A bicycle odometer (which measures distance traveled) is attached near thecq /( fwtb;0077:hsgarq tsnr wheel hub and is designed for 27-inch wheelrhrw(fa7: ; /bg7s 0ctsqtqn0s. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constanth ,ipo:;sj5rdh ui q,t1m)s y5 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 acceyi:hh, ;5p is 51uoqrj, smt)dleration change?

Could a nonrigid body be described by a single value of the angu1hjv e,likr j ao:t;8o ttr8/7lar velocity $\omega$ Explain.

Can a small force ever exert a greater torqq4: ;y-0crgm3nlc czkt p-u: aue than a larger force? Explain.

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

Why is it more difficult to do a sit-up with you cq(-4y0 anpdw8m)ur-lf dw*xr hands behind your head than when your arms are stretched out in front 8f-da-nqw uxpc0)r4(dwl m*y of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rujs/3osz6p wuxr3f5i 0w8rb lr2j8d c4 )a4en ear 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 harder to pedal, a small front sprocket or a large front sprocket? Why2r xz6 3l5 0w88)4/nub wfrsrj4 dpj3aecosiu?

Mammals that depend on being able to run fast have slender lower legs with flesh ni;)6tt5q ynp and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distri)ypnniq5;t t 6bution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) caa 0s7dzgs;3 ubc y4m,kmu4ax ;rry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero? If the nvhyuk)w;o d+qcmz-9 / et torque on a system is zero, is th w)udy+z kh-qc/mov9;e net force zero?

Two inclines have the same height but make differentw 4u.7t)/kbxcx gwf5z ptov;+ angles with the horizontal. The same steg74tw.+5 ou wzbpxk; c fx)v/tel 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 (fromnde+m0; ma)mq rest) 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 greatera)nmme ;mq+0d total kinetic energy at the bottom?

A sphere and a cylinder/.1rt+ 5zbukmg i(brr have the same radius and the same mass. They start from rest at the top of an incline. Which rea/ 1z 5+rmk g.i(rurbbtches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum ia *uwlxir,x0re 38iw96 w (cxand angular momentum are conserved. Yet most moving or rorc 3wrxx uil0869i* iawwe x,(tating objects eventually slow down and stop. Explain.

If there were a great mi x-s, deri5a,g4pl (sqgration of people toward the Earth’s equator, how would lr4,gdsip5s(x -, aeqthis affect the length of the day?

Can the diver of Fig. 8–29 do a somersault without ha dtz mqj4f)6y:c/j aps g6x4ay 8e;ea7ving any initial rotation when she leavzm:af64pt ;dc a6sjy /gey7e8 a)j4xqes the board?

The moment of inertia of a rotating sol8la t n h z90xrl5r:c7a/e(mrdid disk about an axis through its center ofxma d89ah (5:0le/ntlcr 7zrr 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-kgle8nh(fxq*s1u x cy) 1 mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? E1u 8)xqx nsel1fy(ch *xplain.

Two spheres look identical and have the same mass. However, one is hollekroj lycnc(q0lqfh,84 :y(ae ro0 ctu047v; ow and th0toy7oc:8vr4 h(0al; cfn0j4qc lre,uqe( k ye other is solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rxhgsotgn f5 ( c0:oi1u(tkdj(ef1aju3bg ;:2 otating 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 acceltfbga(i0:o1u1:t( f2dku o n( e3;sj gcghx5jeration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of aoe -oj1or9s)e 9 (ncsy large freely rotating turntable. What happens if you walk to-1 ocoseney9(os)r9 jward the center?

A shortstop may leap into the air to catch a ball ;4 /zhmy86k7hfama/ yf26lkczun3 skand throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the 8zmm7u /6/ akfk6 haky 4zfch3s2ly;nopposite direction (Fig. 8–36). Explain.

On the basis of the law of con(z dnbza,;( fvservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second bz,nva;f(zd ( propeller can operate to keep the helicopter stable.

  Express the following angles in radiansp9: nv tuh,yj9: (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 amg dp uovyr:0 yvm/1mkq:if36 (azing coincidence. Calculate, using the information v dgf0:yyrp6/ mmv(k i 3u:q1oinside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,00n m+-ay bp1m: 3pqw5bd0 km from Earth. The beam diverges atnb-1 yq+dpb5:3a wpmm an angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blend hae i.x/ixy1rc+/w:kt tl 10mer rotate at a rate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angulalcrxmwe+a1.t1x h :iy/tk i0 /r acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level*3 k-dnghvtjz/erg) ; floor 3.5 m to another child. If the ball makes 15.0 revolutions, what is i;3 k eng td)/jvrh-*gzts diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutions do ttsw.p6 boyfatfh,so)f7 7p5yk- u7 o ;he wheels ma s5p7ythyaooop7u.-f )w; ff7 t,kb s6ke?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotatesollep ,c *a0d*jm8a(: /exxgd at 2500 rpm. Calculate its angular velo xg0ep ,xe c d8*:oamal*ld/(jcity 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-rouzn klt ,7o)c/knd makes one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m frolz)7tn,ckk o /m 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 Ear,/ag a)gum :mdth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a poi4jg;k th;5l a z90/3bgp*pdfw1 yh pqtgn.3 bint
(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 xv .-:f j6gftoh9o: s*2h,daj9aarrm if a particle 7.0 cm from the axis of rotation is to experience an v6s:.ao:ax j2jr 9hhftr- *d mf9ao,gacceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpm to 289ynka4fzyze (u -, t32iums*m0 rpm in 4.0 s. Det sk43 muyz-ay 9*nm,ei z(2uftermine
(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 6p ,pnntlzqg:+).; 4hbsl *p ejl(d zf$R_1$ 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 spacecraftzw3 k 7nw-1-auqmv9vk 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 thought of as a cylinder with a diameter of9kqn71mau- vk wz-v w3 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 15,00t .8r+zvpf)i /ljxyf(0 rpm in 220 s. Through how many revolutions did it turn inl+zy itfpvr8.f / j(x) this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. C y (gc-m/ds2vrdv89 qlalculate
(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 flyiqaxi7 j p+ie0/ng highspeed jets in a whirling “human centrifuge,” which t7jeq /+a0iipx akes 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly fj((fgdf r7 gk,dn h5a5b ey0p2rom 240 rpm to 360 rpm in 6.5 s. How far will a point e 25ybjn k(gg(0ad7,h5d fpfron the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is 3 rcy tmdpx )j )r5l8nn)jvm,-running at 850rev/min It turns 1500 revolutions bmvxmr) l )3 n,8 -djjn5rtyp)cefore 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 speo ta7iu)acw yo b-.20ned uniformly from 95km/h to 45km/h The tir2 bwutocnia-a0. )yo7es 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 its speed uniformly (ut(n; rkf dggkp 4e;9from 95km/h to 45km/h The tires have a diameter of (u( ; kgt9;4egr fkdnp0.80 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 w u/8m-*ubx;) y fncblmvhj 8;vhen climbing a hill. The pedals rotate in a circle of radius 17vx mubl-y8jvh;u )n 8mb*fc; / 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. Wha; 0swi4 /cjxr5lz* ditgad+, lt is the magnitude of the torque if ;wr*g,x 4 dsl 5tazi/c+i 0ldjthe force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of tn,0 7u vmzsd,i jfs5-ohe wheel shown in Fig. 8–39. Assume that a friction to5v ,uiso-0zd7 mfns ,jrque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached tnl3rx.;9a .2fg : cbpks1rrby(c u3jfo the ends of a massless rod which pivots as show 3b.u2n3s:b9 .ajfyf1rr( klxp c;gr cn 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 cylinder hea/q/,zknog msr. rp05,,iaiau d of an engine require tightening to a torque of 38 r,, 5n0q/k.ipso/m agru zia, $m \cdot N$ 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 00lkr*p2 dn kg7kv a21j.648 m when the axis o 10k7a2dg*p k rlkjnv2f rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of if -jze*sek22vuc y89mnertia of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can bc e8u2-9 f*2zjkme vsye ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a xl0ss/1z)3d (n yacgv thin, light rod is rotated in a horizontal circle of 1yczd x va0sn 3(g)s/lradius 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 ed wdazeq9u::w,tn :xf.t7ren3x 2 +pbowl on a potter’s wheel rotating at constant angular speed (Fig. 8–42). The fri2pzx: 7ed,x:+nqr u9ef w.:d tt3n ewaction 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 pointv v/ ij8jm0kg)( o2 ,i2ykljebbyl6k2 objects shown in Fig. 8–43 abouj8ji viky0 )( yvb6,m2obl2lj2k /ekg t
(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 ( q*xltw eka/x n2m5nmwgb9/,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 cylinvp+hol 0z4b 0zfht).s;ai z c 00ob9phdrical satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.hl4b)p h.foh00asz0 ;tb9z i0z vo+c p0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 cm and a mass of/ /txgdqtr. -r 0.580 kg. Ctt/-dx /. rgqralculate
(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, accwwrh6 (k/p;juelerating 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-drio6s 1tz3 s-mi3fz 97totdy 6ujven 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 2j zz3-1m6u7fio y st9d6t3so t.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 l8nkg ,1 y7jt9lqebu) rpm is shut off and is eventually brought uniformly to rest by a frict 8 l9jul1, qk)ebtgny7ional 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 ajm 7 s;g.3 :cl3fge.vjbm4 mqtmv1.db ccelerates 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 solely by the ac 5 epo:jao0qe yb:pz4/2c) mqltion of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45lo:/q j5)co 2mpyeep4z q0:ab . 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 wbm-6fo+ ksvq udo:a w/0a,s. w+0bma thin rod, as shown in Fig. 8–46.0:os mvd.qo mw sb0 afu-/a,+kab+ ww6
(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 masseeh (jc h*bx40ss, $m_1$ 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 accze zb6*waawe dego6 oj, 65d4(85bhntelerates the hammer from rest within four full turns (revolutions) and releases it at a spw5zn od hj6e6ew d,6e4agb8t 5oz(ab*eed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a moment of inertiae rrwm14,db k+t2f3 gn 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 a8/.v*kt yeymtorque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls without slipping down a lam/d bth-j+vy/w v)a 8lne atd+tmvh wvjl -ay// 8b) 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with rb-2w; jvpi).i ne8 ,xqgs(epxespect to the Sun as the sum of two termssvi-2w) . gx(;en8b,qpejxi p,
(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 net wo2y b s6g+bsu2u3 nntc) d sn/(y8kz4jzrk is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a sskg +u n3t624n bs8cz)2 jsbyzn/y(duolid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest andowx8 ;(j hj ysk/ *u)fsw+f. 1z:lulua rolls without slipping down a 30.0;a lls:xos1 8)f. jkuwf( ujhzu w+*y/ $^{\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 itsj a kucj9g9/91a j*2f g4taomp tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end 9amgup cfoa4j19kt/*a g2j j 9of 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 u v/emg /3ku 7ax:h5yion the end of a thin stringa5xie gmhy//u7v :3ku in 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 uniform cylindrical grinding wheelesg4d6dgo c4 ) of radius 18 cm when rotatinsgo6cdd 4)4geg 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 platfo/vn5cr)lrg q4ndu)ib,qk 0h6 rm that is rotating 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 )ricdn /u45,nqv hgk qbl0)r60.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. /k*/b.c v7s 8g v7g jxo(rvlja8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.(/os8v jgbkj.*7lx/gvar c 7v0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial wfnsze:g)1z , e*rem 5rate of 1.0 rev everg 5e,wm*):zfz resn1ey 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 vertical axis through its center at a freiyuw( * vwpl4+s lro usm 20g+((fgx/iquency 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 isxrv(0(+mi (go4 ull sps* iyfgu+ 2ww/ the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum o;28de qt 6mhyhf 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 -soh y77 bh da226usby1s)lkj of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is dropped onto an y ni8 :nont+q-identical disk rotating at angu:ony -tn8nqi +lar speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at /v)rw5 1-nn sd xgns.n2.4 $rev/s$ 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-roux3 1rlqwa c.1d.8umtq nd platform of radius 3.0 m and moment of inertia 920rtdmw qql811a.xc3u. $kg \cdot m^2$ 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 rot1g.eoas6(k zh4 vn3 kdating freely with an angular velocity of 0.8 63vk.hn d 1z(kgsoe4a $rad/s$ 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 o gfzq,mw:9t. ;zq yc0;*oy nqits mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carrz,ztg.9oy w;q*0mcony ; qfq :ies 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 yalvh(wk6y(+k /xh. lwinds in excess of 120 $km/h$ at the outer edge. Make a crude estimate of
(a) the energy,    $ \times10^{16 }$ J
(b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 $kg \cdot m^2$) of radius 100 km and height 4.0 km.    $ \times10^{20 }$ $kg \cdot m^2$

  An asteroid of mass in3xye z54hm /j0*lx c$ 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 plat ;h7rl,cnv 9ji-e u86/t .naaclxzy(o form, initially at rest, that can rotate freely without friction. The momen(9zi on,/j7yu;r taxh8l l -ca cve6n.t 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 personaf raw.t4+-z s stands at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inerti f4.a-taszrw+ a 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 roll jpeznv lbhx3e d s2.v15(i1 2n1d2zmfs on 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 11 xs23 jpved 5vdien(z l2.1hz2nmf bbehind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the samzjhb;5*)qwdtr/ l dy;e side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In b)l t hz y*qrd;j/;dw5the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate ofud*m(+* qrje 79deiwc 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 uni j w-oz;i2fzp.form cylinder of radius 0.20 m acquires a rotational rate of from re zopf.w ;2ijz-st 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, o/o2zr 0wxpopovistd/. ,2a: each of mass 0.050 kg and diameter 0.075 m, joined by a/sr2awov/:oi p, xp 02ozt. do (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 ;fnn(m y 1es- fd g 86-wjfj8mqklz(:fangular speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 timkysgv. a,7k s8es 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 speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries ofyvg.8k,k7 sa sf no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobiltijbpej:rw8 t,t 0,f;e-o 0 tle is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 ,w,; ttti jbojt :fel0-r8ep0kg, 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 anp-p;pi6 sy3jf $H_2O$ 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 speed v=3.30 taxdeda4m sa b4 8h0i3b,/vj $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., we ignore fdm(g9a/ 3qa 5xtlk6n jm p*j36ejlv+ griction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hi ng36 3*tpvxm l5qdmaa el/g9kj(+j6j nt: See Fig. 8–21g.]

A wheel of mass M has m,/5 5didym6 8ardkuovu*g i5radius R. It is standing vertically on the floor, and we want to exert a horizontal force Fg6 *id vuda5kiory dum,/55 m8 at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with x/1 usa7wgjmkmi;)lt6 ax9h dcn8-0 s vqjg7* speed v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist and cycle are the qmi7v0 n /sxg; hg8*a )7d6-9sujc mjxwatlk1normal 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 length 1.5 m and bex. 4n f*0narhpk6b;s agins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the tonx a.4n hp;sr*ak 6f0brque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s bp/xe9( . gksl(ux 7uonody 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 a spinning skater with outstepx.o/unxs kg (u7(9 lretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindrical xaw-kn76t/zt plates, of kt-wt6xanz7/ mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 loop .8k9y6surttr 7ii) zmed rough track of Fig. 8–58. What is the minimum value of the vertical height h that)ri 6k78 tsiz .rumty9 the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not-o -iuj *s3.rskvz7a zl dyl.0 assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speem2,qsvfe ;v6ed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular accel q6 sfve,em;2veration 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