A bicycle odometer (which measures distance traveled) is attached near jlp8m g;j(ik br2*cw+the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 8jmrbwp (k2li *c;j+g24-inch wheels?

Suppose a disk rotates at const 0gvxq yu87hi3ant angular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the mq8 y3ihvg 07xuagnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the angular vel0u0qe-4gcnno 8may;x:7bns iocity $\omega$ Explain.

Can a small force ever exertl rrmoss/7l/ 5hg oaa 1yn+e0v/:(rah 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(1 0(ojjflsfz sit-up with your hands behind your head than when your arms are stretched out in front of you? A diagram may help you to anszof(l 10jj (fswer this.

A 21-speed bicycle has seven sprockets at the rear kzrkbx 8,w mj(xx6l3. wheel and three at the pedal cranks. In which gear is ixxmjkbwl86. z(3 xk,rt harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower legs wi1ryo1d9g 4qd 2/bk dsuth flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why tkqg11 4 bd9/ru ddyso2his distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow b1icfm k-pb0d(eam?

If the net force on a system 1fh()lho0 ahisq z7zqn831x v/tew:bis zero, is the net torque also zero? If the net torque on a system is zero, is the net ot1vh0x7 z h slbw3q hq:aiz)f1n/e(8force zero?

Two inclines have the same height but make different anglxca qw8 ,rdr:6es with the horizontal. The same x,cdrq aw6:8r steel ball is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down anvf:w wak -.z)w9q6bsl incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at tz vqf:w-wb9a6 w.s)k lhe bottom?

A sphere and a cylinder have the 3o9-7umx eg5c6k q6lm ritax9same radius and the same mass. They start from rest at the top of an incline. Which re73lx 6tar ix9ugem6-9 qc5mok aches 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 angular momentum are conserved. Yet most movingp,i9m f 8owo)/riwvwcol 7qu7e.3 q z7 or rotating objecouwwifw9 8l qp o/z,rq.v m77)c7 i3oets eventually slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equatorio 7r/nth+3 zu, how would th hu37rontz+ /iis affect the length of the day?

Can the diver of Fig. 8–29 do a somersault 0oj.eot xpq 06n/t6yh without having any initial rotation when she leaves the boardty0t6 .q0 o/hnjxp6o e?

The moment of inertia o)zaf x;ld t:4tf a rotating solid disk about an axis through its center of mdxt4:tz)lf ;a ass 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 rotatc2pbz s lh2*+/.kwjqw +5b fkzi72v hding stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will you szkq22 dp2k fw bjvhhz/.b57+*+wcilr angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mas1f;ea lnr9 )d8wv5e tmmd :b*is. However, one is hollow and the other is solid. Describe an experiment to determine which is whichlw;58ame )df:bn dr9m vte*i1 .

The angular velocity of a wheel rotating on a hg 4q95dph w. ,lzhay,sorizontal 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 the top of the wheel. Is the angular speed increasdhw4 yh,qs5.a , pz9gling or decreasing?

Suppose you are standing on the edge of a larg hqm/ftf-s0;gonjr qy. yl (i q60c*v1e freely rotating turntable. What happens if you walk toward the centyviog r y ( nq1q;j .tfsmf-qh6l0/c0*er?

A shortstop may leap vj y / q76shi) 9jguazfzc82x*8fe:jainto the air to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If y/q:8 hiyuzjfjecfs*xg j 6) z29a8v7aou look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the lawk ghd( 2vqb0w1 of conservation of angular momentum, discuss why a helicopter must have more thanqh2g 1bwk(0 dv one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following anglesq,hc9im isht u9u1wsb .+f26e 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 ofge s)1n2aal 84wlp: of an amazing coincidence. Calculate, using the information inside the Fron4wp:lla g 21ao8)efn st 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, 3t xjlo-s8c .6d80,000 km from Earth. The beam diverges at an 6c 8-d j.lxtosangle $\theta$ (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 650. zha)w7pbwo.n: yaq 00 rpm. When the motor is turned off during operation, nqpy.hza ):awb7ow .0 the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a leveienq 44 )mhh 5fv8-j53uoncm0 ubn8uml floor 3.5 m to another child. If the ball makes 15.0 revolutions, what ov 3)uj4bihec -qnumnm8m 5084hn5u fis its diameter?    m

  A bicycle with tires 68 cm in 7 k,u,6-m ygi xz1mo: ean5rwldiameter travels 8.0 km. How many revolutions do t6mkm,owi1x eg5ay7: r -zul n,he wheels make?    $rev$

  (a) A grinding wheel 0.35 m in d;3 cmjj0: 6 sqpzm,tuziameter rotates at 2500 rpm. Calculate its angular velu 6;,zpc3smztmj0 qj :ocity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.0 s (Fig. 8–38). )4f3 b; b8kpia peo03tvgl dq8(a)bltedkv8b8 a3f )p034o;gqip What is 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 velocit n wmgoz9vk13dk3/jyyll2 :n2y 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 -uzm cghs/rv/ ;-9)j x7mswqaa point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particpt8h7l w+ a;scle 7.0 cm from the axis of rotation is 7;8h+ splct awto experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel acceler4evrewl e-rfh. d.x+e2h,)s fates uniformly about its center from 130 rpm to 280 rpm in 4.0 s. Detedv es. h fwe,xr -+f.2h4l)reermine
(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 radiuc*8aat sobf9-ssg, 7as $R_1$ 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, r3h,yo nb(cpue cc;13sje -, z 6c7djkastronauts aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the stpykh 1b ,c (c6o;eucjc-s7 ,z j3dn3reart of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpm inf9h kfgv9l o4v/8xa 2n 220 s. Through how many revolut2fo x9/va9nklfv8 hg 4ions did it turn in this time?    $rev$

  An automobile engine sl r u86c*v4 af;t1crih6adb:u1ikx pu+ows down from 4500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for ther/b4ywy z(v;z : ru8n6x.s cuq stresses of flying highspeed jets in a whirling “human centrifuge,”4zzr;/8rqbyv.ux( wcnu: 6s y which 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 acn4 i.oz*f sos2tllo. (celerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far wii z. s(o4lo2.fnslo* tll a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is tursu k -:me : (tv 5)k-boyxsuh0wqen3e2ned off when it is running at 850rev/min It turns 1500 revolutions u vqns0hekwxto5()y 3m2-ue:e-:bk sbefore it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the card9gfdj+q e;wgxpk gwh 932 z-,wioq: 5 reduces its speed uniformly from 95km/h to 45km/h The 5oz9wp,f 2kx9j3 w;dgqq+gg wh-ie :dtires 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 uniformly fia9 3,w5u b+cq-vriwu(ul8-yf qxo:from 95km/ qa,3bic+fyf(ui8 rx5 wlwo9 u qv-:-uh 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 weight on each pedal when ct8z/ gt5gvr0.wysn i9limbing a hill. The ped.tzy vtnr5 9i0/ggw8s als 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 end of a door 74 cm wide. mt; 8vpy1e9f /d 1dpwrWhat is the magnitude of the torque if the force is exerted8 vydtppwfmd9e1;r 1/
(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 5q yov/vj.;w1nu k 3wdof the wheel shown in Fig. 8–39. Assume that a fd1k .v/5w 3uy wq;vjnoriction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of m eqv+ y-sz,x))na*d rgd*iu j,ass m, are attached to the ends of a massless rod which pivots au*ie)- j+qs,dnrdxgz ,ya)v* s shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightenikshb6xvlznq p9qy;og o67; )5 ng to a torque of 38bghkqv;o z x 669ny5po);q7ls $m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of radius 0.uqp,;e i8yraea )de e4*s*3 bb648 m when the axis of rotation is through its center. ia e;3*ay48qpe)er,s ueb*b d   $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diamety ;o bge8sl+w4sq8ow 8er. The rim and tire have a combined mawwqeyl 48s8 s+o o8b;gss 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 rotated3-ngklp7m w l0 in a horizontal circlel w3n0k7lm- gp of radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating at c5t ;nb fkwqf9xpbd( 2ka92u.eonstant angular speed (Figkktb5b.9q(nae2 9x wd;2 ffup. 8–42). The friction force between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point objec-er 1-ynahf: pzik6x:uc 0+zy ts shown in Fig. 8–43 nkpf::ce1-0yz zxyu+ rhi6 a -about
(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 consiss y ooy*3zoracgfpg00sbr2b.; (tz ; *ts 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 sp*v vi y-8nljsb;b p0t)inning at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a ma*p-nbsji v0b;8t yl)vss 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 wiq9:myh-y 6:g;g * dlw:hk pfebth a radius of 8.50 cm and a mass of 0.580 kg.behf w-m :d;l 6 hkq:gyg*py:9 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, accelerating it from rest to 3 3gm-dy ;ifmh ,$rev/s$ 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 onark(1 ew3rf;y) h 1diw a small hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15dr1h3f yww)1r iea; (k 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 acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating r9 berm10zhl 2at 10,300 rpm is shut off and is eventually brought uniformly to rest by a frich r10e29rlbzmtional 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 accelerate3qqo 2y0nde,zvow 9(b s 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 by th9 tp,dfl8tzz+v7zd ers+w ef+7b-fq 0e action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rpz7fesvqf8r tz- ,tlz 7 d +9+dbefw0+est 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 loncu ey *a11 82vdi3okreiy3nf9g thin rod, as shown in Fig. 8–46. inko91yc3e1 vfi3da2 ey*r 8u
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine consists of two masses;ae)/oesl exhjs1jvl3:, 3gojum(6 d, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest within four full s* 9uutvl gmqnu(0g9:0 rv0b rturns (revolutions) and releases it at a speed of 90n0vvt9u(0: qmblug*rrugs 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 yq7wrj;k rb61moment 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 tz6slujf *s0wi56fmi,6eem9 worque 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 9jxg0t h7su,ih)+,j9,u uzjbk.0 cm rolls without slipping down a lane at 37kihj,b u,g+h 9,j0)ujtsuzx .3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kineticu9 bqs q1*8pkbw5r-g nj2 .ird energy of the Earth with respect to the Sun as the sum of two terms, rikb28rgnuwp s*j5qb-1 .dq 9
(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 7.5 2hkox3b9 u,gd0 m. How much net work 9hdogkbu2x,3 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.    J

  A sphere of radius 20.0 cm and mass 1.80 k i(5ehqb chob(*c6 geub38yj,g starts from rest and rolls without slipping down a 3j hcb*o,(ige8q3cyu b( eh56b0.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

  Two masses, $m_1$ = 18 kg and $m_2$ = 26.5 kg are connected by a rope that hangs over a pulley (as in Fig. 8–47). The pulley is a uniform cylinder of radius 0.260 m and mass 7.50 kg. Initially, is on the ground and $m_2$ rests 3.00 m above the ground. If the system is now released, use conservation of energy to determine the speed of $m_2$ just before it strikes the ground. Assume the pulley is frictionless.    $m/s$

  A 2.30-m-long pole is balanced vertically on ita-r b if et/e)ke)w6:vs tip. It starts to fall and its lower end does not e)abf-/:ekr6vwt) ieslip. What will be the 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 rotajw chkd6g (7i0ting on the end of a thin string in a circle of radius 7cj 06wdk(i gh1.10 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 unifor fh;-.ouvui- 9r fisk2m cylindrical grinding wheel of radius 18 cm when rotatiniu -fuv9kis.2f-; orhg 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 rotikjc8i,z1 g+bkd1o8) 2 ceyw*)uylpeating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–ypld)wc 1b 2co j88z )kgy*kee+ 1i,iu48, 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 ipc(l4+ju+) njx;u9uw3vb va y )il5kzn Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight positionuvxj) )3u95(+ aibnjl4vwlyk +z cup; to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an inr;bfi *mxz ts 5o;o4g4y.6m71vpgtywitial rate of 1.0 rev eve;4go;4 o. 6zrmxy1sit5y tpmgbfw 7v*ry 2.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis tjw bl0. ypbwyxs t0v9j3ju e3s94a)p5hrough its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel aftv j9bsb uwlp330we yjxsap4.t0) 9jy5er the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater sw1; yqo.nx 2jqpinning 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 mi+f lg2t7lg 8lgv ab)7nf3b- domentum 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 av,z+dr 1ku:qx(3 quw jn identical disk rotating at angul q(:x w+1 uk,dquz3vrjar speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4d6htirey( m zq8-ei6f 28rhh* $rev/s$ 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 cent6u(l8ulrgb*tugr7; gj pz6 e7 er of a rotating merry-go-round platform of radius 3gp77t z;g ( lur8l grj6b6ue*u.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotati*agqv.1e9 j ldng freely with an angular velocity of 0.8 l q.eg9 1v*jda$rad/s$ 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 ). n6q5jpl eshinto a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearestj 6h .)l5qpsen integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in excess of 120 )auyeqnidk)1a4 -gm2ht( qg2 $km/h$ 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 zi *4bjg6h)kep.x5i s$ 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, that can r yb 18-xyi-qxqq5m) :;f snidrotate freely without friction. The moment of inertia of thq5-;x-yri mfix 1: sqd)ybqn 8e 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 8.is*/g) kat ugql,nl-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment os alkg) qg8i *tu/n.l,f 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 ropwji ns65n ct,0e rolls on the ground with the end of the rope lying on the top edge of twc 60nn5jt si,he spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always y1f17u,c k kxt2 32qjiro-ldbfaces 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 thxi tkr1c-o2 jb2f17l3,kqdyu e Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 12ptn5vfu 0obi0adugo) +*:l5+bxs o w 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 j 34o az7c n/r,f-jsr wcd:y*kof a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Cw7s j*cfcy:rj,4n/dz-rok 3aalculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 kg an1ydnom4jj*d) c*h *e bd 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 ejjy n)cobd* de*m h1*4nergy 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 ang9 *wkbi5ksy / 0 erxaxakz6+b,8*uoc bular 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, burmm b 3:7bgy6kt 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 spbgy3 : m7bm6krhere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile ipv+.mf.ovw - ds for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywhed .-owvpf vm+.el 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 illustralcp *zsm1 9rv: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 sur re bg9lkc)j2:face 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., weuv b2xrt07o e1c 8m6gf ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the o u7xtb602r gv 18mefctip 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 st8 i6fqs abk1g/anding 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 ak6fq1 /gi8 sb(Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making adig; rus(aep5 7y,-s pg-,sa n turn with a radius The forces acting on the cyclist an,(a5ae,i7u ysn; pgrss -dg-pd 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 of len8v8 8q t;iqybsgth 1.5 m and begins whirling the rock inqy;8i vt88q bs a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylindegc jr7a )xuk14/ fhzp*r and her arms as thin rods, making rea *7ucgfa/hz1rp)xk j 4sonable estimates 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 conf:3s osku 5d6 46n vuapjmvu*,sists of two cylindrical plates, of : s*ju u 63k m4v6vnfoad5sup,mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and ra qz.kl2 (s,iwmdius r rolls along the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving the track? Assumezmql (,w2.iks $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, bu cg7ua sv6bbcd:78a t4t do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces4bjfps5+q n0f 2 7fkih its speed uniformly from 90km/h to 60km/h The tires have af20njqfph b4 +5 7ifsk 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