A bicycle odometer (which measures distance traveled) is attached near the wheekt 8ht9+asr5 apo* wx7l hub and is designed for 27-a8tt5 ahor7*pwxs9+k inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at cutuwih 94,l .bb*pd r1onstant 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 linp,whi tbubd 4.*9rlu1ear acceleration change?

Could a nonrigid body be descrxu:bdqu x r yks/31,t*ibed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger forcuo) z-, )yrdhbe? 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 yon, 4w w 5:rabn gzjd+4 ,4x(bw-jbjucxur head than when your arms are stretched out in fnxj : +r -jgj,x4wu(bnw,d5zc44 bbaw ront of you? A diagram may help you to answer this.

A 21-speed bicycle hasbse-k9rsy3y ( seven sprockets at the rear wheel and three at the pedal cranks. In whichky (-39rbes sy 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 sprocket? Why?

Mammals that depend on being able to run fast have slendebd1pmij0l1el6p wmy4 iaq z2 nk2;6+ir lower legs with flesh and muscle concentrated high, close to the boqe;pily +z0bdp62lmi4i11m kwj 6an2 dy (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers zn8- h2:ee sndgz -f2a(Fig. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero? If;:lg sd.md;gm la+. 0qu5wpcn the net torque on a system is zero, is the net force wg+:5 q.d;spma.dgu0l;mc nlzero?

Two inclines have thcmj1tmkr z ; /c79bmt3e same height but make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speedm /c;zm t3km 7tjbrc91 of the ball at the bottom be greater? Explain.

Two solid spheres si pta pg-n6i 3fw.(kaj8multaneously start rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has thpgntai8-. 3fp(kaw j6e greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. They star 5*uszvny-o h*t from rest at the top of anh -n vzyu*5os* incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum i(1 3kt yiwdaoy ;i9i8jo3;j land angular momentum are conserved. Yet most moving or rotating t3 d io18yywkili; 9j( 3;aijoobjects eventually slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equ;mfn qe63slhu1n fcuzk .21k6 9ma- vmator, how would this affect the;1 q39 k6s a62n -mncz v1h.uulmkmfef length of the day?

Can the diver of Fig. 8–29 do a somersaul-:zkwbt ueuaqge20 4 9t without having any initial rotation when she b :au-w9e0 uzet2gqk4leaves the board?

The moment of inertia of a rotating solid disk about an axis throu vx.8vjr+d z:(ar d3rpgh its center ofrv8 dp+3:adjv.r( rxz 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 holding a 2-kg mass in eat2qnwg ,e k93zch outstretch2 nke3wgzq9 ,ted hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical y6 0+j(a pu 01ggtulkcand have the same mass. However, one is hollow and the other is solid. Describe an ecgju0gy 6+luk t1(p0a xperiment to determine which is which.

The angular velocity of a wheel rotating on a hor ro yitgoxlw1/r:4 g05fby26aizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If tgiw: o gb0624l1/aotyxr5 yrfhe 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 edge of a larg0;b7e m7xn- -osuacuefd5 s mf84 ijk*e freely rotating turntable. What happens if you walf7s8-uxnb74m mu5 e ice0djak*f;- o sk toward the center?

A shortstop may leap into the air to catch a ball and throw it quij85rgtyk w3.rk ) o4qpckly. As he throws th3r. pg5q r)kotkyw 4j8e 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 (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, disc 7euzqx )ya7 4i8dldh7uss why a helicopter must have more than one lyzud)84ed q777iahx rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (a) 30rd,d2ny z.c/m8c zdcx9wx::dq h, ze, $^{\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. Calculate, usiq0qi ;p/i rzp)ng the information inside the Front Cover, the angular diameters (in radians) of t)p/qqi 0z;prihe 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. Thec 3;r:x(v arnch z+h9c beam diverges achxn hv(z ;r:c93rac +t an angle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender *y2ne (v.r. mxyps/b fu- s0dlrotate at a rate of 6500 rpm. When the motor is turned off during operation, the bladesbv.y0(u-f*.el sr 2 xn/pysmd 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 childwcm.o ;s) gu(u+v1 )a jtfkd5j. If the ball makes 15.0 r5+d.go)j(vtum;c1s jua wk)fevolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutm fhvi5iaa8:b k / zb4e 3m3d+rx8xtw2ions do the whe4:8v dimx k8 it5wxb+far2e/3azmh3b els make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate its anstrq6wcv: f2foea 5/f0sz; n 5gular veloc/:as;oc6 f2frfwe 5 q 5nszvt0ity 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 complc76 ktccb t-ye*1 sya/ete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1c7 y c*a6sty/e-1ctkb.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orbit arooo 190o3*g9q(gy/ dr mgmbx,hqn ylz (und the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of 9ugv( )ny2h iqae4hb*hg 1sb (0hu)vr 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 centrifuge rotate if a particle 7.0 cm from+ iaf dd-2sz-6gwtjh/ the axis of rotation is to experience an acceleration of 10- +fw/ ddz6is-a2gth j0,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 r9v)i5 c ruvshz*mq:586pcgb 3hd hj5ipm to 280 rpm in 4.0 s. Determine j5vz hvr:9 d5mb*i65hup qhc g83)isc
(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;bt9kgdpi6 +hlhb *9x $R_1$ 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 spwkiw/uwd*n p8 n*s)j9acecraft put themselves into a slow rotation to distribuk9u*ws*np wwi)n8 dj/te the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The 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 wu+m8y07il o js. Through how many revoluju +wim70oly8tions did it turn in this time?    $rev$

  An automobile engine stc14xoc0wfjmshbx 11 z:y (2/8mxeh j lows down from 4500 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 highsp5kzxc9sh v02eygr0c6v slmh.32h f psibj,7;eed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its 67,m hv;2skc95r s 3xp0.c jb2vhg0lszhfy i efinal 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 accelerat-0 -rwm .0a0zu(nc hv*vubbgees uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a pu*gv0bbn.z0w hr-- uav ce(m0oint on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it mne u86 6gv6 3uwnlit*is running at 850rev/min It turns 1500 revolutions beftl e8 6n6u6igmvwn* 3uore 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 sp ,kve6 vzq7yg h6kb ;cy+3w4theed uniformly fromztvke y6qwb3gh4 c v+7, 6;ykh 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 revolution xdle2yn4 ul57dt/5vc s as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.80 l4n5cu2v7e 5dydxl/tm.
(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 eack)v; zchw3p u8jo6(s5:i rr vdh pedal when climbing a hill. The pedals rotate in a circle of radius 18p ;c3v)i:jzvw d(5r rhko6us7 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 thj,b-rp +w ibad i69pu8hlku: :e end of a door 74 cm wide. What is the magnitude of the torque if the forwihr8k ui p+:bd :9p6l-uaj ,bce is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel sh 6ws71xy;, n5gjgd 3pxxaqe8gown in Fig. 8–39. Assume that a friction a7jx5n,6gg8edsgx3w1q x p; y 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 t e(cax/ -;xmjptn2k5/k ov/fqhe ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is xn f( /5/jqp x-am te2v;o/kkcheld in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening to a t x35z9kbo/jzq sqz9z2 1oilk ,orque of 38 l2 o i5zzkq kq9oz,9jx3zs 1/b$m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of radius 0.648 m wh v3 cp +wepc-gb.qwb1, 0int3ben the axis of rotation is through ic- pgibe+0b1t.wb33v ,cnq pwts center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. i zeq wkofo9; -74iu7t 4( sg3djzr9btThe rim and tire h3j47tifd ;k or-btqs9zz( o9wgiue 47ave 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 rotated in a horizon 1dh90vcu4q24dp gl) l)kn bhrtal circle od vuhg1r42qd) lp9h40ck)lbn f 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 ajlij;t+k 9+lv potter’s wheel rotating at constant angular speed (Fig. ltl9j ijk+v+;8–42). 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 ikjh0a606 ijjrz pc9kz)1ro 3kn Fig. 8–43 about)kz6ahzr3 0i6p90co1rj jkjk
(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 atomsr/3qsx s5i us0 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 sz2g4f ,iu3*bp4fafe batellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If b2uf4p *g, 3bfz 4fiaethe satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 cm and a mass o j 1 0wijwd;s nbxx7f 7iawjako v*(+ns0*yk)6f 0.580 kg. Calcu0kfwya nxxiv) (kj0 7jb o*j1i + s*7w;swand6late
(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 ev/3,w2d-rvobcska ) i7ef+s a bat, accelerating 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 tangentiant9 tw7-s w 4edcz.r3(nk4ykq lly 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.stw-4 9kd(y 7wr34tnk zc eqn 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 rotatingr55y-n ,cqifd at 10,300 rpm is shut off and is eventually brought uniformly to rest by a frictional ,f qdcyrin5 5-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 3.6-kg b /vx50f nhfahtn. o,.hall 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 ba76sbrhuo x; i+ll is thrown solely by the action of the forearm, which rotates about +;6ubr7sih ox 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 bee5uk4ys d.; dvalxn 51 considered a long thin rod, as shown in Fig. 8–4 4xlk5 ve n51das.;udy6.
(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 b (:/x*vie mitconsists 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 reij:n 4 0 xjce,ha+jk6bst within four full turns (revolutions) and releases it at a speed b4a,ejjj icxnh+ 0k6: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 inertiayp(,icwxn2+bgn5gn6 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 de(:23b*4ukt+ q+ lfe bdqp tydmvelops 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 and radius 9.0 cm7 jp3-freol gs+zg9w ,id5/gb rolls without slipping down a lane at 3.3e+5 oiglg frgd- 9b,wj 7zp/3s $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic en)vl)ts hw*+z96qf jkiergy of the Earth with respect to the Sun as the sum of two ter+stlqkhw 6f)j9* vz)ims,
(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 am d0 bq,no c(7u(teid2pcnc85 radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cyl in d8(cm0o,pdc7 tqc 2n(5bueinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls without ktn7gm hl9ya4he 803ucnx8u 1slipping dow8 lcme3 1 n9nk07yxhu thag48un 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 fall and itn .v 94j czba7;n3vpc.dg2hp.tcxr7 us lower end does not slip. What will be the speed of the upper en7x7; udvp3nh t4jpvnac.z2rcc .b9.g d of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotatio s+.rjc+7,ehxb+rwpng on the end of a thin string in a circle of radius 1.10 m at an angular speed of 10jo+ .w b er+s,rch7p+x.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg unijajp jm9b3 :ytxt7k0+form cylindrical grinding wheel of radius 18 cm when rotk+37t9a:bmj yp 0jx jtating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotating at a rate )dve*o h, ;z vio 8/s+6zr4*ep4malbat;dkqyof 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed4ad 4ihapoyl;ozq8*et,vz)6 ke; bm r/sd +v * 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 osuf ,i1 9. cjsqhrdv38; 4*y2lcmdhrmne shown in Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tucd9fsr; 2siqymvrc lm 3h1h.u*84,d jc k 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 rotation rate from an initial rate brtwj8h* -jdd d)*.ee of 1.0 rev ever8 e.-d tw) *drhj*dbjey 2.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertica-.v;(vqwm)rc id o: ifl 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 frequr ov-ifd )ic:(.;vmqwency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure sv5l6qth 3p x yzav; ht+8;(s vkxdg9+wkater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momentum of the Ear7 l vb(vx4vw ;yvvy-:pth
(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 qb ;midt,ct ehy, 8s6i *s,g2cof inertia I is dropped onto an identical disk rotating at angsteqciyd*6c ,,2t;mh 8gsbi ,ular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2 fj1qx;m:*- fiia9hmetjy8p 2 .4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotatin+7i3fcgow5* w2 h 6iao kilbj.g merry-go-round platform of radius 3.0 m and moment ii b7o3.h5*ljiow +c62k f gawof 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 ou9x odh+ew14h f 0ehh9w1u+dx4 o .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 into a white dwapvtk dl2f cc/;9:xn9vrf, 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 the Sun’s nelfpn dt:kvv9c ;x92 c/w 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 excetc+14 o2iqrk ess 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 hy5b7dt 3zdi-zo99 si$ 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;u0o/zxgj77ao m)n oi, initially at rest, that can rotate freely without friction. The m)ou70 o xjma/;n 7zigooment 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 d4mjahb -9 n3usc(zvm+6a(sfthe edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearin 6v(4dha-c + s9buamznfsmj 3(gs 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 rolr57b h9 yrucub*:rda)ls on the ground with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center o5ayrb b97 hr:r*ud)u cf mass move?

  The Moon orbits the Earth such that the same -grku 6n6(1x c 5e5ksukssksizh+ :5cside always faces the Earth. Determine the ratio of i kn 1xk(5+sshugc 6crzks6:e u s55k-the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates froajb xr 0)hiy:9m 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 uniform cylinder of radikn 7pd*ur+ .hs x a/xf8m7h0rhus 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque dekra h.8+fu*h msh7r p7/dx0xnlivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each o+ ;0np;u3d kpklvqe)m0q -jyif 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 1.0-m-long string, i3pi;k0 dv-nk p0qyemuj )ql+;f 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 reagak zjvl+(u5i ,f y,r9r 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 lf 16g7cnu pv,d+ xm2nSun, but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and2fmdg1l7nnu,c 6+ xp v 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 strfyg(,m 41 0pyoc;z hmtk) au0ored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should pg(zmt0fc1 yr;y au4h0,mo)k 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 illustrat/yz f-wr vat;qnt+.;des 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 rollingeg-a-y+/el.p)hsdj) al jix * on a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., we ignore fo0vr+ ylkca/w. u+*am5d3y ulriction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then releau.0/l w ym aol u53cad*vr++kysed. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M haa vui-ow:0,2swhsfsw *2gzi1lh v 69ls radius R. It is standing vertically on the floor, and we want 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 isv*,o6ahhlf 2vis w w1zug 20 9l:s-wh

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a turn v3t u29kdw)k lxo-p4uwith a radius The forces acting on the cyclist and cycle are the normal force23)lvtkx -du 9pu o4kw $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ 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 s ,m ted8jll6ibl(v 1 ;i8via0vling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where (v,di 8l a1;jlmbv6livt08 i edoes the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cyivasvkf*)fgs32g. 8ky + qr 3hlinder and her arms as thin rods, making reasonable estimates for the ds.*vigk8y3)fahv fsr2gq k 3+imensions. 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 cyli+tvw2 k d8 s*mvgqhnbq)( -4zcndrical plates, ofqbzd-n+) g2mv qhkv*w t(48cs mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 tx:pgdt 39j 38nslw0uche looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving the track? As xnlg3c930ud wts :p8jsume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, bro . hc9*hu9k6a-mibqut do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revo f(k. agsoxaznh* y)( (zva ds*2b70njlutions as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) Whan zygsh*saa(k0n f2( *xd) z(boaj7v. t was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s