tv 5wp2*q xof point objects shown in Fig. 8–43 aboutx5q *gpwvt9q 2
(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 consistsp2*( f h,2qozqk;ttua dph)1t1 qb f9q of two oxygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spis;s.:tryk/k m nning 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 if the yk.:tsm;k/sr satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a unifoq-x+it1r 3q-q sfp)airm cylinder with a radius of 8.50 cm and a mass of 0.580 kgfqi-p stqra-31qx ) i+. 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 4g4r, neov.bgbat, accelerating 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 s.b p,4kdy j7symall 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, neglecti y74pkysj,.db ng frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is eventually im ntrm:6n3fbp zhixwo73+i 8;, (sgfbrought uniformly to rest by a frictionaliw :n sxmb;o z n8+fp73m,ihgtif r36( 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-kf 7 p)m +plyjdks-a+4yg 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 ba t0q wrg),;j96luuuz tll is thrown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is acce gtj0q6t uruzl9uw),;lerated 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 ror chts9y5+ 5h(jkt:j id, as shown in Fig. 8–4t (hcjh+ s9yr55ti:jk6.
(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 masseb1hqu-l 1 ua8tp6th d;s, $m_1$ 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-.6gvwaowg07bah r9tr ; tnt( n+sux+ hammer from rest within four full turns (revolutions) ana 7na+rs -0 but6xvgttgh w(9n;o +.wrd releases it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has:d8,wf7j o9xw hv6w ce a moment of inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque of 280psmfh8ydy69) ijb 9vs:u )tr1 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kgpn m 8yadf*evmvq7+bu* *f)x- and radius 9.0 cm rolls without slipping down a lane at 3f8 x-ynpb7vd+vmq *auem* f)*.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic ener ixnediz;* cr9-0yu0 zgy of the Earth with respect to the Sun as the sum of twozr izn u;yc09 die0x*- terms,
(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 164a/bnl 0ph09icp6*v g k0 kg and a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate6na 00 c9ip h/gbvlpk* of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 zsxd)s+/-u4lii x u3tcm and mass 1.80 kg starts from rest and rolls without slipping down a 30.03i ti4dx ss/+l -uzxu) $^{\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 i8sf)xj na(9g h1(qotzs balanced vertically on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the grf (8zqas9h)n jg(1otxound? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end of a 5j;fa4 l(zd4wongff4fy h-; mthin string in a circle of radius 1.10 m at an angularnz4fff4a;-l d4 5w(gfyjmoh ; speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentumsh/mpsi4 l:7/ ayig2i of a 2.8-kg uniform cylindrical grinding wheel of r4:l ma7 ps/hy2isi/giadius 18 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rottam*t1a r*bn ;ating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, m*atbn;ar*1 t the speed 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 Fig. 8–29) gadi17 .nuu-xcan reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she maad7uin -1.gx ukes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin pm)rf 3(4ouzr8c0b16pzz4ssij r-pt rotation rate from an initial rate of 1.0 r- 0b( 34 cju8pmis)zo4zp1fzpr s6t rrev every 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 r usfa/oc+l3j 8otating around a vertical axis through its center at a frequency of 1.5rev/s The wh8fluco + sa3j/eel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular cbfkn)lj f 4o-y1+ v/s1 cu;hxmomentum of a figure skater 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 momn f 7+y-*dlwt+cxlgexas,fb/l0 xr)4 s0, urqentum 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 ofjim qgx -ntiv2;yn3u 18 ew8) neyf0y. inertia I is dropped onto an identical disk rotatinge18j w2gy)te ixiyn;yu n8-v f0n.3qm at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4 y2w/k( :kyszb$rev/s$ 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 -e hmnhu-qi jy* :k: 4w3+nirecenter of a rotating merry-go-round platform of radius 3w-i: r m3h*qu- yninje+4eh:k.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating frg ian)1r+/ucjeely with an angular velocity of 0nac) gu+/i jr1.8 $rad/s$ 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 l q:yo5- ,vdzq 40mosu52h9 ycg sbh1dinto a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass cardvu,1q5odm5b9s h ql-: y2y 4gcz 0sohries 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 eygp78*/bm h x7mxk0uo*y mguh*.mh0j92 hxubxcess 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 (qtiz 5a.z 48, ukmyh2jecon.$ 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 atecl05no o+f s* rest, that can rotate freely without friction. The moment of inertia of the person pf5 o0 *secl+onlus 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-god7g s ng7y 5e;tb*tn9u8ug /s3ij d3xb-round turntable that is mounted t3 uyd;g gbsesnu 37i5/xj7t98*nd gbon 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 lmwcpyebmcr0:*3u -5 l ying on the top edge of the spool. A person grabs the end oflr5c b0e3ypucm * w:-m 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 move?

  The Moon orbits the Earth such that the same side always f0p*7s* hv 7tj,lbukioejlw;e-+de: m67g lbbaces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) tob7-il * lespkjg+u,wt7lm6do 0 hebj7b:ve*; its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from reptyye5; kurf2c(af 2hs. 0 xp/; ie8kost at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder/i fn5)uc3p d v+yehr(3)fsp y of radius 0.20 m acquires a rotational rfhd3n3c)s+ry e p(5yipv/ f) uate of from 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 cylinfx(ah /j.,d*i.epenfj:vaqe;zi3s ;j (ev8kdrical 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 is released from rest. ieedk8v j*;xhjfzfjnees v3q,p/(..a; ( ai:   $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular u cmbsc:+/h5k speed 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 t.e*q76re + z e9hqu 8mfr8cstfimes as great, were rotating at a speed of 8f.te7*re 6esf +qu qm89zrh c1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a 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 automobikwii4 n/xhr2 :g7b 74ucvh/uqle is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total hihqrw 4g 2:x4/bv 77k/u cniumass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrat -x4(tqh bg 0xgfu p+xcsq/0e:ihw2s2es an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizonw;szvr8 o/l6ltal 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 pw;pvk q/)4 1cpxjy q/wivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rowv/;)qqpyc /pk 1jx4wd, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is s; 3zmg/f dm6nr)pf oq1tanding vertically on the floor, and we want to exert a horizontal force F at it/p g3n;qrm z6) f1mfods 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 sp iy, s2+thwhv/ r7p1mjeed v=4.2m/s on a flat road is making a turn with a radius The wit2 7sm hhp+r1y/j,v 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 0.5bcy5vd + fh62wnr65 pg0-kg rock 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 after 5.05p6wnc 5hv 26r +yfbdg s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater8thm(zgv.8 gx’s body as a solid cylinder and her arms as thin rods, making reasonable estimates for the dimens ggth(mx zv88.ions. 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 ass95xy jja6gok/embly which consists of two cylindrical plates, of ma6 59kj/ yoagjxss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 track of Figa+)55locri8 9f *z odrs +egbp. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving the trac)5r+ldzi cf *pa re 58+gobo9sk? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, buto9f,eoq,kx (m do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uniformly fs40- ; kivwunarom 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What wak-4sw n0 ui;vas 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