A bicycle odometer (i6qehv(+ z4pw which measures distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wiq+ep v z6(h4wheels?

Suppose a disk rotates at con) 5:ownna +dcvkzhb8)h, g 3hsstant angular 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 radial and/or tangential acceleration? For which cases would the magnitude of either component of linear accele hd)5akn: sb3 8now)+c hv,zghration change?

Could a nonrigid body be described by a +dmmkin ) /-m9lrn6tu(xhj(zce , 2cv single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a laxowx+xh+-ev t( n7s n/rger 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 w/ q5/dkbnjj if.i1+hh ith your hands behind your head than when your arms are stretched out in front of you? A diagram may help you to answer this// k5hi1 .q+difbnjjh.

A 21-speed bicycle has s/6xfeu)/ 3 vxv ctzbf)j7 icc(even sprockets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harde) b cvxcevf()3 ufi /6ztxc7j/r to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have smlr3b-5by0sm*jz nioc .6t,slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why thiso305lsibn.r szmy,jm* -b6 t c distribution of mass is advantageous.

Why do tightrope walkerscwx54 b0hdi4h (Fig. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the net 8ewd irngvx*ar ,b 223torque also zero? If the net torque o rgd*rnv8ab ie22x, w3n a system is zero, is the net force zero?

Two inclines have the same height but make different angles with t(erlkbr-nr 8 g5svy+ in87 eh-he horizontal. - 7ei8rbk ne+nh5ysr(r 8gl-vThe same steel ball is rolled 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 an incline. 0om926jj v 88xhd 6xuzuo j+txOne sphere has twice the radius and twice the mass of the other. Whi6zdu xvtxm8o 92ox 60+8 jjjhuch reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the samff,5fo;fg4t)b6,mkm ohq g3k e radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the gfb,mkg) 5mffot4 qok6 ; g,fh3reater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most k0jq)k2 c .rg1 blutwgi(k( 5gmoving or rotating objects eventually slow down and w ugckqb.rlk)g2tj15 0(gk(i stop. Explain.

If there were a grea)lw 8nq2o,-t uu.ovvu t migration of people toward the Earth’s equator, how would this affect the length of the).qwt,u vunou2l vo8- day?

Can the diver of Fig. 8–29 do a somersault without having any initial rotation w.*s8 itxs77f)zygmmxhen x*)x7mg sy8 isf7 tzm.she leaves the board?

The moment of inertia of a rotating solid dum697ju c: eoqisk about an axis through its center of jq97ou6uem c: 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 i; iags3cjd1.5+t /tym/iib gpil. j4an each outstretched hand. Igt pigj.14s ijdab;3i lmy c/a./ i5+tf you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look iden/ xuz 477ajyqqvlb/7 utical and have the same mass. However, one is hollow and the otvqy77 //l7jzbx 4uua qher is solid. Describe an experiment to determine which is which.

The angular velocity of a wheeahm,0:mckhx8 l) im(dl rotating on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. )8: mh kacl xim,d(h0mIs the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freelyl+5em,vla :b zibl7l n. yfhg2rn;60 n rotating turntable. What happens if yol5nl2mrln y eg :ahvnb,. 6i0 7zbl;f+u walk toward the center?

A shortstop may leap into the air to catch a ball and 4qpn8g5lv l :nl6qht ,l bzs .j6i*4jpthrow it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly 8:4pjvjt .66pq4gbzh, lil5n*qn ls lyou will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, disco(*yqnb) 4g; g( usfzzw-1d snuss why a helicopter must have more than oned (o 4qg*);usf y1sng nz(zb-w rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles injlmosm 1*ts6./ trwknq2 2ho2 radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Ea ojj q/dbi0 iw;+1tz w.(;voyd;i5lhu rth because of an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Mooyio;5i.d wj0tv;( 1wu/z q;i+jhb l don, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380x4790yj )a wcy qf ,4ej3mscnz,000 km from Earth. The beam diverges at an 9jqm73)4w,s0c yz f ay j4exncangle $\theta$ (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 r33/d)bx4y m)f0ev4uics fk h wate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angula/43xwumby)0si e3 k 4f)fhdvc r 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 -b j*r(iic6kx- m1p /k ,xjogumakes 15.0 revolutions, what k/ p ji,i-x-ckxr gb*juom(16is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolut hk4h 7k )ouz:/3zv1+f aduicsx4gia9ions do the wheel41kzgh 4+/9v cxih u :)3 dsoiz7akuafs make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 r (gc l7bhu2so.+, mqtmb+7q bc 2klek-pm. Calculate its angula+g tqm-bk ehm ,o bk(bc.sq277l2clu+r velocity 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-rxvrkrc 9t3*m;i:4,bgtqz i8 aound makes one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linearzxmt3bck;g :va r8i*9 4qrit, speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) i7: vq9 gm6ftd(bu3r i3)4v*kvaedd ten its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a poii2/yy+iqacsj-l;6c 2 lfw4je sk6o + ont
(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 particlwg2 t(z.c6ga oc z7uj-e 7.0 cm from the axis of-6 uztjcw.gg 2 oc(za7 rotation is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its r qe7evq*8s6g 9 tr;vvr(7cflcenter from 130 rpm to 280 rpm in 4.0 s. Detervlq* trfese6vrv7g9r8;c7(qmine
(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 l9 t)7razia7 6ds0i sw$R_1$ 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 sp 8mrxt dyb6+v(acecraft put themselves into a slow rotation to distribute the Sun’s energv +xdr6 byt(m8y evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought 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 wf x5yi7wue0(: z)yslv5caa5. Through how many revolutions did it turn in this ti fl cvizy55a (7w 5xs)yu:wa0eme?    $rev$

  An automobile engine slows down from 4500 rpm 5yui6e fup/r; 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 highspeed jets in a whirl ;y f7pbb skzrgu8kz 4t53.z/aing “human centrifuge,” which takes 1.0 kk7r;zp/ fazy3tz5g 8ub b. s4min 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 to 360 rpm in 6 4vugw -4m-p993ny pfwg c4thi.5 s. imw4 -39vygwf-p hg4 ctupn94How far will a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/1wdhr3,1v azu ja/k(f6e xk k,min It turns 1500 revolu6uevkjw3(x , ,/aa 1fdkk 1hzrtions 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 uniformly )axv ub 85f0q 3tzjm2ffrom 95km/h to 45km/h The tire)t 2maf035fq v 8bzuxjs 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 r2z ;zwl7n.(bw.uvdl5)n zacz6ttt * veduces its speed uniformly from 95km/h to 45km/hu);.*t5w b z2vznv anl t(tzc7dz6l .w 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 on each pedalnqeghqcohw(:h lf .8al v;07xms8 s43 when climbing a hillg h m0384cq( wsho; vfa.q87lxlhe:ns. The pedals rotate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end c / )-4afkfbyzw6 g*ced,in b9of a door 74 cm wide. What is the magnitude of th kgw69ae/ b-,czfib 4* yncfd)e 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 c tkzq) zyc1s1*rh.r9 sn l6s*of the wheel shown in Fig. 8–39. Assume that a friction torque ohzqs61s)lz.*t crsk1 crn y*9f 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are at8l mj iq siu:5i .uos+vw.ds++tached to the ends of a massless rod which pivots as sd.+siuo+m8s5w .sq i v:l ui+jhown 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 system.

  The bolts on the cylinde6 i 4czy; 7fzzpl5gwp)r head of an engine require tightening to a torque of 38 f5;wcyzp764)lzzp ig $m \cdot N$ 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 3y+1kb 5kcp z-v tqty/sphere of radius 0.648 m when the axis of rotation is throughbcy v k/q+tt-z5p3y 1k its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bi 6llwlz1 .hai4cycle wheel 66.7 cm in diameter. The rim and tire hia zhl4lw61.lave a combined mass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball ojbhb 440 n 65i/flowsun the end of a thin, light rod is rotated in a horizontal ciru5hw0 /sfnobilj 46b4cle 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 whe p7, hv9q,rbmoel rotating at constant angular speed (Fig. 8–42).,mb h, or97vqp The friction force between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point objects shown in rgp-n (h :m6nv anj3 -vy(rqc+Fig. 8–43 about -h(n (rmrjnpgv+y:3nqva c-6
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen atoms whose total mas)n 2b+ncc8 qhxfit(bs/ g.g m+s 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, en;(mkf588nh) p kt).mga oxotsgineers fire four tangential rockets as shown in Fig. 8–44. If the satellsf;x .amto5() n pmo)k8tkg8hite 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 of 8.50 cm and a massxs -u9 ivdqj1c2l c2o6:y7vw( ac gc4h of 0.580 kg. qa1 9iv:2duos vlj4xgc7(2hywc c-6c 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 aqhr8d3yae1 )s bat, 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 sx1r12kbs a:df mall 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. Calculater 1:xd1 bkfsa2 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 eventuallh3909 b scnv(* wwjq dqs+4qlny brought uniform9(9 43 qsbn h0jncvwl *+sqqdwly 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 acceleccndg -t)j8)a rates a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solelo,avm 8c(ooc6 y by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. 8,oo6vmc( coaThe 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 Fik:f )akxsb8bl(y06p yg. 8–46.as()xyf: 80lb y6kkpb
(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 ma(x p(.s5bb ybtgi eb4xq(g82k sses, $m_1$ 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 th 1ygg(b)0tb2tm1v tqj e hammer from rest within four full turns (revolutions) and relea jvb(t) tg0y1 tm1gq2bses 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 a mi3c jp , :2gx3;eq:wfxnp-rm 6azgh*qoment 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 28; v 641xbdr-ecm)a,rvsz6whz0 $m \cdot N$ 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 slippindd-+7w;6:7 7( ov s4fci8q 9nljpjzjgab aspd g down a lan9 jgp 7l+8noqp6abs(:7vaiz4 cj 7fs-;ddj d we at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy ofjn 8/oeo mh/0akn,xz* the Earth with respect to the Sun as the sum of two terms,/8,km x0/ooj*nnha ze
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radbrbq qyja5mz8d(mtth ea6. s ;6s k;(1ius of 7.50 m. How much net work is required to accelerate it fr1 ra tss. ah((;dz 8;5eq kb66tymqbjmom rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 p.-str1h00mbp ;.bby1q uexwa 2ef.p cm and mass 1.80 kg starts from rest and rolls without slippingpe.0b0qx ;phmr.ayf1 2-bsp.u 1etb w 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 vertictry(mg9qc 0j 4ally on its tip. It starts to fall and its lower end d 9(r0qjmc ty4goes not slip. What will be the speed 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 rotating on the endww6ukh o0r7t; of a thin string inw ow0ru;6h7tk a circle 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 unify2i ns5t1gi5l//z 5n-yo: abfufcp ,torm cylindrical grinding wheel of radis1/ z fnyilbcu 2gtoy5:5 -ftn ,i/5apus 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 rotating at a rate(v.*rj)8 .gl qyf.1cd mpz cov of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreasel)mdop 1 .qz* .vjgcvyrf8.(cs to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce her moment of inertia .i:o2 f h,ix1njd (y3ke /hdffby a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s wf jxi /dyo,n: 3k(he1f.2fidhhen in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial rate of 1.0 minsflot-1 kt5.7 8ocrev every 2.0 s to a final rate oft7 moscni.-o 1 fl85kt 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 arou3ip, k2ozr:osw 33 pcrnd a vertical axis through its center at 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 center of the rotating wheel. What is the frequency of the spz3w 2o3rko r3i,pc: wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure 6xb;p,xvx4a5omw d,o :/ kcrw 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 Earthl-gb5yav pm *kmf 4j,8
(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 dro4:jy v u92rcbzv )uvh:pped onto an identical disk rotating at angular spc4h:u v)b:jvuz2 ryv9eed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns a;4c3 ggefp pb)t 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 platfmwifq3+k+ (+esi ynxv yrlg);x+x8p /ormgsqixnvfx+l+m yky(/e ri x;8p w+ +3) of radius 3.0 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 freely with an angular vely .x 29kqua p2m*-uiacocity of 0.8kuxq9 2 p-y*a.u2maic $rad/s$ 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 hj+ , h trkghy7 v)nyx-.02vjymass in the process, and winding upvnrv+)- ghty 7x0hjkh2j,y y. with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest 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 1dayd gm,g3d0$km/h$ 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 h/s z,kmo3d6m$ 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, c(fneq4-p +. m j:u*:vgx, 2yjqrfofjthat can rotate freely without friction. The moment of inertia of the person plus the platfxjuy nvqc. rfj- +m,42q:fjopf:e( g *orm 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 dnpbgg(blo5vn99j + r5 iameter merry-go-round turntable tha9o5brvb+nlg g9( jn5pt is mounted on 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 g+j*f)dd9h j 7wzc5mw5q )n zzr lig4w(round with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person withozc5 )l)q9j7hmwg(z 54jfdnzw*rw +diut 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 faces the Earth. Detc(mv*nc.o t:x7cb+n5y. siht ermine the ratio of the Moon’s spin angular momentum (about its own axis) hc :tncn (cbt*5y+vix sm.o7. to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest oa:jenw,3v 3ljcz.x 3at 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 shnnt7gj6pld w-b- ng:1j.o : vmape of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor:71mnw:p j6- jnd gobv-gtn.l.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical diskxr*n a-wde /o+s, 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 ofe-*a xn+odr w/ 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, ho21* jesec9vs 3clmgv0d,c o)mw is the angular 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 ok5 54*axwgy +tvh9hmr9/ al stur Sun, but with mass 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 the process, what would its rotation spee x 9gy*9k/5t+wrhs vt4am5ahld 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 fo j0xd;/7 u wsmumfh t5qo7th;6r it to use energy stored in a heavy rotating fl h0jwx; s u tqh/67dum5fot7m;ywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 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 illustratesya/1gm v :b85g 9mj xs+zbhr( mvb;lz/ 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 horizontal surface at spee.)nom4;rtu2ge w uds6 d 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 leng) c d5izv0-xrjth L can pivot freely (i.e., we ignore friction) about a hinge attached to a wa0cvx rdz )5ij-ll, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. Itm7wr,:/9kwhv a/rj je.feym 1+o:tkn is standing vertically on the floor, and we want to exert a horizontal force F at its axle1evnk/ ewm,hj am/ :j ky7+9r.towfr : 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 as /x.as.a6.hjdv,5czg y:dp z turn with a radius The forces acting on the cyclist and cycle are the normal5xcsp/v zsd ,:..6gyzd haa. j 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 n0lk so n-rtxu c* c)qc86c5x7a 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 ofxc sx-okt57cc u6 )8nncrl0q* 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 skaterg8 z(ig 6w9vdsx+8 gj w:dq6aqz0xb1z’s body as a solid cylinder and her arms as thin rods, making reasonable estimates for8:1 gav0gddqi(89wqbzg w+6x6 z jszx the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which c wo6 :wh.+p x3n*pdnlyonsists of two cylindrical plates, of mass xlo+pw3:6n. dphwyn* $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 F -tsuvg4,pn p;ig. 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 the4;, ptvgnp-su track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do notnpwcyj8e w9fak1 wj 7rk e7c(8ho:1l- assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions d1 (ohyxlg ry4; y;:edas the car reduces its speed uniformly from 90km/h to 60km/h Theyh:y 1olx4 gr dd(y;;e tires have a diameter 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