A bicycle odometer (which measures distance traveled) is atsr;3vph3qrzwmo +.u 2u :r .lxtached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle w mqrv3 s3zxh:;p2uror +.lu.with 24-inch wheels?

Suppose a disk rotates at constant angular veloci 2 y 24ise+cvuxr:7exdty. 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 erc+u x:4xd iv2s2 e7ytangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value; msbt3hhh5 nf79qgm- of the angular velocity $\omega$ Explain.

Can a small force ev-f r/qkh(xv0d6qp6 wsp:;vil/: npyb er exert 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 thu+ 7bh aov18eo 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 thi1a8h7h+boevu s.

A 21-speed bicycle hib b7wrzt)/t/e b1j4was seven sprockets at the rear wheel and three at the pedal cranks. In which gear is it harderz1 bbrtw 7/)tej/ wib4 to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run 1 nze .fqkx/gvzq.z7/ fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of r exz. f7g.q//1qvkzznotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 0ffxrz - oso258–35) carry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero 202 t,3 mk3 jml it.coprg6c*fskst/l? If the net torque on a system is zero, is the,ko mltfckt0r/t 26lp s2. sjig*c33m net force zero?

Two inclines have the same height but make different angle -bwx6t )0y+ 5tuhoc9fa 7cgbis with the horizontal. The same steel 0o5)titccby bug+ ax 6-wh9f 7ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an incline. One kgcx3cp9bii(r1pea9sw9. v6h75s fw sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the great3w c959ws7iek1fa p pc hi96xrv.(gbser total kinetic energy at the bottom?

A sphere and a cylinder have t vku2(2 oelb5trk f:x,he same radius and the same mass. They start from rest at the top of an incline. Which reaches ,kvu r:et2bolk2f 5x(the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and :52fw cjm:wvs angular momentum are conserved. Yet most moving or rota wcf:w2 mv5sj:ting objects eventually slow down and stop. Explain.

If there were a great migration of people j+w61ps ym+f+ 6oovj gtoward the Earth’s equator, how would this affeo+v6+w1 gj6 m+y jfspoct the length of the day?

Can the diver of Fig. 8–29 do a somersault without hav8bj6nlh2z .q2tp)0cb kvr qe- ing any initial rotation when she leqn v8rlc h.zje)bt60qpb - 2k2aves the board?

The moment of inertia of a rotating solid disk about an axi;sa2qtt; jcw*v+/ch x s through its center o+wc;tx/ aj 2*tcvqh;sf mass is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool w7c4q b k ntk5ze5an c34n0wr+xg6/vmholding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velt +6m/nr c wn4 5 xk5v7zbqeacwk0g34nocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. Hol jzlqrx( m//(4u gxv:v e(qsl:pi/k)wever, one is hollow and the other is solid. Descqxrx::(q) ( /ge/jil(u m/ vz4vplsklribe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle poigfrfe+3 2;llknts west. 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 a2f gkrf l+l;e3t the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are stan d)8 haj,p8e3 xgdzug9ding on the edge of a large freely rotating turntable. What happens,hxgg3zjap898d) u de if you walk toward the center?

A shortstop may leap into the dndhvc3u09 ,j:opbfa )o-aha m1z *h0air to catch a ball and throw it quickly. As he throws the ball, the mp oohd913 )jv an a,bcza-:00hf*hdu upper part 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 conservatio)fewk8m34;rkt 9mo7rpa/bzi2 dfv y9 n of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operatr; k2 3 wz9/eokyrv8bfp)i 4fm9d7am te to keep the helicopter stable.

  Express the following angles in radians: (t,gewh b hf+o90vb76 kyzd+tl- l4kn+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 coincidence. Calculateto)b;rc1;kfmm6ax *2pw /eol m1gn- zp 32okd, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and n1ar-kfmmcoz;t ;)/o2g2 6*xwom pl1p kd3ebthe Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 3(:nl z 5ifo:; salm6ga) 8klbmhdqr :780,000 km from Earth. The beam diverges at an anglosmhldl5 b)i kma8 zlnr aq:g: 6;7f:(e $\theta$ (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 oz8* 2p - /; y1atbv; pycxonjzwvxh,3kff during operation, yc ,z;1xb2yw;a3 ok / p*-npvvt xj8hzthe blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child.vxe: i i9b(d:w If the ball makes 15.0 revolutions, whab: wivi(d:xe9 t is its diameter?    m

  A bicycle with tires 68 cm in diamet4yzcs r;9o 9vtwkat; :er travels 8.0 km. How many revolutions do the wheels mak4 z:o9tar;9 yvksw;tc e?    $rev$

  (a) A grinding wheel 0.35 m in diamesz u,))kb9xq:a; bpu gter rotates at 2500 rpm. Calculate its angular bspq)9kag)xu,: ;z buvelocity 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 complete revolution in 4.0 s (76f3f8wh kn ltFig. 8–38). (a) What is l68 f n7t3kfwhthe linear 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 velocity of the 6h02egg* cado 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 ;neri7dv:g (lfv ./xa8mr.zj of a point
(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 celu7koky 1hpk7,h97nm nv n:6 antrifuge rotate if a particle 7.0 cm from the axis of rotl7n 9h1moyvka,h p76 kkn:nu7ation is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpm to r6mxejea)7* .*0 x7tgh ltybf 280 rpm inf0br7h*) 7*6.ye ltm xtjaxeg 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 radiusjh7,w 2wm/d yyc+2jpw $R_1$ 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, astrd+ ycg nm4/m4ionauts aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start m/+y4m nc i4dgof their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter 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 220 s)aobv 1g.-ekc e)4ga8ux* tfv. Through how many ree- x)aaouv.)g1gbe k48c* tv fvolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4d*/pu5t;x j7 mrlr:b o500 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:+y ++oo mrwwi a whirling “human centrifuge,” which takes 1.0 minw:i++r+yw o mo to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 360 rpm i3f1om bauk .8qboz; j3n 6.5 s. How far will a point on the edge of the wheel hav.1m;o za3u 8oqk3bfbje traveled in this time?    m

  A cooling fan is turned off when it is runnz)as5ki)ch6sivc o (:ing at 850rev/min It turns 1500 revolutions before it co)ki os vs a65:h)i(zccmes 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 mze 1 9ycnsi7tp13:c muake 65 revolutions as the car reduces its speed uniformly frn7pc9iz ys 3tmeuc 1:1om 95km/h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as thec+ahqn/ 0sxq9yf/ f2r8fzrd 3 car reduces its speed uniformly from 95km/h tn /sr0fdaxcy3 h+f9fq r8/q z2o 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

  A 55-kg person riding a bike puts all her weight on ; l)z1,7i w .ekj6 jhmf8+ifucl nmnh5each pedal when climbing a hill. The pedals rotate in a circle of radius 171 k+)ij.f ; wiuz5nleclfh nm6,8jmh7 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a forckp7g+ gobt9) de of 55 N on the end of a door 74 cm wide. What is the magnitude of the torque if the force is exert+t 9po )dggbk7ed
(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 we0wc9u h-*7m c)sjwlnheel shown in Fig. 8–39. Assume that-*m7 sjc clu0w9when) a friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a masslxvv:e o f;9rg;ess rod which pivots 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 torquvxr9:o; ; gefve on this system.

  The bolts on the cylinder head of an engine requirjp zw8 :/yz:ike tightening to a torque of 38 :8 zwkyjpz:/i$m \cdot N$ 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 4 tfim :pot6,sa 10.8-kg sphere of radius 0.648 m when the axis of rotation is through its cenit s,46f:ptomter.    $kg \cdot m^2$

  Calculate the moment ooe8i0ygtvqko(e3nhx; - b 4y/f inertia of a bicycle wheel 66.7 cm in diameter. The rim and 4iq;e tex by n/o vyo-k83(0ghtire 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 th*dmd - c*n8+y7g1uscwqgm1hoe end of a thin, light rod is rotated in a horizontal c-1 8wgcgq d*o*c1mdymns+7uh ircle 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 pott e f6c9 n 2uaa2ndrv( x3s;tb3vcuz-e*er’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and the clay is 1.5 Nn; tu3- *9cvaeu6(bna2ed xr 3vc 2szf 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. 8* 56- pbgv)jst*xavbb –43 aboutbs x-pvj**)6 b5ab vgt
(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 atoms whf, 1k0.xa uyufose 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 pi;;)qs//b6gcwehem satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the sate;gh/ w;)m icp /q6seebllite 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 amy 9v d)uh+db: uniform cylinder with a radius of 8.50 cm and a mass of 0.580 kg. Calculhvdu y:)+bd9mate
(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, accelev bb ;0ottwu;3rating 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 tangentiallydw5uapc;o e .yk(w4k75ygb.g on a small hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 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 torqu5y5b .ake pd4ycw7;( uwk.ogge 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(i(p ( p,pxvq -kwgqv520f gls 10,300 rpm is shut off and is eventually brought uniformk-iwq0qf(vx,plv pg2s g( 5p(ly to rest by a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a 3.6zt pmfa7(6 n1.i1k thr-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 tj,*/ q nxx+cxphe forearm, which rotates aj*+x n qxpxc,/bout 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 blade can be considerekqz*j9cmenm/k 3j4o 8d a long thin rod, as shown in Fig. 8–4nkq9 o8m *z j4/3ejmkc6.
(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 massik9zy p68.bevu 4gvw+g/v nn/bh 4 eu9es, $m_1$ 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 four full turnsda d,*92aom 0huy9qse e ) :*mijqoy/l (revolutions) and dao2u)o9e**:aiqjy,l0sdm /9 ymqeh 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 moment of inertia oh )jyuy,;c) 8gegwdubn 3qs6*f $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque of 280v0 6(ms rwlpx( $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3jw vedoz w 6.;rx/)e 3 ijltr(t+0jbv0 kg and radius 9.0 cm rolls without slipping down a lane at 3.3 vwtlz.oew ; er6b0r 0d3)jjx+i (/vtj $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the s 7.pt-ldpi p i-44psteum of two termsetdl i-sp4it -p.74pp,
(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)oj7+khr6 j)nps 2l tn.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution 72n+ ) spr jtnjko)hl6per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from reskmdql)ujq.n gvjd3 11 a5*4qpt and rolls without slipping down jv3u5 pgdk4 .laqn11 d*mqq)j 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 baaf9dwb 7efux( k h,6r5t38tpc lanced vertically on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the grouner9aut8w5pf(dx f7b ,6hkt 3c d? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotatt/c+-l vip*e ding on the end of a thin string in a circle of radius 1.10 m at an angular speed of 10.4i tdlvp+/e *-c $rad/s$?    $kg \cdot m^2$

  (a) What is the angular mmph-( v;rlmxj) /ise 5omentum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm )vh- (i/rm xelsjp ;m5when 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, t;(te(ju5mwbt 4 uyw1v ,,bq don a platform that is rotating at a rate of 1.3rev/s If he raises hisqd u5 ((4,wy,vemb1w ttuj;tb arms to a horizontal 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 rgrryorl ;rnp,1i em; uu74cor+k; 6ev p1d0 1Fig. 8–29) can reduce her moment of inertia by a factor of about 3r 1ul;vcuok4;o rp 07,d1mi 6regy;rpn + rre1.5 when changing 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 rotril(g e ,;g5 :j clfx,(;ogeacation rate from an initial rate of 1.0 rev every 2l ec:5 ejax(o,l,c( fr;;gggi.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis through its ce5bz 3jl5 h6cc* :u/ i0xylubgsnter 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 bs0ci /ul56xu5ygh3j:zlc*b the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum ofq13d cctgd9kj :;ymk 7 a figure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum ofd* x; dz20bg27l:avtwoyo e7 f the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is dlp-ky*91 aqfs ropped onto an identical disk rotating at a p9 f-y1sql*kangular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 241mkk2o8n9a6 v kebis.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 k).:u9p3 0fht5:vxr.gft o pirxr)m va rotating merry-go-round platform of radius 3.0 m anv hx:g v3p:u9k. )mtirx.orpf0fr 5t )d 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 velocity of dy btl84 j7by7 5ptul/0.bp8/ll74y bt5u7 jytd8 $rad/s$ 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 uk,i l3 93gukqzrz m;t22goj)z 9/wal collapses 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 lost mass carries away no angular momentum, what would thurt2,lw kmq/kaz gi ;u2)9z o39 j3lgze 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 s2yx94bl 5vu;xtwo 7jin 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 k byipnutq)4xv/5ah jp-, b 27(gvya5$ 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, ;xmt0q6ie k .rinitially at rest, that can rotate freely without friction. The m6mitq xr;e .k0oment 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 edge of a 6.5-m diameter mermz (0slgfcc fd:yy1) ;ry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia 0;(gcl:yf f1d zscm)yof 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 rn rdygb 1t 3xsuyoz wsl9(g7y5m95k9. ope 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 rope unwinds fxw9m sd( 9sluko g5ty1b z5ny9rg73.yrom the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth suchqg40pt 5qej i1 - c. ;olzt4wohd1rr6n that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum.0oqi4 e. p15qrttj6c-lh nwr1go;dz4 (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from 2m .i3szzaxd u tzxl(9m+ fc;7rest 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 uniform cylinder(dgp6ec ud4 v/ of radius 0.20 m acquires a rotational rate of from rest over a6vcped(u/d4g 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 soc*fbhgjv1;nsftv xe)-4 3 0 c;vvt1mjlid cylindrical disks, each of mass 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 calculate the linear speed of the yo-yo when it reaches the end of its)m1vj ;g 3tv*1e jh0f 4;nvs-x bvcftc 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 th asy2tq *ty4pn/2dp8ge 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 wik2krm,jaa0,6/g q k4obd0qm mth 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 stk oa,bjmm,6qkr0gmd akq20 /4ar 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 itn4mzwz(j3 bxlo c.5 gp+;w gb2 to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass l24m 3wgb5z pn(gxowj;c+b. zof 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 illustrat4kr i)x ,yl/yles an $H_2O$ 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 surfacmf:g 4zwh s)itk f0gwv5s.hae21p/ u*e at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot faf9/ p 4.u:c3dzn1zpzrff8qh reely (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 acc 38qpf.zd4/zucph aff91znr : eleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is stanr8..evhix k5 kib,f)h6og0xsding vertically on the floor, and we want fg,)i.kkoe0 8h.h5 ivxbx6rs to exert 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 p*+kp,teygal re o6;/ with a radius The forces acting on the cyclist and cycle are the norme/po rkg +*,6 plyea;tal 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 (iix 3;fmwp n) h2hu8h r1wkyvw,y j86sling of length 1.5 m and begins whirling the rock wiy f)krw3i(y;px1n6mw v2 h,88hjuhin 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 skater’s body as a solid cylinder and her arms as thin 6:njv h;xf nxhk mh:9*rods, making reasonable estimates fojvm:hh f nkx*h:xn96; r 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 assembly which consists of two cylii tl:berb*99bo wp,c/cqd- 4lb37u hrndrical plates, of w3b d hul b:icbrclbq/7 e9o4* ,tr-9pmass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 ro2ee)pnbk ,p7ql+ ou o*lls along the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point of the k+pnebu oe,2l7)*oqp loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not qvri-f kfcu /vzz5(r;f* 9u2dassume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car ref;tqhwj d2py3h +lo)5 duces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each +h35t2)yo;qp lhdjw ftire? $\approx$    $rad/s^2$(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