A bicycle odometer (which measu9g5 rwl8psgvcxa:(z5 res distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you usggcav 5:zs xlp95rw( 8e it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a po h.d5x1aqa p+9k 8ws; ql s(dlz8ea-uzint on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration+1ae(q;al wh8da9q8us5sd z lz.pk- x change?

Could a nonrigid body be describsn;e1jv oqlic749d q) ed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever e729 fvu frbytba ,0u2wxert a greater torque than a larger force? Explain.

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

Why is it more difficult to do a sitrkub :)y1( qkp-up with your hands behind your head than when your arms are stretched out in front of you? A diagram may help you to answeq:u 1(bkykr)pr this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the pe6 yrrgokp2 loq2,bw 24dal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? Iny krw rol64 2g22,qopb 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 slender lower legs with flesha9v4k) p)fa*gwt:f ag and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageouapw avg)gaftk9):*4 fs.

Why do tightrope walkers (Fig. 8–35) carry a ln./m c5l+ 7 4 **ouc-rfadisto :7z yvfni+yfzong, narrow beam?

If the net force on 95ynvc .pbe5 8o 1aa e6vtd 4uerq9dy-a system is zero, is the net torque also zero? If the net torque on a system is ra 6vecbutne4dya9-91o5p 5. q8evy dzero, is the net force zero?

Two inclines have the same height but make different angles1cq4skly t1z; n6mki 5 with the horizontal. The same steel ball is rolled down each incline. 14q ikk6s1nz c;ym 5tlOn which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start r 0wr3r h vuuhg-g1 ;xm4(/kanpolling (from rest) down an incline. One sphere has twice tu/4p-g0uh h(k; x3amnv1 rrwghe radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radb*6 rkccd a(0scgy5 8dius and the same mass. They start from rest at thc6a( gcc * 0dd8yb5ksre 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 rotational KE?

We claim that momentum a r iptq.-r5;ktnd angular momentum are conserved. Yet most moving or rotating objects eventual.q k- tr;5trpily slow down and stop. Explain.

If there were a great migration of people toward the l)li2 *3 fkomlEarth’s equator, how would this affect the length of )ol 2killm3* fthe day?

Can the diver of Fig. 8–29 do a somersault without having a izs xspv4y ej2+n/.-pny initial rotation whenx isnpz2ps/4-ve.+yj she leaves the board?

The moment of inertia of a rotating solid disk abo5xm2 h5fse-k ju9jg 2tut an axis through its center of 2jfg t hk5uexj25sm-9 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 sittin,x- h*vbnzsj6.1qz zh g on a rotating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decreahb.z *zsnxz1,jh 6qv- se, or stay the same? Explain.

Two spheres look identical and have the 919m:fc4pt/(uwi hng psbz+ msame mass. However, one is hollow and the other is solid. Describe a/:f( pmcnis+wbu 9 zhgp1 t4m9n experiment to determine which is which.

The angular velocity of a .s sa b h)q/ /xhiz9(m-sg+bxkwheel rotating on a horizontal axle points west. In what direction is thssi)sxk q/-mb+gh/h b(9.axz e linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating turntablenw-(nl6qdq ( f:wz*9xgd7 cqu. What happens if you walk tow *xq-7fw gd9un:(lwzq( cdqn 6ard the center?

A shortstop may leap into the air to catch a ball 5ex7tlp.jq xihef a s cg753.,and throw it quickly. As he throws the ball, the upper part of his body rotates. If youf5lj75 .h x qg,e7ei.ct axps3 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, discuss why a araad9:ab m5x7dg( l 8helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operat5gm ad89a xadl7r:ab(e to keep the helicopter stable.

  Express the following angles in radians:ku9v f, *f:co0siycu6 (7hgh t (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, us(- od+rj zs1mb)9gryying the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earthr)g z(bsmyy 91o+- jdr.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from E 1b k x 0w)m50w;r bdtbditxw0b)*ppl;arth. The beam diverges at d it)xbk510wmdxp*bbr0 ;)w;t wbp0l 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 rotate at a rate of 6500 rpm. When the motor is turn kv h3/ut caxwp-u y+h:tu43/ued off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow doc ua/k+w/4ut xy3vup-h 3 uh:twn?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the ball makes s3wfd:s)q8 bxz8yt+zhyv 2 .k15.0 revolution+v q 2k w3s8 fzz8.dxb:thysy)s, what is its diameter?    m

  A bicycle with tires 68 cm in di)024v vjzh7afjtfkv( ameter travels 8.0 km. How many revolutions do the wheels7f(j 2vfh)kz 4 0tvvja make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculas9w ngra 4f tx3+e;enzft., 2yte its angular veloc34e.en,29xzysfg+n wtrta ; fity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolutio vkyn:s7u)a 1x7uz3 ut/.ktar n in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated zu an sxyk7.atku t)1r73/:v u1.2 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 veloci07gn4 -pamxwipgb4 4;c0 qtao ty 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 poi5rk*dcty 5k s4nt
(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) mu)-kxdjab2 *oag g+to7 2ajnq2 st a centrifuge rotate if a particle 7.0 cm from the axis of rotation is to ed-2nt+a2gq) oj*ka2bgj xa 7oxperience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates unif2ldlm* 9u2fhr1 t kp/p o*6nruormly about its center from 130 rpm to 280 rpm in 4.0 s. De1r hd62up/t ln*rm9 okpu*2lf termine
(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 radiu 2-6it4d h/mm mxe)ngas $R_1$ 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 spacecra;:35 sm sfaj a.8lx*b-iipqyaft 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 revolutb ;.ji i mxasp8qaal-fs:y5* 3ion every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpm in 220 s. T* uj :f p5krym18lyg*4pt.qcnhrough how many revoyj** 8kl up4. qmrf:tc1gnyp5lutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpmq.vqugq94mi ; 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 h cs oa 1ije5kh)2fw dh8g6,h*the stresses of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to t def,hwk5)oh 6 18g caijh*hs2urn 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 rpr: 3gmj 1wk(psv:ks0 p; algp)m in 6.5 s. How far will a point on the edge of the wheel have trk p0)r;mw1:jgsk(3valp gp :saveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 1500 revol: (yy;f, p: enliv48pkv0jpn butions before it comesypn il:p ve nf8:v;4,p 0kjb(y 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 tro/e 5ndi2 v,speed uniformly from 95km/h to 2,ed 5 rintov/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 re36zsacq 6i7r qvolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0.80 m36a6zc7 qsqri .
(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 al)rw-sz9 l-+ps ti psoif-*.t el her weight on each pedal when climbing a hill. Tltssri --+e p *.-9tizof)wpshe pedals rotate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the emib8wifzo4 34h+vx-w5 3wt d vnd of a door 74 cm wide. What is the magnitudewwift5v8 + vbowxz-dm3h3 i44 of the torque if 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 wheel shown in Fig. 8–39*b 4jult 3mq2n;c+v/h*m9tvsmyf( xd. Assume that a frictionhtdx;*lm+qb/ (muvt3*fjy v 4sm 2n c9 torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, ay 4:5yyvvn/rcazij ; *re attached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and di/v a*y rcvnyy;5 :zj4irection of the net torque on this system.

  The bolts on the cylinr itqjbm-h q ( d/*ybb:e2e9(lder head of an engine require tightening to a torque of 38 h- qdljb:/e9yb (b*2(rqit em$m \cdot N$ 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 whv n6v)w -9fla 10.8-kg sphere of radius 0.648 m when the axis of rotation is through 96f) n-lwvwvh its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a biao482rgb-frz5ot+: hfec ,mfcycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.5z8ea 2f-, gf4rt:b oo+mfcrh25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a t,17pk0+hvm .;oz v3p onoain thin, light rod is rotated in a horizontal circle of radius 1.2 m. Cani,h 73tn o0;p+p.vv mok 1aozlculate
(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 p(o/iin/yq;. k 4veevp :dh-p wheel rotating at constant angular speed (Fig. 8–42). The friction force between he eiv /-4/dipqh.;o e vkpy(:pnr 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:-p-z. arlpbumnp gu8./mk q* inertia of the array of point objects shown in Fig. 8–43 g.8pbuqu-zk* nm-lp.:/r mpaabout
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen atoms whose total mass i11zeva jz5 rz5a-; 5ef7 zicyes $5.3 \times10^{ -26}$ 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 coi;vmxy ( 1e22odxt-ts rrect rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to ret ex(2ym-2x1is ;vot dach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radiu)c4w c8, y*qo*ihvu85ykv( bedopehs*7c(f ps of 8.50 cm and a mass of 0.580 kg. Calvyds7eb4,eco8kp )5 (wi 8yv*h(u c*qofhp*cculate
(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, accelaks qciyto1 8rc+*1y/erating 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-go-round0.mbu 18s9pmb i b pdbz34pa(y 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 childre yd8 ai30 pbb9b zb.(mmp14pusn (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 shutyv 3l; e1vmbr4-a: ztwfd ;s0stm16na off and is eventually brought unfwmltnma t b-6sv3se: 0rdy1;v za;1 4iformly to rest by a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 acceler1cdyy fco9wa0.x2vz 9ates 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 +)k(p .be4 skl1idqnt thrown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated 1pe qsdb)k(t+lk4.i nuniformly 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 can7t g xbfdcjfpe*0wf2(i +up*2h1vr w: be considered a long thin rod, as shown in Fig. 8–46.t wpfd w2 0veuj*grp(*fb+721hx:ifc
(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 consistsxbpwn ;crf75) r.c*qt8 c ct 3arss4h7 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 wi1/ 8 l5f ign3b g.8+tbneaxzaothin four full turns (revolutions) and releasesgxzb1af +otnal ge3n./5ib88 it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a mome2n8lfr 4 ;hzw qsov1t8nt 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 developsi3 ab0k r4qf ugd9f,k9 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 cm :mc ,hu(wp9de (issfo6 k.b9erolls without slipping down a lane at 3.3ss9ow e dc6e(im9fkub. h(p ,: $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun aszr .8.4zdssxzie9we 1d1f ;9 exrvq/ s the sum of two terms.qeexs rzvf8 4/1d;ex 9sz. 9dziws1r,
(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 7.1.nlg2mcn57-lg tm wm 50 m. How much net work is required to accelerate it fg2. c lmgw-mm15tl7 nnrom rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 2:h8 nt0 iuno7m0.0 cm and mass 1.80 kg starts from rest and rolls without slipping do0 tnm:ni8uo 7hwn 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 ov;nt:f 2wect6 n its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [H6 e:;tf 2vntwcint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the,pc i 4vk7e67x e,zexnis:u7jt9prv* end of a thin string in a circleev pzj,x 4uxer677i,evc9spkni7* :t of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg unif9w holq, 3bp+kcua86q.0 mifaorm cylindrical grinding wheel of radius 18f+ 89wc 3bham,a.l opqq 6uk0i 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 thatqs52rsm(xi z-p,, c ty is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speesc zs,qr m5,yi2-tp(xd 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;nr5n:h9qpxvj 5+u wq c*q-iuvrs/8j–29) can reduce her moment 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, what is her angular speiu-8 pn9q rh+5j*n sxrjw /;vu q5:vqced ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can i* -uq .)o4 q 6,wtzympasalh5bncrease her spin rotation rate from an initial rate of 1.0 reva)5 6slqwbhat.pm*-o, zu4qy every 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 : t3cgp16es8a u:typ paround a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is et:as 8p uypt1: 63gpcthe frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentuvzbg9d(k(we6y 6n e;yuo37 b vvuqb-3 c do+1im of a figure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular3j-7sc +hl 3o 5gtepxejft9 (j momentum of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is dropped onto an nnq1s5yid(u-rh 8h+aqb (e x0 identicalyrh+anbs81xq(de iq n u( h0-5 disk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4g8ic s2f z //hmh2idq4 $rev/s$ 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-g4jnnl /:smv+r o-round platform of radius 3.0 m and moment of inertia 9+rj 4lnn :mvs/20 $kg \cdot m^2$ 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 velo1 l)byfuqs32)w4m +x2ov frj ecity of 0.8 12 4)rouyjw32xbvle+ s )qfmf$rad/s$ 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, losingc6row3**i h la about half its mass in the process, and winding up with a radius 1.0% of*c*ha ri lw3o6 its existing radius. Assuming 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 winds i u-1qs bs35iu m9-rglqn 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 tl csrj/6g 99s kqml,2$ 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, initiakm:0 uras: *k4-1f rg3jrg vnolly at rest, that can rotate freely without friction. The moment of inertia of the person plus the platf4-u0 rm g:n :rokvs3a1k*grjform 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 pers hb x1 sgfnm+,xz.zq15on stands at the edge of a 6.5-m diameter merry-go-round turntable that is moumz1h5xq+z .1 sbng,xfnted 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 rope rol+cl d.fa9be8 fls 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 distancedcfla+9f8 b.e L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces t) mskip/8z) : lnyb,wzhe Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital anguliw ) :zs8pmz l)y/,nbkar momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of nr mgc:r0c +.m1 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 e.5su63fk elya uniform cylinder 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 delivere 5e fys3ke.u6ld by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 kg m +26c gbbppfsf-oj/ 5/hyh.eand diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.00jp + fgh/p/e.5 fycb-sobmh6250 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, h9m(4ftbyb2 )5 eor.lr qh-hfnow is the angular speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but withd90y4 vzmg)oa cey) 5c7afbj.l(3a xf 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 ul x amv eo5c) dj(ybfc)7y4az3g90.a fniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use energy3cmh,3)fpf3irw z 6sq awdd8, stored in a heavy rotating flywheel. Suppose such a car has a total mass of 14dhsw)a3m6zidp 8rw3,ffc3 ,q00 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 6d3jg.6*ghlsn p*9ak r w-ltran $H_2O$ 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 horizontalhr/z8;dj rsp+u)-7cwxrmjx 8 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 masst3c2q 0r4k nvct( smt9 M and length 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 horizontalt mn4c0s2 3(vt rtq9ckly and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius2xei xe+* hng5 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 agains2xge5 x+n*eih t which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling wir(fgwsrgn:+irf7x *35k-t ll+) kn cg th speed v=4.2m/s on a flat road is making a turn with a radius The forces actirgtl) wg l-gr3ck+*i: (nn57fs k+fxrng 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 sy da8r yz.fdw.7nc7n9ling 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 achieve7z8 wf7.dn9.drcy nay this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid;hp1tu -lu q, pqwt.*x/ieyn6 cylinder and her arms as thin rods, making reasonable est1;wu yp/tqh e*l- iptq.6x un,imates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindrili 6jr o8o 9h9,ouy6hjcal plates, of mass ouji86j l,9o6h oh y9r$M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 track of Fig. 8c0rvy gd-bf v3rx 3t-:.8v cf8giv ci.–58. What is the minimum value of the vertical height h that the marble musti..8r8g3cfbv 3y vv -xvf0 irdcctg :- 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 do sypb/7iq u-.+ xcz7sp not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the ca .t mmch(uy0r2:s 4;kbr g)ador reduces its speed uniformly from 90km/h to 60km/h .tsbm4rk2y) 0ac mg :uh;ro( dThe tires have a diameter 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