A bicycle odometer (which measures distance travqd k/w:y1rm ,wp )aft6eled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if w,:)/rqa6p dwfmt 1k yyou use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a wlg uo(6f, m5b3l,ddt point on the rim have radial and/or tangenti35f,(llub 6dowg ,tmd al 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 5k 4h9x *heknlbe described by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exertym-7z-qye g++p2o lro a greater 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 to do a sit-up with your hands bed-*,zjza+deq hind your head than when your arms are stretched out in front of aj*d,zezq+-d you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the pedq, +zs iz9 h0kavk,t /f:,vadhal 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 ,zi,+hvz tv 0aak/:fhdsq k9 ,a large front sprocket? Why?

Mammals that depend on being able to run fast have szbqv dwa 9nhac1)zbz7:8 rr(1j pz:8blender lower legs with flesh and mu71 rj bwz dqb:9n:z88bapv1h ()zzcrascle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, ni* o 4q ez+re4hj)7 ;d*uou:pwgms y.carrow beam?

If the net force on a system is zero, is the nepup;zvfm5 yr; yv2w va 8.bx/sxp96-ct torque also zero? If the net torque on a system is zero, my 5wvf cp;9p 2vvp.suz; r-6 8axyxb/is the net force zero?

Two inclines have the same heighkn6x4ns(wjke7alo0p7 /*.w nf n l 5agt but make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of thesfgj47n 6.(07a/en aokl *nlkpw5 xwn ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (fromu2rqm jlb; 0ndc-wol)k.r/ ,q rest) down an incline. One sphere has twice the radius and twcb .llw k2u,om/r ;n jdrq)0-qice 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 hiiat9df 4dl/5e/l.rle w94djave the same radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kineti9 .ai/5l9f ri jddedw/4lt l4ec energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conta*4 )maxwlz1i.k/xld ep 6m-xr/- uzserved. Yet most moving or rotating objects ew .*xmmil1/xd z6p4x/ )u kla --etrzaventually slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equator*t- koa+i q8sd, how would this affect the a *tsd-ko qi+8length of the day?

Can the diver of Fig.stnj9g6g .9 oj.pww) 7fv*j vj 8–29 do a somersault without having any initial rotation when shnvgw9)*v9 fjgtjp 7. .sjj6owe leaves the board?

The moment of inertia of a rotating solid disk about an axis through its conp xeu;u)bh,;2( w.1-rubbfn lo c,zenter of mass is b.cur nb-z2npuew,xu 1olo f ;h)( ;b,$\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2-kg mass 1xx m-tvhsv4 l9m:y*vin each outstretched hand. vvt9m xmhs*y-:x v4l1 If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identiwvkghyj tyn8 08f;8,3 y mhr,kcal and have the same mass. However, one is hollow and the other is solid. Describe an experiment to determine which is gy0nv8 y8,3trw;mkyh,hj k 8fwhich.

The angular velocity of a wheel rotating on a horizontal axle points west.(ihyo r:ulnz1-ws wyos/05)f 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 on )oy/y5u-l fzssi1wr:w(0 hof the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating turnt2 rqd, td7j(.3,fuqmg aukt e2able. What happens if you walk toward the cenu2mrtd,.3f td,kuj7qa(e2 gq ter?

A shortstop may leap into the air t akv afoou 6e.0on:ka57+vhf,o catch a ball and throw it quickly. As he throws the ball, the upper pa 6fou fkak7avvh5,o n+:.a0oert of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, disingz-asga(wv *6i2 t8p do q.5sa -0oycuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the se8g*ngoap(qosi iz- a a ts65d.2 -vy0wcond propeller can operate to keep the helicopter stable.

  Express the following angles in ral6)5 j9sct *b meeb)n/ ivbai+hl3xf7dians: (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 amazing coin;b9) *)rp ciec6xvhyqcidence. Calculate, using the information inside the Front Cove)h69q cxi ebpv*r;y )cr, 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, 380,000 km from Earth. The beam dioehq;d1skbd5r) s /jz)rc., yverges at an ancs1)d ;eb,)dhoyrj/ q zr5sk .gle $\theta$ (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 turned t9dxw7u 2axp vx1,3e zoff during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the b1ztexd2px7,xuva w39lades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the vgl wx2ni//-oball makes 15.0 revolutions, what is its diameter?ngi- /vox2lw /    m

  A bicycle with tires 68 cm in diametx h*eo oz1txnwp5-8+:mt:b a w9it7wcer travels 8.0 km. How many revolutions do the wheels makee :pbcozx n8 :tx*17wtw-awto9h5m+ i ?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at sqc/ pm5zm /q ,jobv0jzk .. 6-nwy-vt2500 rpm. Calculate its angular velocity p-josz vn.tqybzv6c,m /q m0wj5 k. -/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-round makes one c43tw;1.uqj e:me h,,r 7n- ywsrtkhm pomplete revolution in 4.0 s (Fig. 8–38). (a) What is the lqm,7:4em;kh.-3 hnr s yu,ewptwjr1 tinear speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular veloy+gf: t:zj p2pcity of the Earth (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 poin- su;/7zdh8wh; vwrjr t
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.0 cm ,m dd akk)hr)..eh l+efrom the axis of rotation is to experience an acceleration of 100,0ekh,) k.dd .mel+)har00 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 10, 56fu/dqgtd *f waf3/,chw(s he gzo30 rpm to 280 rpm in 4.0 s.d c thuw z3sd/(,05,h6w*fq/g o afgfe 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 radiuse1gowqox9.d 2 $R_1$ 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 spacecraqi)e 4z.m *cxnft 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 int ce4iq m.*z)nxerval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpm in 2:j3yp *v) n3f-inlohg p9w. ag20 s. Through how many revolutions dij.ypg3)lg *hp-vo3n f9w ia:n d it turn in this time?    $rev$

  An automobile engine slows down from 450nmt a/7;;h.q 3 niwoov0 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying11orhbk-xs1 m highspeed jets in a whirling “human centrifuge,” which takes 1s-1 rbmoxk 1h1.0 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 diamenz /;;8cq4x gt -udgntter accelerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled in this 4 dtn/8qt ;cgz;-xug ntime?    m

  A cooling fan is turned off when it is running at 850revddclo,1on,z* fc fy 52/min It turns 1500 revolutions bef,nfoy2 *d5fl ozcd1 ,core 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 spy r5u.z1awraze d lc0-0 w-is;eed uniformly from 95km/h to 4zlrw wczu y0r0 .d5;--iaae1 s5km/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 revolutpe/nxp s3wyg zr+:7 l0ions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diametz7 r/xs:w3en0 +glyp per 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 weio 8hwfqhy2;x9 c/ mun:ght on each pedal when climbing a hill. The pedals rotanm c fu9 8xw;y/qho2h:te in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 Nqx bpev0 .slq8i90 ck+ on the end of a door 74 cm wide. What is the magnitude of the torque if the force is exe. v0qb cqk xl9i+08pserted
(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 wu d3w-3dg hiqe4v a3)/te9f/ is*:k bs zw5dodheel shown in Fig. 8–39. Assume that a friction torque of 0.ugosf h dz /a*d 3es3w3-t:d9iwkeib) 54v/q d4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are +ghm -u0vs+cm 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+ h+-0mvcusg m then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightenxboeb ,sk;a0* ing to a torque of 38 xkb,bsaoe0 ;*$m \cdot N$ 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 sphereoml 3-o2).mtshyw0ub3 7 zgp o of radius 0.648 m when the axis of rotation is through its centero)wt3y.mpo- bsu372 m0hlogz .    $kg \cdot m^2$

  Calculate the moment of inertiat ji2bu+qi;, n of a bicycle wheel 66.7 cm in diameter. The rim ;it+, n j2uqbiand tire have a combined mass of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotat7rznurb ee7iq-s 7m3 n:kuc .m3q+.ijed in a horizontal circle of radius 1.2 mn7r ks.eu3u.b i-j7z qrm:e +nc 73qim. 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 bowl6/g46 wqf/glned vb(o on a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and the clay is 1.5 N totawvln6 b//6q dfeo4g(g l.
(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 ob*y fo1v3epd1x jects shown in Fig. 8–43 a3p 1efx1 vo*dybout
(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 v-a4s b*yj ibo: ;kv3ddv+4a.wx zm+m 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 cylindrical satellite spinning at the correct rate0r18mqg d/wmz , engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radizq0/r w1g 8dmmus 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 unklz6 wl,ehc5. zf* ed(iform cylinder with a radius of 8.50 cm and a mass of 0.580 kg. Ca5 ,lc( zld f*.hzew6eklculate
(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, accel, du)i bztzt8;erating 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 tangentiallmjcw df+9)uk:p0 llf5) b2vm jy on a small hand-driven merry-go-round and is able kbwu2l+)m9:lp5jf df j 0)vmcto accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. 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 -.l lx i6z le1o6u927u lfiztqat 10,300 rpm is shut off and is eventually brought uniformly to rest by a frictionauo7f16z z- 9lixu.6l tl2iql el 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 ac;+)xqu o bhro** lpck-celerates 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 action of th 1vj; fr-d2w 3kjrifk2e forearm, which rj;v r -r2i13wjk2fkdfotates about the elbow joint under the action of the triceps 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 b5xke8 io nmg4 1bz:riz osn,zed1x78/lade can be considered a long thin rod, as shown in Fig. 8–46. x zkedi/5xm84gz r1n1ze8 os7:o, bin
(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 cotetoel9vzho/ns 93 62nsists 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 rio;to1cq3d7 rest within four full turns (revolutions) and releases it at a speed ocqdo 1 r73;iotf 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 63cw8a 6pwlra 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 dev8covpsj n7/ w1elops a torque 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 ,v pb30h*yk gld8wpe* and radius 9.0 cm rolls without slipping down a lane at 3.3g whd 3**vpy08bkp e,l $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to3; jcjdqlc6b . the Sun as the sum of two tjcj.q;c3db 6 lerms,
(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 16jje q4pt56+cz 40 kg and a radius of 7.50 m. How much net work is required toj6 p+5qe4z ctj 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 rest an ck aqrxo, k+b6:n-xl4d rolls without slipping downxra ,bo4l-xk: kqc 6n+ 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 0qaqh u4jzu3m z6n05k j)j(mu fall and its lower end does not slip. What will be the speed of the upper end of the pole just befo 4)qmna6umzz50qj u jh(03ujkre it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball vr2( 3gmqj +j9ohn5qp rotating on the end of a thin string in a circle of radius 1.10 m at an anguj 9ojhq +g 2nqvp3r(5mlar speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylinjv84*iy-tebl:mb v e:0 zn 2bxdrical grinding wheel of radius 18 cm e y*e ivt8-b:n: xb lzjv4m2b0when 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 c: poy kh;akwx12lr5z7:ft-u 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 rotatz:p 5wl2;fk1 x7 tryu-a :kcohion 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 b 4l+b zsa,)648s mz3hz rwdb 4a/htwdin Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If /b)szlb4r8,hzz dhwm+ baw434s6 d ta she makes 2.0 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/twr9os1iu (i0+az k z her spin rotation rate from an initial rate of 1.0 rev every 2.0 s ti(roki0z z9+/sutwa1 o 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 freqzd56gru vey64 uency of 1.5rev/s The why z 6dur6eg54veel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinning at 3iyjv66 cnwv *a:50 ckc.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 oaif77:o 4f wptqv0 s7z1d od3if 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 inertjlt2sg;yg*w366je4l li r iz6ia I is dropped onto an identical disk rotating at angular *6zlewi2y3 t6j;jgi lr6lsg 4speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns o*i2p,t: x 8bvfbz0o fat 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 thqasl;3rv)2 /q;cepf jxa k6 r+e center of a rotating merry-go-round platform of radi3se qa; ;jrfxp)krcla/2+vq 6 us 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 isj -b,k k 6o3aca,(qluv rotating freely with an angular velocity of 0.,jko -6,3 kcbaaluqv (8 $rad/s$ 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 i .26jp mgnrc*ns 4r3agnto 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 lost mass carries away no angular momentum, what woul6sm jn g 3rpc4ra*n.2gd 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 e 1ohlkgq2e5,lf ,9ylvu.uk2 axcess 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 2urvw3)b 95lrf(oewy $ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, tm05kh l:u h1sqbcxiiuh 0i)78hat can rotate freely without friction. The momenhqx0k cul8)si i mbh0157 i:hut 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 person oy)rp q)qxv2 + rp+r0tstands at the edge of a 6.5-m diameter merry-go-round turntable yoqp r)2tr r +p+xv)q0that is mounted on frictionless bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of ropewi882szkrc b4 66clsbo 3t5yo rolls on the ground with the end of the rope lying on the top edge of the spool. A person grabs the end of the rozosb2 k5ilrcy46s3 o8wc 68tbpe and walks a distance L, holding onto it, Fig. 8–50. The spool 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 that the same side azlbzvv9+9 o1oe q ,: xfo/uwp3lways faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) toelq1/ouozxv w9 fv+ b9:zo3,p 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 oavql +b/o5ydb jbz,7(f 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 p8(t h7ok9 +tcm. xdwep ph)x1of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the t1t pxdxpc8+te) .hp(m7kohw9 orque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of masas 3n0 tj-66zl jx6qifs 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to cs6ja3zqjt-i66f 0x nlalculate 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 rear wheelzksvg j.k; qz6 dm770i ($\omega_R$) 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 Su 0):x.lhydfvx z.fa7gn, but with mass 8.0 times as great, were rotating at a speed of 1.0 revolut yz.xx lff)v d.0hg7a:ion every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use energy st6y)-mn67l jlu0kwb naored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a unif- ly7knnj a60lb6m)wuorm 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 xt0jbd h1e fiw-0x ;7ian $H_2O$ 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 m/k2m)q9gbme p(cx*o g +)ylqspeed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., we sgry1y3.sy6b6l 2l4(ryvyq f si /vq3ignore 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 (ls6bq.6y3yqssvvl1 ygi y2y4r3 r /f of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on the floor, andr lg :a es1 wxe()38x44rzstfw we want to exert a horizontal force F at its axle so thwxet4l4: se8wsrfra1(g 3z) xat 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.cy( ;;utsoo 4av yolg4. n2w2o2m/s on a flat road is making a turn with a radius The forces acting on the cyclist and cycle ao4ntu2w s42oa ol ;o.cgyvy ;(re 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 z5;oajvjq-z3 f yqd:zq-iu7* and begins whirlioq dq*-yzi 7a5fvuq3zj j z-;:ng 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 torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater50 p4zkl slej-’s body as a solid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular spee-50ps zlj 4lkeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindri6ih;)i (-mx,- kv b i5orxkffyq7jac0 cal plates, of mass vx,k6( f-qhamixykr)7-i i5 ;ocb 0fj$M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 t8xk2 v6-vj c+ k /pz4f+imcxpuhe looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drm+ck6zcux8p-/4kvpfi x+2 jvop 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, butt(p1gnco ;r+f do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions xt5 4v(v vdixawxg/d 7ant86)hm5)yj 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 a65v8xvx5d( migtaw avx4ydn )7/jh)tngular 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