A bicycle odometer (which measures distance traveled) +*/pfc :2bd rozt:v iais attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheelst2a :+b:/*izpfod vcr?

Suppose a disk rotates at constant angular velocity. Does a p84h+os )zcp uwz:ym; r.ifp1coint on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does thecp um c1f sipo4:.8whz ;y)+rz point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value o 9qq*m3 c7aasjf the angular velocity $\omega$ Explain.

Can a small force ever exert a+ cy; m*wzdtk. 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 difficfc j +9ckc0s/oult to do a sit-up with your hands behind your head than when your arms are stretched out in front of you? A diags 0k9cj+fco/cram may help you to answer this.

A 21-speed bicycle has seven sprockets au7dzj1xvx( 3+gr2cr3wxnz w 6t the 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 sprocket or a large front sprocket? Why? 1r(2jz637xc vw3dunxgzw x+r

Mammals that depend on being able to run fast have slender lowvb,h .(n ju:.4y fqdyper legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dyny.dbu ypf4h n:vj ,q(.amics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry ahvbi;d*ya4g xxfc*pz3.z d 62 long, narrow beam?

If the net force on a system is zero, is the net torque g 7;ooq 2rz9a54ksb jqw3:(mwlza g 7mn gb35ialso zero? If the net torque on a system is zero, is the net gg7 ab24kq g 5moljr7qwn (moi3b5a:;ws9z3z force zero?

Two inclines have the same h1 h8 fu4 8wecs6haun; nzz3(hmeight but make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greah; ens zhmu8unwc(6 38ha41zf ter? Explain.

Two solid spheres simultaneously start rolling (from res*er0x9a x) qcyt) down an incline. One sphere has twice the radius and twice t xr09* qca)exyhe 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 sa+ dx ggdec;dwkmw*fy 2q)0 8f6me radius and the same mass. They start from rest at the top of an incline;)*xfydd q6 20 +w8dmkwgc gfe. 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 and a: j8/b:gm, kkgjhcxwxl2 (9w j7khkb5ngular momentum are conserved. Yet most moving or rotating objects eventujjk,wxkbj9h x8mh:g:2l5k w(c7g/ kbally slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equator, 1:jj( (tv;da 2i ewsazyf /rr8how would this affect the length of the jyi f2artw jsa(e:d /8zv1;r(day?

Can the diver of Fig. 8–29 do a somel -8s,5ag/hj orfvh*x rsault without having any initial rotation when she leaves the boa*h5agrj/s,h vl- x8 ofrd?

The moment of inertia of a rotating solid disk about an axis thro .. xhv+qn0aoqpw;zx,bk*3r k0ct df7ugh its center of massbf;3phq0x. ntkq,*c7kaor zwd v.+0x 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 rotating1q:t;.g:g/ ycpu fh3i 5a nwuh stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your ang;t ua3gpq: fn15h iuwy/h:gc.ular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is hollow and tpgl z4 ir0p+.azi0u/u he oth+z0pui4rapi0 /l.u gzer is solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotatiyp 6+xe74a ee 3x9f3fld(sdvv ng on a horizontal axle points 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 at p xevsfv4+6xe l(y97dd33 fea the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating turntable. What dv*) stunrc-tb*sp0 ;r+8rx xhappens if you walk toward th0 rt-x+*s; pc*ubxn8s)tvr rd e center?

A shortstop may leap into the air to catch md;g ox/q3 satr9xa;+ a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in thd9q3sg;; rxao/ ta+mxe opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation o3zpy m*jfgbd,* 0nwj 7f angular momentum, discuss why a helicopter must have more than one rotor (or propellb3w, 0* g *yjfzpmjd7ner). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (3degy7usx 8m6a) 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 becausll. tshy6fz o/;jd-/0 qxn oj7e of an amazing coincidence. Calculate, using the infh 7l.f 0;nyjoojlz s-qd/6t/ xormation inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km i;-,:l *blurgu ,0wrhk,4zdlf ioq;l from Earth. The beam diverges at an an-g,,o0k l ;i4 l*bquri ;:,lwduhzrflgle $\theta$ (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 moy ,oe,-m2euf :hndo7ca+bg l3tor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blade7n, c oyeo3 l b:fheu,ad+gm2-s slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. Ifddojx *zpql.) j*tn 9- the ball makes 15.0 revolutdxlj- tzd*9 *)qj .npoions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How bfdh9nqc( atf5znsi9 )c 4onc 6 p:-l.many revolutions do the wheincbnd: sn95c-a6)zcfh.4l9( tp qofels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 r vmol/ h52,zijy(9l ihpm. Calculate its angular velocity in 2/vihi l,zjhy (9 olm5$rad/s$ $\omega$ =    $rad/sec$
(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 revolutioxp;lab cw *pd,ts.4 1gn in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seag.pbp; sxd,* w 4al1tcted 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 angula3a 92pbq5pjpy*c tq;ur velocity of the Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a y(zg +7 nkextvo3k65opoint
(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 tx1e t5e f+mphd ;6 14bawyh0tk.yi: gthe axis of rotation is to experienxth1ika 1 5p yy.d h;f0wmg+4tbte6:ece an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel acceleratv8bxwg:ug(g 3 es uniformly about its center from 130 rpm to 280 rpm in 4. bgg :(3ug8wvx0 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 radiu508r)p4*fyqp*7oay q; b g tcdz,ly rns $R_1$ 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, aslw lqo9/9f.k*yi l uc0tronauts 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 rotk/u l y wqlf.l09o9ic*ation 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 n5q uzsp9,h x6from rest to 15,000 rpm in 220 s. Through how many revzxuhs ,n 5pq69olutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 120085n/qnl /q7pu)upy us rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeed jets in a whirling2wt5 kepi* +ly “human centrifuge,” whiche5 lyp2*w k+it takes 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 360 rpm in 6.5 .dld +eaouu,5 s. How far will a point on the edge of the wheel have traveled in this ti,. od+au5u delme?    m

  A cooling fan is turned off when it is ruz lw2r5ywewc;btzrxcv(),3 ,nning at 850rev/min It turns 1500 revolr3czw;)z2xe,r 5 v cylt,w wb(utions before 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 a1,v u,kj0/aqeky n tfd9kf4 9is the car reduces its speed uniformly from 95km/h to 45kq ef1,9i n4u/k0jk9kvy d aft,m/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 k6 s78 5jkzhwd65 revolutions as the car reduces its speed uniformly from 95km/j6wdsh kzk 578h 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

  A 55-kg person riding a bike puts all her wp3,) 5r,;jwgz emvp dqeight on each pedal when climbing a hill. The pedals ro)wmq, 5 zjv,epg; rp3dtate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end r,(cdk pm1lx(of a door 74 cm wide. What is the magnitude of the torque if thec p((m1lk,r xd 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 theg5sb.q twsz bb(;o -jp 0vba7. wheel shown in Fig. 8–39. Assume that a f s(bgvq;a.bb5pw 7bt js z.0-oriction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are 63mb jmvhdq4x ict9:o*)wdk3 attached to the ends of a massless rod which pivots as shown in Fig. 3hjwocm4: t9vk mbdq*i3 d6x) 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 an engine require tightening to a torque of 6/k d;g4 od,(k 8p *vbqdsxoc6dhm 9.:qaqxea38 mq x :sdek4pd/ho;( vdqa gcq. boda,96x8k6 *$m \cdot N$ 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 momentrc5 fc;q+ q-gb of inertia of a 10.8-kg sphere of radius 0.648 m when the axis of r;5 +rbqccq g-fotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle jem0 gy5q1dq5 wheel 66.7 cm in diameter. The rim and tire 5qje0y1dmq5 ghave 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 hrg,z3qv fk2 0lorizontal circle of radius 1.2 m. Ca,qg2 v0k rfzl3lculate
(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 potter’s wheel rotating at constant anguv9x(ob thvt91h uoc +:lar speed (Fig. 8–42). The friction force between her handsuh9v1o tvt 9c o+xh:b( 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 in:d(a9pt pr/i3nk l0rcertia of the array of point objects shown in Fig. 8–43 about9ac/:i03ptrnl rp( d k
(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 tr6;u -11calp np -:bxgv:wqfdwo 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 o-h +y /t. cctr+gy*2fkpxn2t at the correct rate, engineers fire four tangential rockets as shown in Fo-t kg*nf./tx+ rc+y2h2p yc tig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder g6i - 0s:hgbcf(c+9ml 8fjnegwith a radius of 8.50 cm and a mass of 0.580 ke-0m g+clb:(cjh gfni6s 89fgg. Calculate
(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, ac diuj,zii9 iq)n1*q63 )tq gekcelerating 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-driven merry-g*a sm9uyn6 ,luo-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 torque required to produce the acce9 6nusm*yalu, leration, 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 i kgke6v1ts8jhzs 8p5( s eventually brought uniformly to 1ezpskg8k6 (hjs8t v5rest 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 3.6-kg ball at 7efw2aljf1v l f9:5ls. $m/s^2$ 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 the forearm, which em5z.o4; :erviqc8d srotates about the elbow joint under the action of the triceps muscide;e45cqmzo.8srv : le, 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 considered a long thin rod, as shown in w3qwx g62 z:qoFig. 8–46. gx:q26 zqo3ww
(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 consistho6g 3f p3 loathhg0-rgwo;-:s 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 a;deohih +t++ from rest within four full turns (revolutions) and releases it at a speed of 2 do;hah+i+et+ 8 $m/s$ 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 inerti acg4 z -ni1pp:mg(lt3k ,9chba 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 7gvz:dx cz07kjc -c,q develops 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 rd evttj6sf6h5kp8 uk 9.k :dg-adius 9.0 cm rolls without slipping down a lane at 3dkfv :ps-uke dh6 9tjg 56k.t8.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with resco vm28r:/m tmvok17v6jcjlav- h7q;pm 3.f vpect to the Sun as the sum of two vm:k7cljcm2 pa7m/ r1qvt8h -.jm;3vv f oo6vterms,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius of 7.50 m. How much net i4rshb, 9.dcwwork is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder. r.wsdbi 9c,h4    J

  A sphere of radius 2gsw *b+de qv7 mo;.lr10.0 cm and mass 1.80 kg starts from rest and rolls without slippingdev.b*rlso ;q +17wm g 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 verticall ui4tqz9(:j 8 ou( d q1kwzsxv+t+x:hby 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 hitsuh(4k+(xdx:u:b o9qqv8jw tt zz +1si the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end u+kp88 o wg 1lv5)a(susg6qmp6mko ;vof a thin stringgp8u6s8ksqp (mk6ov)m 1gl +o;5 wavu in 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 momentk rzy o/:lvyy*2 m7u)jum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotating ayym*j/o7)ruylv z :k2 t 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 platlkx, -m:p2fpkaje8 4d form that is rotating at a rate of 1.3rev/s If he raises his arms to a horizonmf a8pek2 j-plk,:d x4tal 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) can reduce her momenu fjl )exg f3y y2-rquz8;4/fqt of inertia by a factor of about 3.5 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, wh4uf u8 fxy/2)g zyfq;-j3lq erat is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can incre0szy1 *,kmlu rase her spin rotation rate from an initial rate of 1.0 rev every 2.0 s to a f m1uy k0l,srz*inal 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 arorxin gqbjavl50ps; * 3*7hgpnlb : e9+w6kp (aund a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniformglnk9:7*bh p a+ w( xi v5;pgqe3a*pjb0s lnr6 disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figur9w/ks*uz.qf9ncfu/ h w6 (egye 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 momentu6caqynj0 9hj0zds2m 84)c 4ud)cnx sum of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$

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

A nonrotating cylindrical dismu od( +rj):j-dq 6hujk of moment of inertia I is dropped onto an identical disk rotating at aud+jj mhr(-o u)6 jq:dngular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4 pzu)j uie8k,3 $rev/s$ 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 cent/v3i,sp: f m,re bp1iowa2d 5ier of a rotating merry-go-round platform of radius 3.0 m and moment of inert v o3d,5 m seip:wpr,i1/2abfiia 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 rotatx. zfw9 2anzqqibqhp2 f;z7)3 ing freely with an angular velocity of 0izpzf ba2.9zh nwqfq2q 7x; )3.8 $rad/s$ 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 dwarf, losing abv / k6j*-n0ztwz;kg eiout 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 woul/ eig- wz;kvtkz 60nj*d 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 +t j(xh.zqgb-3wgprrm ;:uu9 winds 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 l jfxug b//g5+t x:9yx$ 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 rpjpu*bbe3:6m i e6eu 8ur1mc0est, that can rotate freely without fricti3j0b r um66epbuic1u8pme:e*on. 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 edge of a 6.5-m diameter merry-go-rp 14+kdjhd2v.fl aom(x g)w4g crw :/gound turntable that is mounted on frictionless bearings and has a moment of inertia of 1paww o 2.lg mf)gg :+kh1dcj/44r(dvx700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of the rope lying onn3pe)v ;ca-r j 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 or)j;3-p ncave f 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 alwar 5b(i6:njzywzd j ,9jys faces the Earth. Determine the ratio of the r (z zjjjni by956,dw: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 from reslt/gvh o :l qr7:kfjm0 ic0g(:t 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 cylind9)3wr4(d4i k u yryxigp:bzkj1j h6 r9er of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor9iy43y z1irkup9hx dbk4gw: jr (6)jr.    $m \cdot N$

  (a) A yo-yo is made of two solid cyli jaq81ema;prvlp 3b-8n* ers /ndrical 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 ar18 p-vp* a3rljs;mqn/b ee8 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 thegxh)+ 8mb;cqh 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 with mass 8.0 times as great, wer 30 blhyb+u-vle 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 tb+l yvblu0-h3 imes, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use energy sto lxv:jkt-jf73a; zd4mred in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, us3kvj 4ftadml:; -zx7jes 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 illustrav 9cq0mz8a2 ;tcp) qsgi:nqfq-v vz2-tes 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 sjh;/r3ah*l dyw4 n (vgurface 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 lenug87 o 9vwnq-tgth L can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and thenovq7g9n u -8wt released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on the floor, and we 15b;ihpq qg2kwant to exert a horizontal force F at its axle so that it will climb a step against which it resq1; gkp qi52bhts (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat mkkpo xd 4r;20awn.9zroad is making a turn with a radius The forces acti0z opr;2k 9xdwm.n4ak ng 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 ro71ddla v(w( nrck into a sling 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 1 nva(dd7rl(w. 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 h9hvz.rmc (p..p)p(p3xisd n thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for pn(( p. .dz 9hvsx3r chppmi).a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch as,/hs*3c +;dk imq+eagcig-r esembly which consists of two cylindrical plates, of m isgca+mdh ;,*gcrqe/ik+3 -eass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 rollsq6ujo- x fgsssm76d j9r8zlfr-b15sx.m ,jg/ along the looped rough track of Fig. 8–58.6 j-8sjx 9 7,q db.1muz -osf6js5gf/rgxrms l 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? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but dov5)i 3kn*o6n zvlq jc/ not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uniformacc w:qbn 18lvb gm1/3ly from 90km/h to 60km/h The tires have a d8lwgq bv31ab n1cc :/miameter of 0.90 m. (a) What 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