A bicycle odometer (which measures distance traveled) is attached ne(qcd/ rf06)nu8jt nb7. 2frksr8kvf g ar the wheel hub and is designed for 27-inch wheels. What happens if you use it8djkv8t62 rrbq7(u)nrffk .gf c s/ 0n on a bicycle with 24-inch wheels?

Suppose a disk rotates at 2y9qulkr, te t 8qy(6yconstant angular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear ac8y tekyr 6l,qt q9u2(yceleration change?

Could a nonrigid body be described by a ova tci tmd h22qh/t7/l) b5o0single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than zcyvw g(//s t8a 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 behind your h:y hlan x.6o:eead than when your arms are stretched out in:nl x: eoy.ah6 front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three a)tg 9(rdkl47( bobmh ut the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprock(hd)4ubrt lo (9b7gmket? Why?

Mammals that depend on being able to run fast haf hvo7oe9x(9ne6,iqcy uf6x. ve slender lower legs with flesh and muscle concentratedeoov9 f,q9i hey x.6xc 6f(7nu 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) car7vo5 y ( tq)bi2k4jl g9w-fbbqry a long, narrow beam?

If the net force on at yzbv4mz sb e8pepl;- :5(rgzih*0g3 system is zero, is the net torque also zero? If the net torque on a system is zero, is :4zgle5y 8m(3hti*0r zb;pp bvg -ezs the net force zero?

Two inclines have the same height but makz((uy4q,bd mze different angles with the horizontal. The same steel ball is rolled down each incline. , qb4(yzdu (mzOn which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (tdy9- yt ag10cfrom rest) down an incline. One sphere has twice the radius and twice the mass of the o yt-gycat 190dther. 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 ehbp91qn (jz.the same radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic enej9zn ehp.(b1 qrgy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular mom106p* aylrisqentum are conserved. Yet most moving or rotating obje *0 aqpiylr1s6cts eventually slow down and stop. Explain.

If there were a great migratioms08gy(+s p ckn of people toward the Earth’s equator, how would this0 (csmy k+8pgs affect the length of the day?

Can the diver of Fig. 8–29 do ;-mc )5pwo7 vrr0 ()lovka.l tybi4lia somersault without having any initial rotation when she leaves thrlw)bl ) -ooiml70a;tvpirk 5v4(y c.e board?

The moment of inertia of a rotating solid jkr z 5kk-a.dk;f5z- nbo 9mz5disk about an axis through its center of masz j k z5o;rkk.b-fn9 5d-m5akzs 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 r eaq:*zhpg( 1tsi w(4fotating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will you 1q:4a eis (hp(*wtgfzr angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. Howeveybw+;fah jyj5t a4q .9r, one is hollow and the other is solid. Describe an j .yq 49fw+ajy;5bha texperiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points west. Iy.q/;iq urq+(7gc5lc qttv3 in wht;ql. r /(yu t3g+qqiic57cq vat direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edgeijj*y))1jmsy of a large freely rotating turntable. What happens ij jjsmy)*y1i)f you walk toward the center?

A shortstop may leap into the air to catch a ball and throw it + kjqr fsl9ah1jdg1;, 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 opposite direction (Fh 1jkqrjs9,a+ ;g l1dfig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, disctsk,ar pxl7qe9* 3vb/ uss why a helicopter must have more than one rotor (or propeller). Discuss xb3qvk7 p,tr*/ l sa9eone or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles im (xr2 sjb3t-xn radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of ak6 25o ek9wbe,a5/ji,bpjt kpn amazing coincidence. Calculate, using the information inside the Front Coa9 5wji kj, b26k,b5p/tko peever, 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 Eart0ic;0e5a /fkoq5 0g mwj izn)lh. The beam diverges at an zemjlk 0i ig0 5nf;c )/o5q0awangle $\theta$ (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 r:qw7 abh3ddf k b krmo z;-4jv8z0t.(date of 6500 rpm. When the motor is turned ofmb 8q 3o 7k-ba;thdv.zk(j:w z4f0drdf during 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 y.. - o2qiurl,,p0mpae )ekyz a level floor 3.5 m to another child. If the ball makes 15.0 revolutions, what is,piqu z y.ym2r 0e)-ea,.kol p its diameter?    m

  A bicycle with tires 68 cm in diameter travem,4l f ,zmvro8ls 8.0 km. How many revolutions do the wheels make? 8o lm ,vr4zfm,   $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate s30wu9xoymk*y 5g v*rqgn:) h,rsf7 yits angular ve* 3 vy7 smr*ky0h xfwo9g )n:r5ys,qgulocity 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-rgdsm) c1ic2d 4d /xne+ound makes one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m iec)dc +4/nmx s2g1dd 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 t)7 gz0jeetg/9o/.me fzea .ykhe 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 point so7;ee; +tr sjket*z8
(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 6g1guy e+ym1xif a particle 7.0 cm from the axis of rotation is to experience an accele1myg 6+uxey1gration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly abou:vmv 7 bxix-z9t its center from 130 rpm to 280 rpm bi-m9xv x7:v zin 4.0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radius 6 o/tk6m:6hjm n1udmt$R_1$ 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 v/n nk*+jc/1,fj1qdgncsyc4 Moon, astronauts aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they acceleratedn kv q jc,d*+n/jg ycc41s1fn/ from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest t;zmlk(p * e+rlo 15,000 rpm in 220 s. Through how many revolutions did ite *+(plm l;zkr turn in this time?    $rev$

  An automobile engine slows down from 4zq)44pg1y0.jy9)sc qwp sxz r500 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 flying highspeed jets in a whirlinf d5vgl5y56 ysz:zg y6g “human centrifuge,” which takes 1.0 min to turn through 20 comd :g665lygzyfv5ysz 5 plete 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(d4- f9zntv,r 3khrlh50plt o from 240 rpm to 360 rpm in 6.5 s. How far will a point or0-( hh rtt4k p9l,v5ldofzn 3n the edge of the wheel have traveled in this time?    m

  A cooling fan is turned :phul)e7d, t s8/ygusoff when it is running at 850rev/min It turns 1500 revolutions before itu h8/ pgeuyss:td7), l 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 reduces2wt1- q)m6a- bgkvmpx its speed uniformly from 95km/h to 45km/h The tires have avxa- mw-2k1g 6tmpbq ) 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 revolutio6q1 r8qiu rvls j.k47y aezv95ns as the car reduces its speed uniformly from 95km/h to 45km/h4k5rr9y vi szu8v1 6jeq7.aql The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weiti 4 zy9a127kn d)cxsight on each pedal when climbing a hill. The 7i9tsayki c1x nz)d24 pedals rotate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on tt2a lq7is 9 o+t)r6 1toec19icn2f dqlhe end of a door 74 cm wide. What is the magnitude of the torque if the force is exer)ifs2qt l7t 9roqda l1c9e+co1ti 2 6nted
(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 Figt.ut 2d :4j:zn(jagn n. 8–39. Assume 4t::ad nn.tnzu2(g jjthat a friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of ma -5zqr-ugx ,bppj x+r)ss m, are attached to the ends of a massless rod which pivotr gjqu,r p )px5b+x-z-s as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cyl0urj 0c,bop7ta zdq,1inder head of an engine require tightening to a torque of 38trbju1,p 0dz0 qaco7, $m \cdot N$ 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 mq u0a5dsr3 t8ne-9if7r w tvv( when the axis of ro8 f- 9(v wed7qs3r0rntv5aitutation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. -;0 mbw1lc9gcux d vztx 0e26uThe rim and tire have accwgdb1v;e 9zx-u0l 0tuxm62 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 8j vsveoa f)2;rod is rotated in a horizontal cire v)fa;s82ojvcle 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 potte( m70gob q-mglr’s wheel rotating at constant angular speed (Fig. 8–4 (-lobgm0g qm72). The friction force between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point objects shown in Fig. loftm6i 1supq9j p65 78–43 abouj 1uliqp56 7o69sf tmpt
(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 coa-ize)omh7; bjqr6t9x o9t14 8szwkbnsists 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 cylindrical satellite spinning at tharg; 1w3djp3ue correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite ru1 ;w3p3adgjis to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a rad1n+v dhx0x m:bius of 8.50 cm and a mass of 0.580 kg. Calcul+0xbx :1mnvhdate
(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 g.3f6obqu9dma 3x.ov9l1 qgn from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven merry-go-round and i lr kje7gdsjk9t wk jj2-**9bq:1c )jts able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2jd 1jj*j tr7skqlc:kgbe2 ) *kt-9 j9w.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is event0n:y:,+mey 0gyuk9hpt .zf mnually brought uniformly to rest by a frictional torqun mgy:z0yph,y 9f:0tmeun.k+ e 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. l7a;+(le kx;lqg;r- e8 wiexps:s t4a8–45 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 thrown solely by the ty8b35m1wa6 / bywsbr action of the forearm, which rotates about the elbow joint under the action of ts8635/ 1bb marywywtb he 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 bcjmhkba4p 7-19sf z+u1 cvc-slade can be considered a long thin rod, as shown in Fig. 8–46.47b1m1-- u chs+szap c cfk9jv
(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 consists46)0wsn 2c wjxy0 kuco of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest within fn ku;ri ; y:r1obu)a3vhrw(t.our full turns (revolutions) and releases i(t)no:r iwu.k3;vrbah y u;r1 t 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 moment of inertia ofg4kh kx4 j33mn;y ;pru /ubi;d $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque oi0 zkbpwk 30)vf 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7. h+h33c9mxcp ,d +tricjw3n. w3 kg and radius 9.0 cm rolls without slipping down a lane atnxh3dr9+wtc,ch pw+jmi c33. 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Eartz lgq3e,r sx 11eat;, tv;g rqbq-q9v6h with respect to the Sun as the sum of two t;sq1 tg3x,,qvq;19b azlgtr ev6e q-r erms,
(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. Hot+yqha xl/a3+fx l z:8w much net work8ly z3+lqtxhxaa/ : +f is required 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 1xuvzc9mel7(b 0w0r4 g7 - cy efawc8/i.80 kg starts from rest and rolls without slipping do8mg9xwaw yel(f0 z0e/c b -ur7v c7ic4wn 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 bal6xiva yb4/ o)oanced vertically on its tip. It starts to fall and its lowerix6b4a vo)/yo end does not slip. What will 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-k(m 0d) j,rerya 2i(lqyg ball rotating on the end of a thin string in a circle of radius 1.10 m at an angular spe i( yjm,dy20(eqrl) ared of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grinding whee qz*r,zbhd 2(phyanx+f5pro4+6e ,lr l of radius 18 cm when rota f6 noz5r2hre hlzqp+*b,(4ar+xy d,p ting 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 r3a i+)fhlogv, otating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreav+ 3ofh,gla)i ses to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown ix3 b-ckx/3z;aqd c8. gjxixa3 n Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when chxd3 ic8qaak3/-3x b;cz x.xgj anging from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotationjt(ezwsx)s+:w z w(:y rate from an initial rate of 1.0 rev every 2.0 s to a fiwsz(j+ew)z w:s(tx : ynal 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 arig2u*+ /xm3 +zgjjblt.u v4m1d s ye3sly2d9t ound 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, 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 3gz +2gtylej ydi3dt/.+s4u2 1bv*lu 9msjmxthe clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skate d/7ml 0mej7+aduf5ior 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 mom/fnz* tqfesx7 ( +/ryjentum of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$

(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of vbh3vb k99tzf h-qd sl 8b:(9binertia I is dropped onto an identical disk 9 qb(9f8b dk:vh s 9-3bhblzvtrotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4 v pk-giv+cg(7 $rev/s$ 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-g 0mk (7ax/a drk*q::nruawe o4o-round platform of radius 3.0 m and moment of inertrq:wk: e adx7(uk4/ na0r*maoia 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 free2 b,o .qiytlp9fkohrk;sk5 (1ly with an angular velocity of 0.8krils .9ky12oftb;,po5hk q( $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, losing about half itsz5:d0ozjve;d f q7go4 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 would the Sun’s new rotation rate be?(round to the nea5e oq;0v4do jz:zd7 fgrest 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 excesfhvo1ve ;,+ *: lxgrcrg), wdys 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 y)f;y*t e8-wefyjzrz+q f22r$ 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, inpj y w8otb e,-e0/m zogmh,x 6(n9:hnjitially at rest, that can rotate freely without frictionxbgzjy:jmn 8-hnee w/t,6 op0m o(,h9. The moment 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 stands at the 0oojpu5 i;z.dedge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionlesdo.5 oziu pj0;s 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 rope rolls on the ground with the end of the rope lying onc -if34/sdm mf the top edge of the spool. A person grabs the effd/3c m 4i-smnd of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds 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 the Earth. Determ *vx. v*fthoc/w(k udx4t) 9omine 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 p*vf xt4uvw*hx.to kdm9)oc/ (article orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from2,5ii 0zz0hbb,oxyowg/,z m q rest at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniftu8*:mq)um mporm cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate th ):qpmmm8*u tue torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical dlm; 0khkptf.*c lqez3sl8 3g; w1caf6isks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub fclfc3 k3t ;kalg p;hsqe0ml.6*8zw1 of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of ;yz o;( 7v fy(zcu3hbm(gblg -the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but sfv-4 2.gvqi*kmda p )with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. ms*gdp )4.v qv2ika- fIf 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 stored5;klf- -a* sgjfb ri9o in a heavy rotating flywheel.ogifb9 a- k;lfr*sj5- Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates anx2na/kuskei: 7np)km7 6 omm) $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal surface at speed v=32 ryg(7 1m8zdh6qjsm bq n5j k0bxc1;zc4on5i.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 ignore frictix je ,6d6d3tfovx)1qion) 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 mf6xqd txd,1 eijo6)3v ass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is sb3;mdi1bs xi veu )9--lsrw7 wtanding vertically on the floor, and we want to exer l b-r93xwe1id-i bsv)w7;ums t a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a turn widwnx ma:vc2y fv,/yh ) eb43s*th a radius The force:fyvsa/bwn 2d 3 )h*vy ,e4xcms 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 whir1kxu:h0fwq/vh a 6l1hh7; xtjling 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 1 xh ju hlf0:v1ak7t/;whx6hq from?    $m \cdot N$

  Model a figure skater’s bo 1 s/l*g3t i12qvdlxyydy as a solid cylinder and her arms as thin rods, making reasonable estimiy3td1*l 2/sg l 1yqxvates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch l 5*)mo)s6 3asy,3s1y mxa)wzlim9nmc kk7qqassembly which consists of two cylindrical plates, of mask6 )ismmloy lmq )sw9a)nm7x 5*k33 ayszq1c,s $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the loopetm-uqp gp,b2jw p /.q-d 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 withouttqu-j-q ppm2/ w b,p.g leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assume x vio75o6zah3 )+ptyq $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces itq3d37 fd,7 lsmtw9zyqs speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration o3lfs3dtqy 7q9d7 m,wzf 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