A bicycle odometer (which measures distance traveled) is attached pqfpp*8 vmv x 1-uvp6q,j,q;znear the wheel hub and is designed for 27-inch wheels. What ha,q,18fpq pp-xuv*p 6jvqmv ;zppens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on the riy;t t g3 02iy ;xqzon3xujqgn4/t+.vqnb;pd /m have radial and/or tangential acceleration? If the x+ qdz n;4/tn 3y3q uov/g;nbqp0y ijx.tg2;tdisk’s angular 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 describv56t q7: i 7bukvau:jxed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert hu :zguo 6)f vyh 0i4x0*vgsmn. x8ko5a greater torque than a larger force? Explain.

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

Why is it more difficult to do a sit-up with your hands behind your head than. anb,t3( gedw when your arms are stretched out in front of you? A diagram may help you to an (,tg bd.a3wenswer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the pedaav(r kzr/ 7no0 vhu6p4l 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 large front sproc4a h6r z/uv0or7pvn(kket? Why?

Mammals that depend on being able to run fast have slender lower legsp2txa pdd)6n ,wa33oo8xiyw- with flesh and muscle concentrated ,oa d32dwx8iaywnp6-tp )3 xo 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. ;zf ec0hb ./elc vwc5e .ej8w:8–35) carry a long, narrow beam?

If the net force on a system is zero, is tc0 wjz5l a/h(rw:wn( phe net torque also zero? If the net torque on a system is zero, is the net forcep:(ww0j5rzc nhl(w a / zero?

Two inclines have the same7o qtz( /ons r+8d8xgh height but make different angles with the horizontal. The same steel ball is rr/8 q+g (zxodsot8 nh7olled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down anl xxv 4 wc)eik42j-1en 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 greatx 4cwje e niv)lx2-4k1er speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mlt 92mmx f0a(5 -lpptfass. They start from rest at the p x -0 tmffll(mt5pa29top 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 the greater rotational KE?

We claim that momentum and angularaemy+ 0 y ad:13/krjlwxe)l9p momentum are conserved. Yet most moving or rotatingy +3e )l/a pmjek1 :w9arydxl0 objects eventually slow down and stop. Explain.

If there were a great migration of people toward therj: ,z2 ntixgp2bq5n, Earth’s equator, how would this afr :tn,qpi2,n j2zgx5b fect the length of the day?

Can the diver of Fig. 8–29 do a somersault wisn: fy29.gr wpthout having any initial rotation when she leaves the : .2ywnrgf 9spboard?

The moment of inertia of a rotating solid dipqeze-(k7r;: dfkgn):jbj j, sk about an axis through its center ofkp:dj jek,;b-q (7r gfzjen) : mass is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2-kg mass in each outsjtu6 l9au.62aq rhy 3i yk;tq)tretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, orr;92i 6a)u qya 6hk.lqt3tuj y stay the same? Explain.

Two spheres look identical and have the same mass. However, one is hollow and t 3lt6jca y55qihe other is solid. Describe an experiment to determine which iiq y35 lt5j6acs which.

The angular velocity of a wheel rotating on a t7sol*- k s8e.6v1,si+ttdz hbvfwd. horizontal axle points west. In what dirf. sd1v *kl dt8 -wsoeihtbz,t7.6+s vection 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 at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating turntable. What ,1 4pizzs,k:7wk.h yr:/lepjsqe jm9 happenr ,94/sp. swypmjze q7l:jzkh:k i1,es if you walk toward the center?

A shortstop may leap into the air to catch a ball and throw it quickly. Ax )j341qs;4e.)h e tcnoshh iys 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).14shen ).hc4)j; hio3tq ysex Explain.

On the basis of the law of conservation of angular momentum,dbzw s*ad02y 4eb)yx. qd ,9pe discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep th4aeb dwx,pbe0)dq.szyd9y 2 *e helicopter stable.

  Express the following angles in radians: (a2r sr ljvs ye7*:7mu)e) 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 Eart ::)nuiplg- n i95vp/q)wz*4fm s limph because of an amazing coincidence. Calculate, using the inmp-pqnfnv*zi i9lls5giwu:: )pm/4 )formation inside the Front Cover, the angular 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 km from Earth. The beam d+x*h;zqaf *-g y2rk bv gc0lf;iverges a ffgg;bh vry; zx2l-q*ka+*c0 t an angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate 5b )b5foz:i9e, lp+t *78pwkpzmyg haat a rate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest i5 +tpwhypab9*5bi p,) 8m 7lgekozzf :n 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 ano)obkchipdn i8f z 1 xd:(2yybr.z 8*,ather child. If the ball makes 15.01d)ibp hcyo: i. kf,8x 8yd(*baz2zr n revolutions, what is its diameter?    m

  A bicycle with tires w0/ohs * iop3y9ya0p ka;ud:i68 cm in diameter travels 8.0 km. How many revolutions do the wheels my;ookyi0p as3*d9 wu0/pa ih :ake?    $rev$

  (a) A grinding wheel 0.35 m in diamet.le 1 cxsw/)m5 lu;edher rotates at 2500 rpm. Calculate its angular veloc/le 5 esml1dxuh.);wcity 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 revoluto3.py6x q:aw00jzpz9dzn k ky 9 ;ivf6ion in 4.0 s (Fig. 8–38). (a) What is the linear speed of a cy 0p9zjp dio v63x;wa0zy6knk.:zfq 9 hild 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 Su(ifw;v)q2 ntu0*oczb f) j4wp n    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a p5aanv:usdk)+d+pd f/oint
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a parti n-m0gena4onyv;r. ;qcle 7.0 cm from the axis of rotation is to experience an acceleratmn aq0.oe -v ;4ngr;ynion of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from . bpw.(q( 6aol9usvuq130 rpm to 280 rpm6. 9pvsq.a(oqlu wbu( 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 radiumfo :d a3vb:097-dxedu; feevcs,9o y s $R_1$ 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, astronvo/idzx9olf+ m/u .t( auts aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelera/.i+ldzutm (xof/ov9 ted 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 fromm,5hlsvf8r3 t+i pkw2 rest to 15,000 rpm in 220 s. Through how many revopt2fws 5 ih+v m8rk3,llutions did it turn in this time?    $rev$

  An automobile engine slows down p)q,;6ozf uaj 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 v 2wm7 0yej7dyz f.rwsr;/jz; flying highspeed jets in a whirling “human centrifuge,” which takye rdzwz7m/jwr.27sj;fv 0 ;yes 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm t e6)d8chb-ntpkwp- 4n6 :ayxa o 360 rpm in 6.5 s. How far will a point on the edge of the wheel have - pb)dk 86-wa:eaxyh6nn pt4c traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turn myl0/ z6sj5 ak(nj,o31ydixo s 1500 revolutions bemn 6i0 z 15o/l3kyajs,(ox jydfore 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 revolutq0p,rqnzob.e79 an*c z ec+s 7ions as the car reduces its speed uniformly from 95km/h t*+.s07zneo,nqp z7rqcc e9 a bo 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces itm62td; w1 5ku9ybll xhs speed uniformly from 95km/h to 45km/h The tires haveh9xytm5bl ulwd;1 k26 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 1rb:*5e d 2iv b0nfeprb, 9vqtbike puts all her weight on each pedal when climbing a hill. The pedals rotate in a ci bd:1frb5p9ve tre*2iq,nbv0 rcle 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. What is the2ss1vbpl7+ mt magnitud2m+bp1svs lt7 e of the torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel eqdrsl5 5dag ,2i)uh)shown in Fig. 8–39. Assume that a friction t5 ),s d)druh2ia5glqe orque of 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 endk, ri2 a:f 13ovb 8tad4wl(qons of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this systeodqf ko13r28ntwl,v a i:(b4am.

  The bolts on the cylinder head of an engine require tightenint7g;wylkazl9 9;tw :kaku4 :bg to a torque of 38ak:u l:wt bza4kgw9k9;t;7 l y $m \cdot N$ 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.3px5lt ,q2 kqyxad ;g axis of rotatd xqtlyk,3.5agq 2 p;xion is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia4s:ht.zr oj v0 of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The majzsr 04v. h:otss of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end xrbx nyk8uks +hcna79, * i2y4of a thin, light rod is rotated in a horizontal hn7,*x ki+c9unayr x8k4sy 2bcircle of radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel ovwubs pm(d:5k kgy()x+l gv2agw z4(v ;2b 4vrotating at constant angular speed (Fig. 8–42). The friction force between her hanpvk (by (4gvmg k4go;wzl2( )+xw 2bdv 5s:vauds 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 z*z4 n+a :to/n;qjnxmbgyo+. of inertia of the array of point objects shown in Fig. 8–43 abo +.o n:mnby jz*zo4+gtqxn/;a ut
(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 t r8 lw(uh2gf1r(n: xbnwo 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 spinning at the correct rate, e)gvhhor/(j , ongineers 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 revgrohoj(h, / )quired 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 radius y0 rzkys)pwg5* ps3q;of 8.50 cm and a mass of 0.580 kg. Cal p*ypyw s3)qkr ;5s0zgculate
(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 ba rh6l9sr9m d-l 4ieu0yy4f ,yat, 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 small hanw 9gz wo-x u 4suh9.ujlf8d-k6d-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 childr9sl.wzuj f 4xugou-86wkd-9 hen (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is eventually bte) ggr dz9;ovnb;+9x rought uniformly to+gr 9xo n;;e 9gv)zbdt 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 accelerates a 3.6-kg ball a.64jea /qk ptxh0kmjx k6)ku7t 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the action of the forearm, whici rtfi*xolw tct,cq 89,8,3mch rotates about the elbow joi,ow tc8rx*tc9 , 3iqc8l,imtfnt under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod,w(idrr, cv1x,m9zh;jq wt3g 3 as shown in Fig. 8–46.xv,rg3m d 1iww3(;ztq,rj9ch
(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 masse8ka3jk b3vwi:d v 06exs, $m_1$ 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 fro5,p c ,ygozlq,m rest within four full turns (revolutions) and releases it at a,lzo p ,cqgy5, 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 a moment of inertiah +asecbyi:xs;)y0 *rc pok8 1 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 develop0uc*:grkr sbbh*o7/,j hz)y vs a torque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls without slippibp5 )pv,symj 7ng down a lane at 3ys)v7m5bpj, p.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the slew o,va9gm0 6um of two terms,l, wam0 e9vo6g
(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 andg2 ,1ulf kh0qp a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 r, hqulf20 g1pkevolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 swcbhe -;eku6kk37mh mtmu3.c 30ic8cm and mass 1.80 kg starts from rest and rolls without slipping down a 3we- 8 76uksk0he 3k 3 c3iccbmmht.m;u0.0 $^{\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 t g 35kn-abs f+ck2xmg7o 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 gr7c f2ggbx5ak+km s3-nound? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ba, y: z0aaksi ox(2d3jpll rotating on the end of a thin string in a circle of radius 1.10 m at anzyiapjd(320a s:, kxo angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical gripc9cjo99n*oh nding wheel of radius 18 cm when ro phnc9o*oc99 jtating 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 rotating at a rate o5r8o 5z yta)51jedennpcy3c5f 1.3rev/s If he raises his arms to a horizontalyn o55na8p)c ct3 djer55eyz1 position, Fig. 8–48, 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 theu;lsz zw)iu-g bk5f-- one shown 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 tuckk iuswl-zz-; bgf)-5 u position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an f1ar0.l. ysb binitial rate of 1.0 rev every 2.0 s to a final rate of 3 bl r.fa0ysb.1$rev/s$ 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 2d84k b5 a+hv bq*hgazat a frequency of 1.5rev/s The wheel 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 centerh4abqkz+*a gbv2 hd85 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 skater0 bzw*wj:i38 cu 1/fo,keu vkh 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 momentuij/8f6l5dxj1x zdi/g y fua0 x88s4x pm of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is po 9 j)c2mcptbp68x: rdropped onto an identical disk rotating at angular sp 9pmppxo 6:c8jbct2)reed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns l6hioye 2 v +b*qwh, 3v+d:wmvl4y0rsat 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 rotating merr9 q:l/(tzj cdmc82o5b gcm lo9y-go-round platform of radius 3.0 m an8ol/jcgqtl99 d 2oz(b :cmcm5d moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely w(*zlg+x dx2ow/do 4hh5k 4ywbith an angular velocity of 0.h(5 +x/dg wxo2 hkw obzyl*44d8 $rad/s$ 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 aboe0blpf0kdnzb fqu wu+1r-1 1*ut 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 momu1f*p1urzf0 lkwb+q-1n0 dbe entum, 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 windsdw gi)9, h.mry in excess of 120 $km/h$ at the outer edge. Make a crude estimate of
(a) the energy,    $ \times10^{16 }$ J
(b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 $kg \cdot m^2$) of radius 100 km and height 4.0 km.    $ \times10^{20 }$ $kg \cdot m^2$

  An asteroid of mass x, ho9h5b sib0$ 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 wiu/t zn shy9g3+n. imv3thout frictio3iv+n3gh n.z9st y/m un. The moment 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 diameterfrad 8,asp4.379m 4 :axrz pgo1bxly l merry-go-round turntable that is mounted on frictionless bearings and has a moment of inert 4yoxrllrg 8d1p34msa: zab. 7x9fpa,ia 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 on *:q 32d, jgodsg;( ,hddsohp mthe 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 persos( d,hgd32s dp*:q gooh;mj,dn 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 ns 2ry9:xeoa+ zz )k:vsame side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat they+ z):szar2k e xnv:9o Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a raj d0o:7zed,su te 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 unifo jn.ues0ka fe2,p1ulm)wf)s5rm cylinder of radius 0.20 m acquires a ufwm,ssj1eae) p5k .20u) fln rotational rate 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 solid8g;w6 .c2-6m+ik yd hra; ijtbll5h ve5w/kiw cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0del+6vi5gwkmwl 2w/; 5i c8hi6arkjbh;-yt..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 speed avw tkrr n)a7d/ 02hx4of 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 massqeb7b4wu nf.kh 5v2w0c 5mfx( 8.0 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 th2 .q5fxw07bmfcwn4b kv e5hu( e 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 automobile is for it to u8ou ezzhg q-1vq+j11/j 7zdagse energy stored in a heavy rotating flywheel. Suppose such a car has a tot ze1d qzh 7jo 1gg-aj1qu8vz+/al 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 illustratehxh5b 6e-8ya/g s4 fgwfx+5kgs 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 (ho) l:v:yjd5vx bh gyrlpgq-3zp07gh. 4op) is rolling on a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., we ignore fvil7 *cvdnl( /)0b exfriction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizovicxb7n* llf)0e/vd (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 ve7c8 qe blerz+b 9cs0o(c;90c;sukai artically on the floor, and we want to exert a horizontal force F at its ackaqsz9 e7;i+9;acs8c eb rlcbou( 00xle 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=4.2m/s on a flat road is making a turn-qw;poidu4 u q)4pp t1 with a radius The forces acting on the cyclist and ; uup)4pi q woqp-41dtcycle 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 od;j9l:lzir ) oc 5(ktrf length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from r; rdz)9(o: rjillck t5est to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thin rods,hbnky9vk00 7jo7 6ej q making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretchek0k bjv 06oynh ej7q97d arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of b), s0lw uq7c305ji4xir*yt bhxvx -gtwo cylindrical plates, of masb yiit 5 c0gxxvjrlwx,)74s b 03u*hq-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 track of Fig. x5 wbx-+wlu.d8–58. 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 leavinwd x .xu5-w+blg the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do nc f.wb8 u:ihh8fkm1q-ot assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as thx; t8m*ysm3lke car reduces its speed uniformly from 90km/h to 60km/h The tires have a di3mtsm *8 l;kyxameter 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