A bicycle odometer (which measures distance traveled) is attached n hy:iqa6zf1j d)ue -7vear the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch whee- ) ey1vifzu7jadqh6: ls?

Suppose a disk rotates at constant angular velocity. Does a point(s z1dizoy .*u1rfwlb g4 +0fc on the rim have 0*bluczwg(d1 iyzo+ 4frs.f1 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 acceleration change?

Could a nonrigid body be describjlo*b8jbt9p x4 ti 3+oed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a large r;bcfqb35 ,+ek:pwhgr 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 l0zizcar*yai+9s4 c ,u c131o ikm9wrdo a sit-up with your hands behind your head than when your arms are stretched,i9ri 1iccams yo9 aulw4+k31 0*rc zz out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven s, e7acxcdl0y v056sua prockets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a l5ev,s06ax cad l 07ycuarge 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 slep o p*z79e:ol 4zavqo8nder lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass isaz 7op*p 4q o8:9olzve advantageous.

Why do tightrope walkers (Fptl4 ; ij,/v3ochkhp5ig. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero?8: c;a7iej;0hixmk tbc) ya;n If the net torquk;tnac;m: 7yeji x ac)0ih; b8e on a system is zero, is the net force zero?

Two inclines have the same height but make djrae3a u / -wa 2y0nit8nqm:n /-n8tgyifferent angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the w8:nneqg-3/t8auj2a n-amtr/n yi0 y speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) dowbdrtvbz*h+q + t l*:+ hswny2/n an incline. One sphere has twice the radius and twice the bzs*r lv:+y2t b+h+ q*/ nwthdmass 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 bottom?

A sphere and a cylinder have the same radius and the same mass. They start frw;o o6ucwo3w3u y5 h0ejdlga6b lp8*,-( w qevom rest at the top of an incline. Which reaches the bottom fi 6p* y-l,(wovjbahu6o u ew85q3;3log0 cwwe drst? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular moment-jmqo 3(rbz7bo 8yj;fum are conserved. Yet most moving or rotating objects ev;qbm(jrbj o8z yf 3-7oentually slow down and stop. Explain.

If there were a great migration of p ay9 h;jw u e8v7rk/a7(v6iecieople toward the Earth’s equator, how would this afjea77a /ckhu( v9;8iw6ivyrefect the length of the day?

Can the diver of Fig. 8–29 do a somersault withoutb 5g/7ca4xaf8nl2 icv having any initial rotation w85cavfl2 gc b4a/ixn7hen she leaves the board?

The moment of inertia of a rotating solid dihc e p3sv kei-lj89j :ona2h,.sk about an axis through its center of mass .eiko -3j 28:esh,ca9jplnhv 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 rot6agn13xux)); bg bf+4 qvraxh ating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? E3)bhqa6r 1gbx;4afx n )gv+xuxplain.

Two spheres look identical and have the same mh7 ts0z0tid9,h foc2k ass. However, one is hollow and the other is solid.z f2th0 c tdo790ksh,i Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points i4e0+2i nowzj3m *w wwwest. 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+ we zwi2wm3oni* 40jw of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating tu. o(dx xq9(ivmrntable. What happens if you walk toward thexmi (qx d9.vo( center?

A shortstop may leap into qul ztu,ue0 70the air to catch a ball and 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 oppo ,e7ul0tzu 0qusite direction (Fig. 8–36). Explain.

On the basis of the law of cf4do .5zc ex -nau;5ifonservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller e;fazn4fc-i.5xou 5 d can operate to keep the helicopter stable.

  Express the following angles in radians: (d,+1vc rs--+hc6utnx9ar2og+ju z :j i)npasa) 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 amazing coincidence. Calcula.gc)2c qf vsv5xw.grku.al 50te, using the information inside the Front Cover, the angular diameters (in r)gg2ru..fqc0k vsv5w5xa.cl adians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km99ux f 9dro /7sh6gz( u:srxng from Earth. The beam diverges at an axxnr9r(o97/g:us6uz s 9f d hgngle $\theta$ (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 ratn gn;i7 5v0gqve of 6500 rpm. When the motor is turned off durin5v; 0gqi nv7gng operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the ball makb.*5om co,vomes 15.0 revolutio.bvmoo ,* mco5ns, what is its diameter?    m

  A bicycle with tires 68 cm in duwu/k d,32 gyciameter travels 8.0 km. How many revolutions do the wheels makek,g3u2w ucyd/?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates 6y 2-lgsv0 oz-t v*(.qppeixiat 2500 rpm. Calculate its angular velocity in*y2 0 (islt o qvp6zvgixp-.-e $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.0 s (Fig.-; bntvwj 6s:wq89 ncr 8–38). (a) What is the linear speed of a child seatsc9t; - bwj6qw 8vn:rned 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orbit around th+si s2jtekmj,u skuu .( -js06e Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a pointrmpi(( psqu38
(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 m5 q zhfrsti34r3q+.ea centrifuge rotate if a particle 7.0 cm from the axis of rotation is to hqz 5m. 4se frq+3t3irexperience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 zd0ec:pg; wyvd; c2 b*59yp cgrpm to 280 9;2e5c*d wb;c vgg z:c0ypypdrpm 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 radiuz/oj yt9z y3l) .vqq:mb6:sdhs $R_1$ 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 spacecraft put themsewo0*ndjd w( f8. zco3hlves 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 spacewnfd hc( wo*o.jd083 zcraft 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 e )zt f38*.ta+ dnnwd(t0ae g,vg3rjn uniformly from rest to 15,000 rpm in 220 s. Through how jt3aten* gn.8t,fg0 d+enad ( 3)wvz rmany revolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rp7b8sf1 l- .z8v7ewltt+kg hrdo;kxug/ x(dq/m 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 whirliny3rb+unhk yuam- )q;1g “human centrifuge,” which takes 1.0 min to turn throughra+y q uyk;1mb- h3nu) 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 6.5 6htn1hp55a ne s. How far will a point on the edge of enh61ah 5tp5n the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It ,fu) e42c;j 9kifpohnturns 1500 revolutions before it comf c;u4e)f,noi2j9h pkes 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 pw*p wufe89*n revolutions as the car reduces its speed uniformly frop9*w* pn uf8wem 95km/h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revoluti:sg rq 8d6(a:xhe05fdh-zbnn ons as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a d56q:8-hdhgde n bz( f 0:sxnraiameter 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 tjhm1 d9h kk(g/ r2z(oeach pedal when climbing a hill. The pedals rotate in ak9zdh(mk 2togr h(/j 1 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 of a door 74 cm wide. Wyfsqx lp- oh634kt68v (y3f 1qt+krrthat is the magnitude of the torque if the force is ex(6t43pf tol k1vfrx -8ksth y3ryq6q+ erted
(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 thw+leb1w o( ov1e wheel shown in Fig. 8–39. Assume that a fric(olvb1eow1w +tion 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 endjzh:uf:7 (nq xnxf5 wn60yb0 ss of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position andzxu q::jn 0w6(570xfnhybn f s then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tn 0q*y.kk7wo)tw g;h xightening to a torque of 38 .kg htok*0;wy qwnx)7$m \cdot N$ 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.iwv4ahmii)u+ qdd 7 5;o/ j*ok8-kg sphere of radius 0.648 m when the axis of rotation is 5q74kovh+dimdaj/i o i uw )*;through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. The ri6 3 g9gsvnfaofj, 1en5m and tire has1 vn6ng ,fj a93o5gefve 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 theg,d/ e3tk9, z* u8moezr p.ywa end of a thin, light rod is rotated in a horizontal circle of radius 1.2 m. Caue kr/w ag3y9 ozt e,.mzp,*8dlculate
(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:dz9ex 1cle d6 on a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands l:69d1z edecxand 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 /,( ypnldnl81qdp g m7in Fig. 8–43 abou,/mqdp(nld1gny 78 lpt
(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 conm6 c 5p,ek+t5zq6h dfmsists of two oxygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylxkf /n11kc/b vh+od 9aindrical satellite spinning at the correct rate, engineers fire four tangential rockets as shown icnkb f91vxa/ok1+h /dn Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 cm and a mass of 0 ,wjm.i p9nat3k2 pge2.58022ik.n jgept3p,9 m wa kg. Calculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerating it frox8t zagg/2g5hd y fzkv (0y67nsgpnndt+;a))m 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-round and is aby uws:sm 3yj45le 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,wy4y j: 35msus 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 t8w awbrl; mq1z12 cz,q9*aek eventually brought unifor*z bc11a9eat2wq,rwm q8zl; k mly 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–v h4z63.njdnf5+t4 /l ek sqiq45 accelerates 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 thrg4 k ,ynesc7y/aqws+:own solely by the action of the forearm, which rotates about the elbow joint under the action of the tricws/ yagk4 sqyc:e7+, neps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long tq40 9n- gk ao-lw8anbx08ichz hin rod, as shown in Fig. 8–4il8gna-- xc8z 94b 0kqhna0ow6.
(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 cl z;cl/juv r*:onsists 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 red2i87e :z8jvgjvpt * est within four full turns (revolutions) and releases it at tjj e87vp: g8zi* d2eva 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 hased cuv34 eo9sym.(4 oo 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 28065x6w8r .n,vx io7ldzp qgk (e $m \cdot N$ 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 rc2w6qmu2 oi 0oolls without slipping down a lane at wo oq2 i0um2c63.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with resycqtae iubl8 ; +5n)vv6max0;pect to the Sun as the sum of t0t m;c 8uqv;6+)lex5nb yvai awo 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 net wort qy(hh 1plu:8k is requiredh:yq (lpth1 u8 to accelerate it from 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 cm and mass 1.80 kg starts from res +ka0i;fe4nmp15vcgg t and rolls without slipg v+i ;gk10cm5 efna4pping 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 fap+r :f5zxds9up ne+; yll and its lower end does not slip. What w zsrue;ndy+ : +p5f9pxill 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 end of a t pzo;6m*rgv 1nhin string in a cznr6o m;1* gpvircle 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 g;c-om rxkq-4l me;r w(yi+1nd qdm7f4rinding wheel of radius 18 cm when rotating at d r4 m;;+qk(dwlyon qm ifrem--x471c 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 plagt,7 7,y roiistform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–47oit, gr,iy7s8, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) czx+3 pj h8nei*d,g 8 uo0c2sklan 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 2.0 rotations in 1.5 s when in the tud0p88u+e*glc2hj ,nxo siz3kck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotati3m4ls fxy3-s0xl ruu3on rate from an initial rate of 1.0 res -4xyl3xl 33um0urf sv 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 a vertical axiy lv2/kh524 imywfnbna +p9w3s 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 whee2 nn l2yk 3wayb+mfw4/59hipvl. 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 spinning at 3. .c6ew, ioiwt35 $rev/s$ 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 u8bqn ++za(us0iw t n0q d4e.p b;qlo)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 disobs)56x5o bu ok of moment of inertia I is dropped onto an identical disk rotating at angular spbuooxo55 b)6 seed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2. h uvmlj.x 540i)if k*xo3tzufk5th,: 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 ptcj): nvg b)ob/.lgp6 latform of rob l bt6)cn/ pvg)j.g:adius 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 freel zj.+5o5pa lriy with an angular velocity of 0.j5ao+ pzi r.l58 $rad/s$ 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 eventualz c7ozetw8 p o;0w-ka/ly collapses into a white dwarf, losing about half its mass in the process, and winding upoa/o; zwp8ec kwt-z 07 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 r1+ lbenzwrq8 jz(g7* 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 -2er(ac,f z jfi8gci+ $ 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 res.j dw:g:j.v3jcl m:nht, that can rotate freely without friction. The moment of inertia of the person plus the platforc:. 3 ljndwmg h:.:jjvm is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diameter merry-go-round tccy l; x.n*8aburntable that is mounted on frictionless bearings and has a moment of ix y ;a8.bcc*lnnertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of the rope lyingxm5fmz:/ vs. z 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 spovz5mxm/zs. : fol rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such thafrg2o 7y.)riz/c gi+pt the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a parti 7oigp/g i2f.rrc )+yzcle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of i5;r 5ysm7whs1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of radius 0.20 m aa.cgp gxc 9hcc; c:d--cquires a rotational rate of from rest over a 6.0-s interval at copagx.c; g cc9--dhcc:nstant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, eabyves ;xrlnu b0 t20j.cqu( */my (43ocobq5mch of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) vu.5y toqs;uc0b/0omlm(b4q* j23(cx r nybe 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, ho;4ll9cb(* .xvb,phsq 9nccla eppa 40w 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 our lfn* 4f w 4+ jtndjt);(fuxz /ekd0xw+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)dz/4 fde0 x ku tjt+nfw*;lfnw (x4+j three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use energy stored s j iq+5uh9z2dkdu i5-in a heavy 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 wik-9 h25dui z5us+djiq thout 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 an/:ec1kngm/ :f smdei* $H_2O$ 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 k4z;j:825 u 0iybndm etm+hqbsurface 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 len.ybi7f+cpc ,9z9 nro bgth L can pivot freely (i.e., we ignore friction) about a hinge atta +rb .cz9i,obypn97 fcched 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. [Hint: See Fig. 8–21g.]

A wheel of mass M has radiusgu50/9mp b snb R. It is standing vertically on the floor, and we want to exert a horizontal force F at its axle so that it wilbu9b/g5p msn 0l 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 mafpzrqm6pb6(nk mf+t 9+z (qp,a. pb 9uking a turn with a radius The forc6mfb,mppfk( t+n a q bzp r9z.p9u+q(6es 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 length 1.5 m and begins whirp 2;wyvgk2w:)ga. s)c2zxjbnling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. cx. apbgw yw2))2z jvg;k:s n2What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as thin r ;yqlte1d5pg4ods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly agains154plq dtye; gt her body.   

  You are designing a clutch assembly wh oy sw8 sr35)vf9 ;d4zsw+plmwich consists of two cylindrical plates, of mass w; pos yv5l 38zsrwd+9wsm4f )$M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 radhiek;l m+ biuox7yr: 53de.6mius r rolls along the looped 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?: iymk6le ie3boxm d7 5.hu;+r Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but doi i /p6cdeh2f85ydwh 1 not assume $r\ll R$ h =    (R-r)

  The tires of a car mak4zj t4ejzu(6pnxqm:j .h , wqcg8 3qv(e 85 revolutions as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of eac ucpx z6e8jvq t:zwmnjhg(j( q4 .3q4,h 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