A bicycle odometer (which measures 5o7k 4nsmbo4odistance traveled) is attached near the wheel hub and is designed for 27-inch wheels. Whatboonks754 m 4o happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on the rim 7z h 86rp/+ ufb-dcahkhave radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which / +zbf7 8uc-rh6da kphcases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value ofsy c9t8:94i:nt )ko lfoi ai2z the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque thfrhwj7lye+ 1( e z1v7ran a larger force? Explain.

If a force $\vec{F}$ acts on an object such that its lever arm is zero, does it have any effect on the object’s motion? Explain.

Why is it more difficult to do a sit-up with your hands behind your hur3fcc;:us3x1bmj w;cf t3m 0ead than when your arms are stretched out1b f 3: j u3rfmmcc;xw03uc;st in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at t r/iehv /nt8n6/t vr3uhe rear wheel and three at 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 sprockh68 ur 3nivtev//tnr/et or a large front sprocket? Why?

Mammals that depend on lavq.1 /3hg) wmfezq0 o7t0r tbeing able to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig.gt03w1o)hzt m vq qralf7/ e0. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carrywmiyp*x5. 0mn/9yw-ijw xf(m a long, narrow beam?

If the net force on a system is zero, is the net torque also m1/sxafqzb 46zn ;aa7 zero? If the net torque on a system n m1 a;/azf4 6sxbzqa7is zero, is the net force zero?

Two inclines have the same1k.zr6zchml azb /r-v. +2ui efi:-jp height but make different angles with the horizontal. The same steel ball is roll :fml.j6z12.vi+z ahcp- u zribke/-red 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 inclkd7 0rukfzvud7jr5u5r /m4 n 1ine. One sphere has twice the radius and twicer01kuj 5rdnrmu /uzf774k5vd the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. They start froge( :y:tcabci7 :6q gom rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater r::tqy (aoce: 6gg bic7otational KE?

We claim that momentum and angular momentum are cod 3;* pa f)l.2yxsxhvfbgt 9k7 +t*xbonserved. Yet most moving or rotating objects eventually slow down and stop.3+ *;b9tafxgxtx*plhf bkvs. 7 od2 )y Explain.

If there were a great migration of people toward the Earth’s equator, how wou1byr0ob6u6d*v 1 (ge)qm, z lnxqs+xw ld this affect the lengd1os g),nm6*v 6+lr (0ewxxbq zqby1u th of the day?

Can the diver of Fig. 8–29 do a somersault withht4h ulhuct7* i,/km ts5o*i+out having any initial rotation when she leaves the u*isi hk to7t+h4t,ul/ 5m* hcboard?

The moment of inertia of a rotating solid digfq9h: *s4l /1os+q;qaepbf ysk about an axis through its center of h9 sq4lq 1qf*yeof/:p+a bsg;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;ko4)txn 9dm,m( uq a*xnr/h 3ae xbi2 a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular vm)/txaehx ua ,9i ;bnn4 dk2xo(r3q*melocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mas.2f*l1j1zl0 ;( h4iv)qf.k cp5 yp ficuo gakns. However, one is hollow and the other is solid. Describe an experiment to determine v(2q;o kp*.c l1zi0gpfh 5fu1i nf )a4k. jyclwhich is which.

The angular velocity of a wheel rotating on a horizontal axle points we,,a;go3 c- g)gnsjsoh st. In what direction is the linear velocity of a point on 3 hgn;,gos -g)oj,sacthe 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 edge of a large freely rotating turnw 0fdij dk9j7lo 5k,j.table. What ha.kd wij, 7o0kjfl d95jppens if you walk toward the center?

A shortstop may leap into the air to catch a ball and throw it quickly. As he tvw-mzt/5)6 uh urf3fh hrows the ball, the upper part of his body rotates. If you look quickly you will nof)v 36hurtmhuz f/ w5-tice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the lawt))w* icik g:xfe-h+ 5vg1mtc of conservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stwe 5:- *fg+mhg )tt c1ick)vixable.

  Express the following anjr w,7 z(b+v/ptt+rme//x krjgles in 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 an amazing coincidence. Calculate, using d;n7rq +dg6kvcq.w-igc-nj. the information inside the Front Cover, the angular diameters (in radians) of the Sui-cr67.;kn -+ .gnjg q dwvqcdn and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed b dsy:xv zk 7bh.pz+:e6 xjv.o;8c2 pyat the Moon, 380,000 km from Earth. The beam diverges at an andv.vxe:y; yz27k:. 8 6 b+o hscpjzxpbgle $\theta$ (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 th1o36svws.g jb e motor is turned off during operation, the blades slow to rest in 3.0 s. What is tg6vs j1bw. 3oshe angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball.;inod8e vab qao 9d74k(tl8r on a level floor 3.5 m to another child. If the ball makes 15.0 revk.;94 orbnd qai dv78t8e (aloolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter trj m ;usr9g/n/javels 8.0 km. How many revolutions do the wheels make?r;mjn9/ g/j su    $rev$

  (a) A grinding wheel 0.3s: qgy su42 ry7grcw:55 m in diameter rotates at 2500 rpm. Calculate its angulc yg y::7su gwr2sq4r5ar velocity 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 coc 6;phir8fh -k6nlaox 09w)jumplete revolution in 4.0 s (Fig. 8–38). (a) What ira h 8)lxkijp -whou06cfn9 ;6s the 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 Earth e-n 2bt nf93da3 rbsh4(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 p/r* wfh * +uvepua-s+x2bsn0ioint
(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 r8:(zq-oks.qcsl7 tva centrifuge rotate if a particle 7.0 cm from the axis of rotation is to experience an tk-:l 87. csq(sqozrvacceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accel2 mdx+: c5ltgw:/xe xmerates uniformly about its center from 130 rpm to 280 rpm in 4.0 sx ed mw:x5/:2 +txgclm. 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 radiush5 j *bi0iut4h $R_1$ 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 Moo:xoh8h a*lhmm :*c mn2n, 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 accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can beah 8n 2lc :mmh*oh*:mx 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 i;wc*gd6x /h: v0tu+i/ mt5m1c e rfylkn 220 s. Through how many revolu/6 :the05t/ui;x cmd+ wcr*gvy1fk ml tions did it turn in this time?    $rev$

  An automobile engine slows down from 4 4bzx iw;j pu.*focdfmi *1)3t500 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 yj w;wva2r :8iof flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reachiwrvaj y 28;:iwng 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)x2wfp-kz 600it jied rpm in 6.5 s. How far will a point on the edge of the wheel have tr2z-)kwp 0ij60 tdeifxaveled in this time?    m

  A cooling fan is turned off when *boe .4e hkhu9it is running at 850rev/min It turns 1500 revolutions befoheu*4 ekh.o9 bre 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 map ,vl /z4;hnc6tn u:ux jkr:+ qtfur/3ke 65 revolutions as the car reduces its speed uniformly from 9c4/j vrzkp;/+tr 6t3nux,q hul fn u::5km/h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed unimt8dbrc )i3jmt)+ d2ha4lmm : formly from 95km/h to 45km/h The tires have a d:hm+a ) btrmt lcmi2jdd84 m3)iameter 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 eac8dhi /v0c 6p-l3 6w0stwwunv ah pedal when climbing a hill. The pedals rotaw/ utva0n 0 68vplwhdc6s-w3ite 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 fo*g fc -g4e ur1-ico,cjrce of 55 N on the end of a door 74 cm wide. What is the magnitude of the torque ifi4rofcgj1u, ce-c g*- the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheeag;md4rd.xtz , vu*y .l shown in Fig. 8–39. Assume that a frictg, *m ruvx4.add.y;z tion 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 massless rzyjbq l+2r klw/atw81gz *0k:od which pivots as shown in Fig.*r/jl01 +2:8 azbgzwtlw kqky 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 cylinder head of amc(e g *mjb52an engine require tightening to a torque of 38g2 be5m( *acjm $m \cdot N$ 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 z*;l9 aetyeq rq9h,d,10.8-kg sphere of radius 0.648 m when the axis of rotation is throughr;d h9alqy9 q*t,ze,e its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycleb,/aa 85g:2i63iq8qvkopg;iodla p h wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be i85i op;q3 ah2g6kpb:ai8q odi,av/lggnored (why?).    $kg \cdot m^2$

  A small 650-gram ball on th,x)z .4 *u dhm5zdkhyke end of a thin, light rod is rotated in a horizontal circle of rxkyhd . )4zzmk5hdu* ,adius 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 bj1 t/qmn8 d+haowl on a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and the clay is 1.5 N h+njmq8 d1t/ atotal.
(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 sji +k 10 gl8:eu4dfwvkhown in Fig. 8–43 abouk0vkiujg4f1+w8 l ed :t
(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 consix)) spk i3 /ras4otv.rsts 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 the correct )3:0n0endbonmopk4 +wav* d erate, 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 ed: n40bwmpk*on 0 edv3) ano+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 cylinderzuns1i5sb9ow+ga/m3(z8 q-r 8f fzcm v hi.j/ with a radius of 8.50 cm and a mass of 0.580 kg. Calcus+afz iq(wov .cr3 b/msji18zmu-/9hz f 5gn 8late
(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, accele*ekts0;fl yz38t) rlt rating 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 tangentially on a small hand-driveto*;pxw,yeo :n y:fs( n merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. A, (fwsy x;no*t:pe:oy ssume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is eventually /jw a ocj( ;bgdlaovcp c(+.,j:c)b .cbrought uniformly to rest by a fri,vgc co(po(; da.:wbjl)./bc ja +j ccctional 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 /q orbq+:adi 0.keu+j3 lx3xojjuq2.–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 y5jblpzy6g3 ; by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The 3jy5py bg l6z;ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a loyo 3a2( nf-y3br dgv fiw7.dy2ng thin rod, as shown in Fig. 8–46y2yo3f32ivgfnyadb( 7 .d rw-.
(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 cons9v6l -y8a :01eeh jllgbgu/fdists 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 four * f52ohif(-i y/i nnt7mbuuq- full turns (revoin7*i5-n f 2uq(tbmyiuo/-hf lutions) and 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 ha iu,d:ja8wtg; s a moment of inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque of 2804mw9ea+e hg o6c 9xir0.v.tbc $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7l5rn0u jk4tsi z9mx ;9.3 kg and radius 9.0 cm rolls without slipping down a lane a is lzjuk9m; 590x4tnrt 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as w+5 j1f/t 8nnr 5on5zsecaz,e the sum of two tezan1e+n 5wfr,stzo 58/ ncj5e rms,
(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 3;il6fl:s vg q )yr7mlgsg0h -5pi.9m sp+osdand a radius of 7.50 m. How much net work is required to accelerate it from rest toq7 yi5fl 0 rslh) msdsg3g 6m+vi :lp.os;9g-p a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from am,(oi 7cyjm9 ) 8veairest and rolls without s7o(m)8e ayi ijc a9,vmlipping down 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 on9 k/g na;b 7;l9t m0j2m8 of0vdrn-wjbnqkh1s its tip. It starts to fall and its lower end does not slip. What will be tw0hbbn7v8 km2-m9t1 n k nfrjq d0aj;osgl9/;he 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-kg ball rotating on themu12l5t k 6tla end of a thin string in6lk m2at l1ut5 a circle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular mk ub:o1aup0p,omentum of a 2.8-kg uniform cylindrical grinding wheel of rau0o pubp:ak1, dius 18 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotwj(vts bb : vy97*ru-wating at a rate of 1.3rev/s j7w:uts-b9v rybv(*w If he raises his 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 Fig. 8–29) cg)oiy .b+ fasu9les**an reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she) a* *ufis 9lyeso+g.b 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 initihwamco:q)w :jmus eq )c,1j2 4al rate of 1.0 rev every 2.0 s to a final raqqjj,w:c)m ea4 w1m)h sou2:cte 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 l f8/ *zazw1x4gq,ctuthrough 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*84l,z1cqa fwtzug/x 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinning at 3.5 vhs56t9cp+7gp u-gdv1 pa o9c nt:k4y$rev/s$ 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 h v(;a7plur ; a;ygif2the 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 2zgbi 38*,aw+ ;hyunu7-eb x/bir ukpI is dropped onto an identical disk rotating+ i/ba enh2g3zkwy;-8,buui u pb*r7x at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 7m 6.;bo,dvn *x6eg( ahehh j:a el-cd2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotating merry-go-round plmdw73 +q 8ux 5j+pkwtfatform of radius 3.0 m and momentfk 5u+3d +xmwq7jwt8p 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 fr;r g hpv.3j-aetp9kv+eely with an angular velocity of 0.8 vep ; vgprja-3 9.+tkh$rad/s$ 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, losic- wgafk*6je 7ng about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assumingkg 6 w-cfj7e*a the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve windg 6(/8xm0r,salrf , ggietq: ts in 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 ht69gu 58*m j*gt acog$ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initially at rest, tqhs -;t;ya1em jsgwf+1w *h)lhat can rotate freely without fricti s1;f1*gjh;taq we-)hm ws+ylon. 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 edgelsbjrp;w9 ,u; of a 6.5-m diameter merry-go-round turntable that pb wlr9ujs;,;is mounted on frictionless bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of ropl*oxtotz;- m ,e rolls on the ground with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto ito, x-m*otzt l;, 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 fsbpxa. .f4p2c aces the Earth. Determine 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 particle orbiting the Earth.)4p asf.2 x.pbc    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of fqe vd9fwrb 3)qm8 /7h1 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 sh 9qo4 vbe :m2kld-8c /ffvcqf-ape of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at dm qf:v q 9bv2eof8fck-4-cl/constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made oem t;2m 2x 4w,jyl-jvoa* wf4r*v :ufff two solid 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 ;vj4ymwvwelfj24ro*a, 2-fmft *:ux energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of the reas0y+) jyml,az sf +:jar 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 sizvb4 eyw g 7ww-e7(mle,e of our Sun, 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, anwm-ywe7 l(b4eev gw ,7d that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollup:okr 8v +lrvae.6wk0tion automobile is for it to use energy stored in a heaaw0vp8lrke6rvk .:+ ovy 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 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 asn g9u l7;qir6s*: vlpn $H_2O$ 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 su 21 7) xi0d)tp t.+-qltvitjbsrfkc,k rface 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.9cd.lkau 3lz;lqo/,w, 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/kzoa .lw3u, dl;ql9c 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 has rad *:fse/zi5fne))7rx yqztcta8-em x /ius 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 f5 ) /z *)tcaiy 7-reeqxet 8f:xnzms/h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is makioy)e; c a,82kx5 pphynng a turn with a radius The fp, pyy8 x2h5)e a;kncoorces 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 oj)nf.x bs 6lj:f 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 does the tor6nb)xsj lj:f .que come from?    $m \cdot N$

  Model a figure skater’s body as a solid cyl c/v7ua vj14-asvhe1m inder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched armshvcmvsaa1j-vu 71e4 /, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindrical plates i;o gh0fz+x.j, of gx.;f+ hzoij 0mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 looped rough tra2akvndcij66 b;ga ;)s4b vo:jck 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 the2baa:o c6;i g 4jsnd)j v6;kvb track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assuep 3jqqd2 v)lqs:i,7f me $r\ll R$ h =    (R-r)

  The tires of a car make 85 re(h40oqivynli l 8o6 t/-:b lmnvolutions as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of ea l:4l 6(toh vil80oq/niynb- mch 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