A bicycle odometer (which measures distance traveled) is attached near thza+*h 2 w ..tyeies4v*pzxom.)s1 epje wheel hub and is designed for 27-inch wheels.se wox *). *+ze4p mt. va.zs1eyhpj2i What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angulag2bgxfb h j.,dwwa,5/ r velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radh,x 2/fwbgj.ag d5b ,wial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described br085jlm s iw)ak9 cpm qk,vp;,y a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Expla( yr cv 1u2ganfhkt;2b1 *ge(ain.

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 difficul1kk7qg *5 -h1 tcebefzt to do a sit-up with your hands behind your head than when your arms are stretched out in front of you? A diagram may help you to answer eh te1zq kc7g1f-*5 kbthis.

A 21-speed bicycle has seven sprockets at the rear wheel and three at thh2j qm. 2zuxu90kmr(ue pedal cranks. Izmhjru. 9x um q22(u0kn 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 sprocket? Why?

Mammals that depend on fer5e vs7:i n4n.e ok;being able to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34).nf n.s;o7i rv5ek4: ee On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long:r n1tkx tt8h )me-uu:, narrow beam?

If the net force on a system is zero, is the net torque also zero? If p-eln 3tw4r +hm2v4qothe net torque on a systemt m r+q3w2nvho le4p4- is zero, is the net force zero?

Two inclines have the5llkxqfk6 9slm.m0 q 8:,gyzb same 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 ball at the botto0qq.6b5gl,lm kmsl 9kf :8x yzm be greater? Explain.

Two solid spheres simultaneously start rolling (from rest),bsk,6kqjj g++ ,u abq down 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 there? Which has the greater total kinetic enej gj6a,q ksk+u,+bq ,brgy at the bottom?

A sphere and a cylinder have the same radius and the sa59likrghp, eo wni6.,me mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bowl.69, koi5pi rhe, ngttom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most moving./vwpxb ds lwxf9r, +h(*3ag u or rotating objects eventually slowulhbasp39*w rg.(,vf / xd+xw down and stop. Explain.

If there were a great migration of people towa9qm*0ui f4*y 9 7 )besjjthdf c5ffn5frd the Earth’s equator, how would this affect the length of the da nd7tmf9 *f0j j5*fes)y f49bqhfcui5y?

Can the diver of Fig. 8–29 do a somersault without hfh 5*1kbrfgadp q 32kah9tzn2it fkj 8tq)68:aving any initial rotation when she leava3n6tpr2a fh8gi zfttjbqq9 d* f8kk1 )2:5kh es the board?

The moment of inertia of a 1nz0gcc t/mr;6b-z+fb mf 5 sdrotating solid disk about an axis through its center o mzcc0tg/+1bdr6nbf5 ;-mfs z f 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 a s9mg)gi 6pw: o1x (l4r+tgrqin each outstretched hand. If you suddenly drop the masses, will wq:omgs p9(6)lirg4 +xagt 1 ryour angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same4k72d g6iaka x eqtk i9u,g3w* mass. However, one is hollow and the other is solid. Describe an experiment to detea 6iku 7i4e *9a,2dq3ktggxkwrmine which is which.

The angular velocity of a wheel rotating on a hordto lo *j7c0g-izontal axle points west. In what direction is the linear velocity of a point on thel*dt 7jc0oog - 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 tubd-b 0bzp x2 h,50c:eo 8apsfhe edge of a large freely rotating turntable. What happens if yx2zhue bc:pb p-bdso5 80,fa0ou walk toward the center?

A shortstop may leap into the air to catch a ball and throw it qut/eixn0q; bx:ickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that hi;x:bqi extn0 /s hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of consx1ve(/ q9dr23q3 d18t b iml4akvf wshervation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propellkdx39hav4iem / qbq8fv1(w rd 1ls2t3er can operate to keep the helicopter stable.

  Express the following angles in radians: (a) i0.7a,hrme: 1h0 q fppq54gui5t deke30 $^{\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 coincijj,kd2( 1 4 .hpvgp+ i gtin55gmmbvt1dence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and thetn1i.1itkd vb4 2g5+5jmj mpv(pgh , g Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km ftnxlk,ekib8* dkoi)s 29l qxf8v/ ,7 hrom Earth. The beam diverges at an anglelns,iov* xt8 f/97ikeh)k8,kx b d2lq $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. When the motor is turned2 l js9*rqbax8bpd1e4 off during operation, the blades slow to rest in 3.0 s. What is the 9j er*d b p18sqxalb42angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3. f//1bqn ,aur qb)5yll5 m to another child. If the ball makes 15.0 revolutions, what is its diamete,)l/ 1uq/ ayl5bbqnr fr?    m

  A bicycle with tires 68 cm in diameteuh7 .x fsca)cjp zo7n2 +x9g-ir travels 8.0 km. How many revolutions do the whe+)as- pj7czxu.ix7gc2 oh n9fels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter6*cs f.5hfm3qs tm d*v)oc1i c4wb 4dn yuwh/9 rotates at 2500 rpm. Calculate its angular msccyh)9sh4 d63*. wocfvmq4dt* wb/ f 5inu1velocity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.0 )tgzsc:nvt(b9a +-5 n 9gpg2 is;clcz s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m 5ccbnsag sz-cvg 9zn( :pglt 9 +;it)2from 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) ir g*c149k ;xphu73d2zosfvf un its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a point . (5fo nrbj9h38nbm4tg o3tb i
(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 h joeqg:buy.7g )/ii/ a particle 7.0 cm from the axis of rotation is to experie:eiq7 jhi o/./b)uy ggnce an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 -0u i-bzdyc+dritm- 0 s ;gl5orpm to 280 rpm in 4.idg+--m-io;s 0r0db5uczl y t 0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radius -imw ((x5nqfr 3y8xi x$R_1$ 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 theo lyq;klg+ e3)cg1 u5y 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 thougy;uyc 5qe1 o3ggkll+ )ht 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 ;q 0o2z7kuenld zh;;m,000 rpm in 220 s. Through how many revolutions did it turn in this7;okuz2dz 0n h;m;ql e time?    $rev$

  An automobile engine slows down from 4500 rpmqxi8+vfzxs;yc4w(j/sv(e ,j 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 highspee) .(evz7z dhngm60oqrkoq) t* d jets in a whirling “human centrifuge,” which r7keg* 6t o qqm)o)0.v zdzhn(takes 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 acceleeqzmhr2 weku; 92w 7;+ntmq2 srates uniformly from 240 rpm to 360 rpm in 6.5 s. How fareqm2+z t wrn7sh2qem9;k;wu2 will a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is ruiivxji s31 -1nnning at 850rev/min It turns 1500 revolutions before it ii-xn11vs3 ij 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 r2(-2w o 2nc09dglzzmiei(qd pevolutions as the car reduces its speed uniformly from 95km/h to 45km/h The t (2ml 2w- 29(e0ii qogdcnpdzzires 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 cfo( e87/lo07)q irr hk 6wf4fxbi-qsbar reduces its speed uniformly from 95km/h to 45km/h The tires ho8q k-f/i)f rlre0swi (h7qfo4b 67xb ave 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 on each pedal when climbingasi ; zcc 146uacs1t)t a hill. The pedals rotate in a circle of radius 17 z);1i1us64tsa c ccat 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 the maghax n bn1b5f2.nitude of the torque if the force is e21f hbab n.nx5xerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fig. 8–39. Assume v/86g) rc.h xtr 7 t9ju/ko t4rp+pmuathat uo .tr k)tha9p8/xt g /umv+7rcpr6j4 a friction torque 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 ens tnmsuxkwh(5 v1;n).b;ku x)ds of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and nkmxk1u5;svb hn)s .xuw ;)t (then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine reql,- kr3p tn-r3ds qc)guire tightening to a torque of 38 dc pgr, -33)qnkrtl- s$m \cdot N$ 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 t1 isfas2ilii 33p* uuz/ 1o:qdhe axis of rotation1ia1z/u3odus3 pfqi li2*:i s is through its center.    $kg \cdot m^2$

  Calculate the moment of inertiwp.ca zho fz1,z*c 287j **ej na3wnoza of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combin* ec1*.,co2 n7 a8jz*wf azzzpj3howned mass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a th0b86 ve2 d0itp;pa mflin, light rod is rotated in a horizontal circ8li ; 0em0tdp26 favbple 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 bowz0d0(ox8 j vzk+ea cf8l on a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and the clay is 1.5 N total. df0 oz vckxze88j0( a+
(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 thhz9az2o7 w3ukx+ ulg+: kh/*f9oi +pu yiz6 vbe array of point objects shown in Fig. 8–43 aboutoz3/iauo: i9ul2*z kxhuk h7 fw+byz69gv p++
(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 ohwm s0n z(ocat53.8zv f 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 gypyd/uwvs) av-ke4 w6v*c24sq x* xqn2- hd :cylindrical satellite spinning at the correct rate, engineers fire four tangential rockets v26v e -wqd24dnx*c gs:4x /syupa)qy*vkh-was 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 radius f*k2gknh zi++ n m/*d7bborgwl5tf :4of 8.50 cm and a mass of 0.580 kgkfho *b7:4gf +r blngnm*td kwz /i2+5. 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, accelerating it from3rh5 ktc.2+h uvcc7 un 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 hax2:wvniq .rgtsr6 +j5jr 0*jfnd-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 opp+rrqj fv.*t:2x j6j5wrgn s 0iosite 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 .m17,0dj(l dh jgj g;vrwguxs .prz,7shut off and is eventually brought uniforml vj71g,drw.0.u7rjsl gh(gx; pjmz,d y to rest by a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accer-vuhus n/2o +lerates a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kgube-a,7nvq 7n 83o le:ppt*ad ball is thrown solely by the action of the forearm, which rotates about the elbow jae utnpa3 -b novd*,8 7q:pl7eoint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, as shown in Fig. 8f (d5i-) (dd 2ujnifybkdhglz87p;t*–4;)kfdpf n* d y7i duhi8-((5zgtjd2lb6.
(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 mas::mda/c ajorulcwy ,aqd6e:,d5 c4/f ses, $m_1$ 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(sc2k5v 9-hwj kubon( from rest within four full turns (revolutions) and releases it at a speed of5jw9s k(boh u2cv-(nk 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 mom(88*p(ns brvviadk h2ent 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 develo .pn.bmen6nunb .i)h8 ps 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 wie)q,t+hgcki 7 thout slipping down a lane at+7g )hetc,kiq 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of th0w)k a0w;cvpa7 b pyd7/oi-b ++sfqfce Earth with respect to the Sun as the sum of two termsqds v0yip o0/-fbk)7ab ++awf ;p7cwc,
(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 mask3vgt)a+ pkl 5;+qzx:bgr f iv :o87gej1i p(cs 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 rc3l7i5 1: (px+)bvao gir:tkv f8gkjqpze;+ gevolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts f/ f pno(j n,itrhij670rom rest and rolls without slippingj fn/t(o607i ,njirph down a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

  Two masses, $m_1$ = 18 kg and $m_2$ = 26.5 kg are connected by a rope that hangs over a pulley (as in Fig. 8–47). The pulley is a uniform cylinder of radius 0.260 m and mass 7.50 kg. Initially, is on the ground and $m_2$ rests 3.00 m above the ground. If the system is now released, use conservation of energy to determine the speed of $m_2$ just before it strikes the ground. Assume the pulley is frictionless.    $m/s$

  A 2.30-m-long pole is balanced vertically on its tip. It starts to f3r i;h8.wrgp wall and its lower end does not slip. What will be the speew.3r p ;8whrigd of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotat+7ony+ jf; 3gc tnw2zting on the end of a thin string in a circleywgnt c++ 3 tnzo7j;2f of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg unifjds)s . d 3f tpf,e(, l5lcce3fgwg75rorm cylindrical grinding wheel rd lg55p)el fcf (c,d.7 fsg3twe, js3of radius 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 rotati-t dbj1cl-ezp vzj0. *ng at a rate of 1.3rev/s If he raises his arms to a horizon j* --vce0 zjlpztb1d.tal 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 the one shown in Fig. 8–29). 9/gv vz0enji can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the v vgi/9z jn.e0tuck 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 fi/u/t g : m*5zcse,rqjrom an initial rate of 1.0 rev ev5*rjqm ,zst/u e:gci /ery 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 rotating around a verzj5b/831 pdgpg;hetq oo 8df,tical axis through its center at a frequency of 1.5rev/s The wheel can be 5fed bhzp;88gd,o/tqog 1pj 3 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 angulaf5b.a 2tcvkkd utey+;0r q qx6o+8 l50t*lakjr momentum 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 momentum of the Earth+35+o,pd uffupjy em-
(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 mom .m3(lo r td5g4dohec.ent of inertia I is dropped onto an identical disk rodr5lh(.m cgdt o3o 4e.tating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at x; kty::nmt0 jat0zv+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 merry-go-round pr ,.,9rqegzgm latform of radius 3.0 rqe ,rmg.zg,9m 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 fk:ynbewp6 p d:kny3 973y iyz+h;;zk oreely with an angular velocity of 0;eoinby dy3p:w+npk yyhz6zkk:97 3;.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 into a white dwarf, losing about half its mlgzpuz /,-z9(lx .g,ya+lj pwass 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 th- g+uzpzl, zlw a,ply.(g/j9 xe 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 excess of 120 e:s5 p+cf 3 s vze.(,1t sth gb6fqleek88whc6$km/h$ 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 3lijr8e7ag7qcc q*a0$ 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, initiallyhr 4lb :i+gq* dgzwl/ 0a,vnd) at rest, that can rotate freely without frictvg,:w0/ld)r* ldz ab+n4qghiion. 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 diameter )u ag,-v :gsc53p2z mu9vhqsd merry-go-round turntable that is mounted on frictsvmgda -p9u:qsuvch32g , 5 z)ionless 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 gr x:sm1gvl59pxa /:, dci/zy bjound with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. Themp: adzx c//l,y9iv 1g xj:sb5 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 gcz3j8 u;z yz +wha1i;such that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angulaj au8iwh cz1; +y3g;zzr momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a ratly;quzem.: b og 15qy.e 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 cylinderoac/oyo r+(3vm x0.x yuoip,;:b ooc3 of radius 0.20 m acquires a rotational rate oomxv. cuy+cx oyro/;3:0( aboo ,ipo3f 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 cylindrical disks, each .:o dqs.r.y 0;oe,6dv izyeomof mass 0.050 kg and diameter 0.075 m, joined byd. oodrq yzsyme 6 ,0:v;.eo.i 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.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speedepq:xuj t (/fdzevn*5zw 2ue(o8s3 q9 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 mach e++2cavfvu-v 93xq ss 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo2c-vc +f3euav qx v+9h 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 automobile iqhkl. p0.g, w6bigbv, s for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 bp,kh.60lb, wi.gvg qkg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates a)tsyv40x . ;c:socerufvn7k1 n $H_2O$ 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 rol c qt9ssof9;8fzp4)a iling 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 M5bz0fjd8 f/mqw d*w0rznid(z+ e -/q u and length L can pivot 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 accelerationi05djz- nfw/u(zw+b erz8*/f mqq0 dd 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 flozl8ie7 f;r5gv 6uiam(v f )k8hor, and we want to exert a horizontal force F at its axle so that it will climb a step zkrgfia56 ;iefm78v(u8vh l) against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with spey 0 b:8u;5zeaqgdyrn-d f1 t3red v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist and cycle are r ;8d30e5r 1zq-dab yutgyfn: 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 rocyag0n,lw(mvj2t fn7 9 k into a sling of length 1.5 m and begins whirling the rock in a negyam2tfln j7w9 n 0v,(arly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s-:tego -uwk s84 yn6unva29az body as a solid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for n:9t-gnek4z68ua2-vwoy asu a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cyz cf/lrxlr*khh6,0h.z 1teuh p0 j5k9lindrical plates, of ckrfz0u ph6/hlt , *5.9hl1j zhr0e xkmass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 jxtk+sg p;tqy. c4kob-. 3 yr0track of Fig. 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 wi;okqc t-jg+4 b0 kstpy..x ry3thout leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do ; n,9f 3rg dm.rvgi/vxnot assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uniforipi.gp4;fn p1 mly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the ang1 if;p.n4p ipgular 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