A bicycle odometer (which measures distance traveled) is arx ,fnf *u*7bu.lw+wattached near the wheel hub and is designed for 27-inch w.buu x+*n ,lwaf*w f7rheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant aniq(2j0n. 7brsypsj 8wgular 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 t s(qjwi72.s8r0bnj py angential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a4x x 7j5i3a.o+n4vu,xgcylw k single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a gr dgek-4,m0ymneater torque 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 v4w+itys21 uz to do a sit-up with your hands behind your head than when your arms are stretched out in front of you? A diagram may help you to answer this. w+14 i2zsyvtu

A 21-speed bicycle has seven sprockets at the rear wheel and 3ez6trc*h8 mkthree at the pedal cranks. In which gear is it harder to ped t6mce8kzh3 *ral, 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 lokq+xbg0:*tbt s+8rhph5k u+awer 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 is advantageprt+qth+*5kug s8+ 0bx :a bhkous.

Why do tightrope walkers (Fig. 8:q7m :udlt 7vmzbe1n 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the net torque also z3dvt :t/z 0xxsero? If the net torque x0:sd3tvzt/x on a system is zero, is the net force zero?

Two inclines have the same height but make different angles with7/r0659dby ercoqcu i the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be gror50r eyc6qci9d/b 7ueater? Explain.

Two solid spheres simultaneously start rolling (from rest) d)ai42e,uw jor own an incline. One spherjoui a w,e42)re 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 bottom?

A sphere and a cylinder have gq3;xe)agp hq83iggn ;espuv4 ) /5htthe 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? Which has the greateh qeu3h5/geqpgg3)in8gt;s); 4vxap r rotational KE?

We claim that momentum and angular momentum are conserved. Yet n *e3 crhu0hdooi5( .omost moving or rota0 oh.5udh( *i3nocreoting objects eventually slow down and stop. Explain.

If there were a great migration of people towar8m51 eqgj bopnc3r bsv(*5h.kd the Earth’s equator, how would this affect the les5v*qj 8ne k1r 5h. 3pbmg(bcongth of the day?

Can the diver of Fig. 8–29 d,ql+n-6f5 kkhs ,wg pjo a somersault without having any initial rotation when she leaves thfl5q,6h ,kp sk-jgw+ne board?

The moment of inertia of a rotating solid disk about an axis through its ce.w:phhtbg d*vk*d)i *nter of mass idiv** gwt h:bh kpd.*)s $\frac{1}{2}WR^2$ (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 holditf.s.6sm 1prp mj3pg/ng a 2-kg mass in each outstretchedp.mr/ppt.sjf1 g63s m hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identicaqtz0,;;ny0k9pbko8faw 6t s wl and have the same mass. However, one is hollow and the other is solid. Descrfpst6 k ;kt08z,yn;b awoq90wibe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points wesmtwm7x;ed3z. sr 2q j.t. 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. Iswqemtz xr2j7 ;d.s. m3 the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotatintbqs4/du ea ax b5l8 ,taa7-al,k rb23xe j7u/g turntable. What happens if yt,tua d l2q8xua/3b-a/5ea jx 4ae,sk7 blb r7ou walk toward the center?

A shortstop may leap into the air to catch a ball and throw i(g x/cm(ktca:dkx i,5t quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his :k/gc,x xca(kdi( m5t 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 ae/tbk .t*a0vh helico.t vbh*/teak0pter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (a)ufu7rfp9o*z u q )6ec, 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 becau yn; f hll+mw(i1 xu79 vdf5v2eef8r+gse of an amazing coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radians) of the lgf;n7 (ihfv1e+2rlxm85udw9fv e +y Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed al;ji2z /ptoc:3sd8g l t the Moon, 380,000 km from Earth. The beam diverges at an angleoz32 g/lj t 8c;sdp:il $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. When the motor is turnex7oc r dgm5,((ju/ fsxd off during operation, the blades slow to rest in 3.0 s. What is sf g7(x j(,m/durxc5othe 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. Ifk8db ik4 96p5fedek7n; u7dmp the ball makes 15.0 revolutions, wdkk8dm ;b ei7u45kpen9f6p 7dhat is its diameter?    m

  A bicycle with tires 68 s1hagne,o sl4y9*ym5cm in diameter travels 8.0 km. How many revolutions do the wheels5hy4nl g9y,*esosm a1 make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 r7 bh*v5ffc wt.,uta 2wpm. Calculate its angular velocityt,wtf .*ah fwcu27vb5 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-ro mdxgt e,du,-,)0bhty und makes one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seat,0)etu,m,h tdyb -dgxed 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 aro l ljl5;/)z h6z5ic2 y l w.i:c9uegxkt9.efxmund the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a ig 4jj7k 45e)repo71f9 lk d9/f simzmpoint
(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 rotablvxp65i rs- m.ua; :x4coaz(te if a particle 7.0 cm from the axis of rotation is to ev-5as:xmziu a p4;6.xlo cb(rxperience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly abt8hjp/*zon ) b9 zc. 3z3udqozout its center from 130 rpm to 280 rpm in 4.0 s. Decp /zo.hdzu3 * qo) b98zjntz3termine
(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 radiu+b y x40hvf3io-v. ;astyb*id f*/awcye gh7+ s $R_1$ 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 themselv9 fw f6t5p,9rvw ax1yfes 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 ev59 f a f1wx,p6rfvwy9tery 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 fr +jyw+2bwy3u oom rest to 15,000 rpm in 220 s. Through how many revoluwy3uj+wby +o 2tions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm totn0hojs17ajj*c ,oy * 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 thoqcsd* 36ggnt *zp s*l*vz ;/le stresses of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min t*t ;gc po*s szng*vzl/q d*3l6o turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 360 rpm in 6.5zvo ;qs/h(mc mkvri8l +, 9*wwkb47i/jk) jxx s. How far will a point on the edge of the wheel xwkmh;87k, qbijkr wlviz +9/ sx*m jo)/vc( 4have traveled in this time?    m

  A cooling fan is turned off when it is running at 8 *3piq+gdcmi27hmu vztba*/2 50rev/min It turns 1500 revolutionmm/*2pugb2zchtq7a d+ i*vi3 s before it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolu/7d*lw 6s83p cyvd; bl0 b vdye*vpd8ztions as the car reduces its speed uniformly from l8vd pd;we l pd 7*b 6zs0*vdy/cb3y8v95km/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 revolutions as the car reduces cf2)u: *q-ai ti )5:5wfcskc wubu0g sits speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.80 m) 0 kb) uuu:f-fwq*i52csasc5wg c:ti.
(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 9dak4avt5 * hmpedal when climbing a hill. The pedals rotate in a circle of radius 17t4a 5 *damkv9h 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 55a)jhm xd*l,k;)*cthn3b yn -s N on the end of a door 74 cm wide. What is the magnitude of the torque if t;h *ds-xn), t kychalbnmj3*) he 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 the wheel shown in Fig. 8–39. A 6)3dnj q9la7wj;z-ec/a7ao 3 rhamry ssume that a 7mhc7are3zq3 d6anaraj )y 9lw-jo/; friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass pz,,euzo3f u,ys 18zxm, are attached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then releaxo,z ,u ,1uef8y pzzs3sed. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an mq5vg2hma (4sidjm4am0mk 29 engine require tightening to a torque of 38ism0h2 jq(2k4 dmg4 5m9mvmaa $m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of radius 0.648 m when the aols-0jamgqk p6 89x7pxis of rotation is through its 9j 7os06klp8mapg x q-center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. The ri0q;t fta ly1fq, q7os,m and tire have a combined mass of 1.25 kg. The mo;1,ltf ,s f0q7 yqtqaass 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 rotatedmv2 5.khky y li.j;z/c, oluj* in a horizontal cir., mjvzuh*/ .i lyjy;lk o5ck2cle of radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating at constant angular spee , f/mwltjdap:x i,66wd (Fig. 8–42). The friction force between her hands and the clay is 1.5 N t6p,j ma lfdxw 6:,w/itotal.
(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 pf2el*l m4qs( )wsf:t moint objects shown in Fig. 8–43f mqs mt:*e42flw(sl ) 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 yv(t h (6 8dxzihe*9obatoms 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 cylindrical satellite spinning at the co.lqv -mo y,of9rrect rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the svy-f .l,moq9 oatellite 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 cylindex: v*4 kpmxqu;ws-( xfr with a radius of 8.50 cm and a mass of 0.580 kg. x:qkw x;-(p fu xm*s4vCalculate
(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 from - u0xen-;r4sydd p,plrest 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-gy,7rs x z0g4y,xl/ hweny8xd*o-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the meh wyy8dxsl0,z,4rx g /*e7n yxrry-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 r8r6q-,y tgux hcnve *e etmn2ho1.p))otating at 10,300 rpm is shut off and is eventually brought uniformly to rest by a frictional torque of 1.rt )-ph) gevt* emcyuq.62o8hnnex ,12 $m \cdot N$ 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 3.6-kg ball at 7v .gcgudp fk3/p av.d/ r88qd- $m/s^2$ 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 f) 6v(vt r0tafgorearm, which rotates about the elbow joint under the action of the t(gfat rvt 0)v6riceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, asgp(q6xiuj54fscwbi*p h:9ouik , 6: m/jlx/c shown in Fig. 8–4 k5qj:f x sugh*9 lo/b64pu,/i(6ij:ixwm cpc6.
(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 masses l4ym ow+nv*l)okf c+q(e+qrag2 fc73, $m_1$ 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 withidmgff+;wy 0ha +(wlb0n four full turns (revolutions) andmdywl 0+gfh bw a+f0(; releases it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a momc bm-e l4k29j lsh-d9fent 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 torquy q1eyf, /ji0he 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 rolle i7cwxp(y)h +s without slipping down a lane at 3( 7iy+epwxch ).3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earvdg8)r,hc6 *wsoyr 7 u8,kqmqok xy (7th with respect to the Sun as the sum of twq*yr6do k v hoqks,gy, wc7urxm7)88( o 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 muchy)pc3pe+2g,v btxpe py)d; *u/ls.vb net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s py/. se;pg+2b c,pduxlytevvb )) 3p*? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolvuir,l r .p j+)/gcmu+ls without slipping ,+ mil+/ru gp). jucrvdown 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 d /am) md/iju:to fall and its lower end does not slip. What will be the speed of the upper end of the pole just beforei / amm:udj/d) 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 ,*jn+,mj okd )e5:chwlpzb k j; kn60da thin string in a circl,c:k b;e)5 d* 6,kh pzlnwmnojj0+dkje 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 cylin3:up yzs,sf ;tdrical grinding wheel of radius 18 cm wftssu;,pz 3:yhen rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotating a8.rxa o0grj 24dc4cyjt a rate of 1.3rev/s If he raises his arms to a horizontal 2.xg0y8cjcojda4r4r position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce her mome 5himkzul2m7gmh,t 1t) e.s /6fdmajhf 9l*k9 nt 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.e5*ktmd2hzhmh79s/tijlamg,6 ff9. l1mk) u5 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 in 7v- :hw/ mopsv jc, rgtgi,--,l0etrritial rate of 1.0 rev every 2.0 s to a final- irgvg-r-te7,po ,,mrcs/:lv t jhw0 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 rotatinl)qnu*a 3m,f6h gy qa;g around a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approxima*a laq,n qygfmh u36;)tely shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of 6e sa*xhb+zm,; - x :qpq/oby7)swpb ya figure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum of the Earth54x 6:d pbxvrc
(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 cylindrlvh/:)doz ff tf4sd+3ical disk of moment of inertia I is dropped onto an identical disk rotafz dso4d)/+h3 vft:lfting at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at+.75e h qx*ej znzw(ht 2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotating merry-go-round platkgs v5z5no. 0pform of radi s5 kvnz.0p5gous 3.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an angular velocitypi oq2u8n50az of 0.8o0u5 8piq2n az $rad/s$ 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 collapses1 c/c5uhr 2 mld2pi8r ;dxq1gt into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost22ri hc;m1 pud 8 1cl5x/qdrtg 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 xy ey z8vadz7vv7l-1p.- g b7fin 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 vaguweo :hu 0s83:*nt$ 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, ingz/r / (kmidh2itially at rest, that can rotate freely without friction. The moment of inertia of tz(2//k dhgrmihe person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edh0(k q csn(g luch31b+kr0m;fdh u 6u3ge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of ine36kq0hb 3km0 sf hcuc(;g hnur+(lu1drtia 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 th2v*r zy w6p;4(qdlas:xn,inre 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 rov 2n;wr :pd64 s x*ly,q(arnizpe unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces t7*-ri pw4 pfd 2oasao+he Earth. Determine the ratio of the Moonr42wp*f 7apdosia- +o’s 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 rate of 1 m/50)tvozaxk f wngh 1dw r8)3f3$s^2$ 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 *kyq bk q4r20qin the shape of a uniform cylinder of radius 0.20 m acquires a rotatiq42ky*r0qkb qonal rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 kg and n * anmk8 ghzf-et m;j1n:ga;-diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub oangeg -a jf:k n-;;8mn1 tzhm*f mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of the reau;d/zqeau/ *q sya (9imee0e 1r 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,;5,l6luna-xzh4 et qq but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off l;6 q5x,aqn 4tl-ehuz no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is foryj o80;p iuz 8dcb;*qb it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical88 ob bucijqz0*;dyp; 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 an cqr 3etu;oe:5$H_2O$ 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 horizonf,8w :*-8hg3f ks unxrgn waq,tal 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 g+ozr,e ron( y5 h712mmz* 4qpxqa0qx M and length L can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear 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 F(hqarm410m nx,yoxo27 5ep rgq+qzz *ig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically qwap+ +01u shron the floor, and we want to exert a horizontal force F at arh suwp1+q0+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 speed v=4.2m/s on a fb2s0wtac7g( hd .t4l blat road is making a turn with a radius The forces acting on the cyclist and cycle are the wcgst4 0t7bl(ab .h d2normal 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-k( ag 9*;flgx/qgi re4gg rock into a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm afterg* x grg/q;9fe(ila g4 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 thin rov vh rr (r(y-uggq)x(:ds, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with (uv(xv-h gqrr(y ):r goutstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly whichh a 2;g1crlj-vtmys6. consists of two cylindrical plates, of mass2j; lt.gmcyhsva6 1-r $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 . wov* ,wzh-brradius r rolls along the looped rough track of Fig. 8–58. What is the minimum value of the vertical heio zb,vr* ww.h-ght h that 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 assumeh:(rsj3fvs/kte r :6v $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its ptu )1so21t jg0 jy:xvspeed uniformly from 90km/h to 60km/h The tires have a diametevopt0 jgx 1 :2jy)tsu1r of 0.90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s