A bicycle odometer (which measures dismy3cc-p*o6 x atance traveled) is attached near the wheel hub and is designed for cm*yxc3ap6o - 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on the8b/ qcj2z 6c jae;pod; rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangentialbp6eq8oc/dz j;j;2c a acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be desclh 514 n*xovapribed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torqu.g9,jl5kv (, v5gjueyhalya/ e 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 diffi az(enfi25 tcq xmy ht.(;(px4cult to do a sit-up with your hands behind your head than when your arzaft( 2emxcpn4xyq;.h i(t5( ms are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the pedal xe(j1 /lwxt57hv ;x ie4hylb:(oqs g06 y suv4cranks. In w5;bv4 h(y ljv67lux :4gohsi 0qxwytexse( / 1hich 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 being able to run fast have slender lower legs witg 1w+ mgv;b5mpv u*g d25*vagzh flesh and muscv5* vzgmgum2+awgbpg1d ; 5v* le concentrated 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. 8–35)7klb yg9(lf iw6sm)v q 3kaz2; carry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero? If the lm 91jg0 o -.aacqnmo4djz 3+vnet torque on a system is zero, is the net force zerojo md+vc.1jmnao493g aq0- lz?

Two inclines have the same height but make different anm ns *s9;(c*h+ef(nmyias7 jtgles with the horizontal. The same steel ball is rolled down each incline. On which incef 7sis *y;n( m+nts(m 9*hcajline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an in5ejj2/8p m-g on m+*l9zggqj rcline. 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 energy at the bjonp+2ge5zg 9- j *lr/jqm g8mottom?

A sphere and a cylinder hv - 02rc: 2w rrncpo *3ihjxu82rovv2save the same radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Whicjixvs-2:r *n 3 vo8u0rvrh2w2pc2r c oh has the greater rotational KE?

We claim that momentum and angular momentufa: jb/:y ,ntiz i-f4cm are conserved. Yet most moving or rotating objects eventually jbynti,:4 z -af:/fc islow down and stop. Explain.

If there were a great migration of people zr)21gn;x,y avpst 0jtoward the Earth’s equator, how would this affect the lenxr0 zts,a1;g jy)2nvpgth of the day?

Can the diver of Fig. 8–29 do a qi ,tcxa9w 9k7somersault without having any initial rotation when she leavei79qckx,t 9 aws the board?

The moment of inertia of a r xw1 3y3 p.hn)0/hfmsbisjr +qotating solid disk about an axis through its center of mass /rfs31y+h3 ihx.q0pnj ) wsmb 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 rot5lv udd:a.ys+aey1h 7ating stool holding a 2-kg mass in each outstretched hand. If you suddenly .51usdvd+e al:y hya 7drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical anj.rfu,ew ;v.uv j o7x9d have the same mass. However, one is hollow and the other is solid. Describev7uvoex. ;.9 fjwu, jr an experiment to determine which is which.

The angular velocity of a w 4(b6hrmvuu iam r p(4yk4o1;rt, hjt.heel 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. Is thmhm 1,(tb u i kra(ph44;rv.rujo t64ye angular speed increasing or decreasing?

Suppose you are standing on the edge of a large frc(*z azsfd5(ait4 ;dxeely rotating turntable. What happens if you walk zf*;5dt axc(sza4(d itoward the center?

A shortstop may leap into the air to catch a ball and throw i5 6ak/be a ja4i *dv5pejdgd4*t quickly. As he throws the ball, the upper part of his body rotates. If you look quickly youa5ja p54bd6*vgd/je4*a e kid 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, discuss why a heo0 3u1api+9 y b4ptolhlicopter must have more than one rotor (or propeller). Discuss one or more b41+y0ouaphop 9 i 3tlways the second propeller can operate to keep the helicopter stable.

  Express the following angles ina y7kh *d4 yhm :-uvdzv..r8kl radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazin /5qptt 0np:/mymr(cfg coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Mo0 cm/ qppfmy:5/tt (nron, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The beam divergemg49e q g0):4ho avxyi1;/wn*+izda7rfvq ju s at arh10f g)*:d + a7i9qgyuja m viozw/nv44;qxen angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. When the moe-qh 2 gj f9x(:)acxhhtor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades s-chegjhxf:) x2a q9h (low down?    $rad/s^2$

  A child rolls a ball on a level fplcojx+ b2)7bx :f 8ofloor 3.5 m to another child. If the ball makes +lf:bf ) j78b2xxpoc o15.0 revolutions, what is its diameter?    m

  A bicycle with tires 68 o0e.xz bd0 km2cm in diameter travels 8.0 km. How many revolutions do the wheez 0 .e2x0mdobkls make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate its angul8j2,lprk, 6qubdy ,wear velocity8j 2ye,r 6b,ldup ,kwq 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-rounl l*fe)z8,u.etmk b)yd makes one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear spee*lb8) ,tz )yeulemk. fd of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orbit .bv9vq 4cjdwx ,aj:2a around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a 7e7r1fop2 gn zpoint
(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 particle 7.0 cm frouko 9 pz1:4qrrm the axis of rotation is to experiencero9 qrp:z1u4k an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelera3 .x1 xwkzpi3cj3vo+qtes uniformly about its center from 130 rpm to 280 3cj +p.xo13vqik xwz3rpm 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 radius 8dd7v;do pw9w v)1dc /9r ghf5 zjgl2n$R_1$ 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 spacecrafm at.xaxcm6:j5+acy o/ vne1/t put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of the:om+njm.5t aa c6c1e /vxa /yxir 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 lh+zgs :t.. ln:waw0ss. Through how many revolutions d:wg0z:snhwt.l .sl a+id it turn in this time?    $rev$

  An automobile engine slows down from 4500 rp m y3n.j2dygy6fnksa(w ()6ql m 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 whnh o2wx g*7o1t 2mkgt7irling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutio1wth t2 2o7 xgmng*7kons 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 a6fd6mty w1ef*rxzam ri)77 z -ccelerates uniformly from 240 rpm to 360 rpm in 6.5 s. How farf fm*) x 6ez7dmr7yiw r-61zat 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/min It turns 1500 revo-2 po1ot a1f4o 5eaousa8inr 8lutions before 4ru8 oisna f-oop2t8 ae11o a5it 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 car4ta7jlc/rh:hngn k 4; reduces its speed uniformly from 95km/h to 45km/h The tiresnk4l7g/t;jhn4 rcah : 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 95xamkh/y,a 6c ) 3rddzeuv o kec,u :zdk,v+/its speed uniformly from 95km/h to 45,cz9k,h:cx3/,v)6amke+y ovke ddr u 5a z/udkm/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on ea- e8vto+,h4zkhy5,ma 0 nfj wr6hyd5jch pedal when climbing a hill. The pedals rotathvjhe,atr-f585mw0+k6yn y,z 4 hodj e 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 og ns7c-za67; c 4oilkwf a door 74 cm wide. What is the magnitude of the torque if the force is exer46; 77 gcwczsok- ailnted
(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 s /+lma-ycc l(xt.nr )phown in Fig. 8–39. Assume that a fac)rct -x+nly lp(m/ .riction 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 ends of a massless rod whndg,vasr8 2e o7nv.6wich 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 dv6 8ao 2sgnre w.7v,nthe net torque on this system.

  The bolts on the cylinder head of an engine requirefp8xjlr b4n s 32iw+/y tightening to a torque of 38 bli2/3pxy 8r nsfj+4w $m \cdot N$ 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-kgihdms+4r6*dl/zig 7l sphere of radius 0.648 m when the axis of*slhr+l 67d i4gi/ zdm rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia s(npl, l4aj)yof a bicycle wheel 66.7 cm in diameter. The rim and tire havesnl()p l 4ay,j 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 on the end of a thin, light rod is rotated in a lk +( 4c(g;du,jv czyjhorizontal circle of radius 1.2 mzu,ykj+ jl(ccd4;g( v. 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 rotating at /rpr y,yov1cn /.vlj6constant angular speed (Fig. 8–42). The frictlyv.yovn / prc/1jr 6,ion 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 showkk0c ecib3( g *bi2(hdn in Fig. 8–43bh kc*c2be(0(k 3iig d about
(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 totaz90gqsh5b kim8 q)v llvh(fyv+bu-)3 l 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 rat bt *m +37n*0aurzbxj+h0 zdcte, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m,* +0bz 7muxbjctz0* hnrda3 t+ 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 wi )53wfvu7xsste78kvk th a radius of 8.50 cm and a mass of 0.580 kg. Calcul3fktwk 7 5u)s7sve 8xvate
(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 batj(( ,o8i xoiveo/zv :w, 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 hand-driven merry-go-rou b o7u9eq1k-lbnd and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assumelk1o79bq- bue the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, 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 eventud d.nql-eb,tb nt;v.; ally brought uniformly to rest by a frictional torque ofn;teqn-;l vdb .d, tb. 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 3.6a5zh6fbhcj,,u)p* is -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 solely by the action of the forea4n1+9gn dbx7 07, s 1sr9iym*ohyoiqlhyww v9rm, which rotates about the elbow joint under the action of the triceps muscle, Fylo1 y7obrw9g97n+, hmx0siyi 49s w*vh n qd1ig. 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 sh+8pi4e ej7sky own in Fig. 8–46ksjee7y+8p4 i .
(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 conj u ibztckg(;h q:;5c1sists of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest with2p:51 m iurpomin four full turns (revolutions) and releamui:mrp op 521ses 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 haykxisd;65n * ks a moment of inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque of 280gam/u(-l 7-ypw8q7 imh- whhrbkl)5i $m \cdot N$ 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 withocn0orx(l-u,c3 k.lb y uo- sn2ut slipping down a lane at 3.32-ro uku-ycnlb0c ons3 (x,l. $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of 37imk*:p *voe ydsjc6x8 ;spaok7xc 0the Earth with respect to the Sun as the sum of tjke8*:0ix p x6opvo7ycdm;3s c*a s7kwo terms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius of 7.50 m. How much nea)z0(v/ty ,o - cwtwfdid 6ox.t work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.-to0dayvdi(,f./ wo xw)t 6cz 00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and masem (ltg2 j,z.as 1.80 kg starts from rest and rolls without slippt,e.m2lzg(aj ing 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 fall antu 6;m v,qlhh)pgd4panc +p57d its lower end does not slip. What will be the speed of the upper end of the pole just before it hits th,vth47d5g) ;pl6+am qpunhpc e ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end of a thin stuklz +922*s0c a ukxbmring in a cl9 bzuucx 2ks02*mak+ircle 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 unis (sl6 r)gjnfi6u-;/lk(w mhxform cylindrical grinding wheel of radius 18 cm when rotating at 1500 rpm? /;- kxi (sgm6sl 6(jwlunf)rh    $kg \cdot m^2$
(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 rota uj0: nbspsb11ting at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases to 1jbnsb1p : u0s0.8 $rev/s$ (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 of6/ 8pho-wx2 4dzhulff inertia by 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 when in the tuck position, what is her angular 48uz-d hl/ 2hff xpw6ospeed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increaseu8r9(vc y6vp6ikw(aix6yb ;j her spin rotation rate from an initial rate of 1.0 6xv6(u9wj(y8 yi v ab6kpc;irrev every 2.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around az h-6 iz p,2r2vjv;uo1w;nx6jf2(qpt x dutx( 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 jw fotti2 j;uh((dvzz upqx1xr;v pn6226-,xof radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater ktky0*loymf1ervr6j rsuyx;-72e 46 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 Eartpnr:a9k x:9phe : pt5t h+hc4hh
(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 dropped onto+7n,uqi t px7lbid; 5h an identical disk rotating at angular spl,+7hqi in;7xbdtpu5 eed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4:od)buu8r e1nmo3/ a m $rev/s$ 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-r0 tb:h(pzz6pq ound platform of radius 3.0 m and moment of int :p0 6pz(qbhzertia 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 rotatingzhhn 6tc-3awpe3)dv 6 l. hkh7 freely with an angular velocity of 0.8 d n-73a)ht 6zcwhlh3evh.6kp $rad/s$ 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 collapsocy hp6gk( o9ou6x7/aes into a white dwarf, losing about half its mass in the process,puao66ogcko (x/97 h y and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest 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 exce(1 bakxk: xte6w7 nl19i5ksryss 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 jsm*g(x* su u:c.* usw$ 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 re-m:.t h 07wm 5rzalcrpst, that can rotate freely without friction. The moment of inertia of the pec :75a0mprhtwmr. - zlrson 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 k+sj7)g(hhm/ b qpji i.8se3p of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia of 170psjh3mshi ( )ike8q.jbg+p7/0 $kg \cdot m^2$ 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 onashm as uow7 l0m5k81yg;h4r / the ground 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 without slipping. What length of rope unwinds 7o/s1; ah mmyksw 8gr 54au0lhfrom 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 Envcd my8(hw* b..n nf1arth. Determine the ratio of the Moon’s d*fcy1 mnnh vn(8.wb.spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a raav c6 llx ll vy-7/ tbvbz9(j9jd3 ,at0n*jwm3te 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 sq,q1m x.iq9bz( w*td5 fiuyg)7i: he khape of a uniform cylinder of radius 0.20 m acquires a rotational rate of fq7*zue h9b1)fg:iw.(m q itx, iqdy5k rom 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 - gcv4cdtcd.8of two solid cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string8c 4cgd .vdtc-, 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 sfk:o.z .kwjkn)t5 cds6r z6a+peed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times as great,z; sve : yno7yx2ztgn(l4ai,(5u*mlj were rotating at a speed of 1.0 revoln2g ,mz4 7:za*t yyolxsu i;elvjn((5ution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollutiqk hogi3m9(m )on automobile is for it to use energy stored in a 9ihm(o3kq)gmheavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 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 anu *s khv1jkth6+jphvxh1*l -2 $H_2O$ 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 hors)uhizqsa0rcn.y 2 95)cb4 ne izontal 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 igx-xn wr9 tt 24kxll(s1nore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the fotnxl -lx2(1xks9w r4t rce 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 octm6-1 ebos) utw 1z+dx1bn3rn the floor, and we want to exert a horizontal force F at its axle s t uso )c1b3 tm61wzr+1-exndbo 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 tu *fs* 68vkmrwo 3yk2jcrn with a radiu*2okk s8wf6 3c v*jmyrs The forces acting on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of 3ffc4:e xs(f1-fmf bx length 1.5 m and begins whirling the rock in a nearly horizontalcms14(3 ffxfef x-:bf 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 body as a solid cylinder and her arms as thinf14w ar l0x.y)/ 7akza*uswbl)9vw k / ;jjiuf rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms hel 1fj k0 ul 4.wla)vi;7s/zra/juwfkbwayx)* 9d tightly against her body.   

  You are designing a clutch assembly which consists of two cylindrical platie* :(fk* -s1l2p+ l1t* fjgq p5+akwxb fchuyes, of massu i+g*-f5js*pktl(y cbek1fhqwxp1*f + 2:a l $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 loo);oikaa0agw9)hb - )z ijvsql b-d*;dped rough track 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 without leaving the track? Assuqhlazv;d )i wbbs*o a9i 0d-k-gj;))ame $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not a t *io,gvo -c mk8k:uqy)9;weassume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its s,bhub5jb -h3 1j44he iksggp)peed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular accelerahp3ihb-5ebsk)hjj4g g1bu,4 tion 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