A bicycle odometer (which measures distance traveled) is attached nvdqpj ;;eh cm2g..a/bear the wheel hub and is designed for 27-inch wheels. What happens if you use it ocb q; a../v mpj2gdh;en a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a poij s6ax6k4 h;9; smwytvnt on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformlyv mj49swxah66st ;;ky, does the 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 of the a4dy( wre mkt)k,co-,sngular velocity $\omega$ Explain.

Can a small force ever exert a grx+w :t tk2lbs0/6i vureater 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 sit-up with your hands behind your head than wh wf:59wpvxmjb k*kkvp 3i cv3-61cd3den yod kxc mwk* 3wkb v3vfp5v96d:pjci13 -ur arms are stretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheu whz4.hsn2 +wkn6)uz el 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 inks u2)6+wwn z.hzuh 4t harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lowka sagt5796p l uzqs*3er legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribua5p9g3sa stqlzk* 67u tion of mass is advantageous.

Why do tightrope walkers (Fic.. 83tjt ppoj+lxzo (g. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the nel c61(e3sq an zp2oeba9:pp 7q m6jvk*t torque also zero? If the net torque on ae6 qk oj1pm2sp 73b9a(:vpeq la*czn6 system is zero, is the net force zero?

Two inclines have the same height but make different als0 en j;s.)6va+lglhkhc: sy.y+f k2ngles with the horizontal. The same steel ball is rolled down eachlh:ns.;6skj 2yks0vcy)+a.+e l fhgl incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously starycefy r2u5yq +2;ztr1zed4e :t rolling (from rest) down an incline. One r4yy5r+ze2f 2zeuc q:;t1ydesphere 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 the bottom?

A sphere and a cylinder have the;-ujml kzw/ /a5 o 2ue01gcoag same radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greate z/m5/el2 gcjuaoo1u k-w;ga0 r total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserns x iaa+53d9v2lb77m zqgxl*ved. Yet most moving or rotating objects eventually slow down and stop. Exp bm5xll*xgis7ada3n9z 7+v2qlain.

If there were a great migration of people toward the Earthviqm;b hsu3pi 890k(1ei,wo q’s equator, how would this affect the qmp0 ,isubo i38hq1ei9(w; vk length of the day?

Can the diver of Fig. 8–29 do a somersault wit5z/ky s+jru7 whout having any initial rotation when she leaves the boa ks7yuzjr5 /+wrd?

The moment of inertia of a rotatingfr6,uj:*npe ,4zsj zhn2(h tc solid disk about an axis through its center ofh( rz *scu ,z4h jn,:6tenfpj2 mass is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating storfjnz srg0u905fpep, y,3( jj6u t t,ool holding a 2-kg mass in each outstretched hand. If you sufj ujgn,jtp9o,3 , 6fuy05sr( erp0tz ddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is hollow 5 2ihf(rxxxn1 ,( hte xvvr.:oand the other is solid. Descvt:exiv xo1(r, 5 .(h x2xnfrhribe an experiment to determine which is which.

The angular velocityljfd+,gyw .5 u of a wheel rotating 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 acceleratio +.ylf5udjw,g n of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the e k9;howr ks5 g9z.*o;olnp*+lsv6sds dge of a large freely rotating turntable. What happens if you walk toward thl*s9*s rw6 +spo. hdvo5 ;knlkz9;osge center?

A shortstop may leap into the air to catch a ball and throw it qui44l3i*ytfrtt 2nmf fp0(b .qy ckly. 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 lf 34.yr(mn2ft4*t qifbpy0tin the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discuss why a helimmqgr-5u fx3 7copter must have more t f5mgxq73rm -uhan one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles i;y)* p7 nq42qymc)id4i r5c 0s xnclcin radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincidence. Calculate,ey 8 gqc(+snd.z-ym4 q using the information inside the Front Cover, the angular diameter+ngs-mc4q( .yezy qd 8s (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 from Earth. The beam diverg/mq,tc peb3y+3xc cs ,es at an angle b/qt c ,mex3+c ,sy3pc$\theta$ (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 e n- lw 6a-(qn1mthftm6j.je(a rate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular accelera 66jafnw(mmn(teh-qlj e .1- ttion as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the ballbo84e,uocl3i8zb j/ ih*ax+x makes 15.0 revolutions, whatol xzib8o /jcie,83+a 4uh*bx is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. b(r9esz:n3ofcxd ,x :How many revolutions do the wheelsxbc(s,xze d9r:n:o3 f make?    $rev$

  (a) A grinding wheel 0.35 m i, sq a5( x2aud9uiu9atn diameter rotates at 2500 rpm. Calculate its angular9 uu, xs adua5t2qai(9 velocity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.0 s (Fig. 8–rn.we *b: 6 jgwb 34gi9kgb.n9vre7um38). (a) What is the linear speed of a child seated 1.2 m fwb mr3e ku74n g v9.6n9:b*bgge.rwijrom the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its nmm99pm(t;8e r9 wa 0 osyi7xxorbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a ,f(mv9zbol* jg19 ej8eqhe b*point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.0 cm from the axis ofig 6g,sf4af5 2)qwcxx rotationfgx 6qic)5g xsw2f,4a is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates unifodb wur)g4*kphi0-fv.i 6 vge .rmly about its center from 130 rpm to 280 rpm in 4.0 s. Determi.pv*u righbk gv .)04 fd-iwe6ne
(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 u+, h7p yg 5;ma*1thldk2cqe6dtd-cz s $R_1$ 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 Mooc l/bl16xtzu. n, astronauts aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from6 xuzlt./ lcb1 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 rj .pi xm30;b eqq1)32i,ek wcclf6 darpm in 220 s. Through how many revolutions did it turn iw ax2.p3e 3mcki ec0);q, 1jlfqidb6r n this time?    $rev$

  An automobile engine slows down from 4500 rpmxt43;atq p 3xn to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeed jets in a whirling “h; *z8xrlin 1l* )woc3ba+gyf duman centrifuge,” which takes 1.0 min to turn througwoy x)fn* dz+ b31;cira*8gllh 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 dia4uf.ud:)6c vcr1hm wlmeter accelerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a 4 lfh)c.u1: uwrvmcd 6point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It ofqkt- onus 0s-)/yq2r6-uy q turns 1500 revolutions bef-qoyu2)futq-n 0sq-k/ y6rso ore 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 revolut54rzxyeg cv7* ions as the car reduces its speed uniformly from 95k4 7g5ex rycz*vm/h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces its spevdgk,:d4- ikded uniformly from 95km/v d-kg,dki 4d:h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on ep3 7 n *sgnhm3eshi.t7ach pedal when climbing a hill. The pedals rotate in a circle of radius 17 shmp.n7sg in3h*3te7 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. What is the mafzv y1 e.vbsx4e:7ydt9jm tvi/e k78np; //negnitude of the torque iftnd8 eb n/zvmt4e / x1yvysepij7e.9fk/ 7 :v; 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. Assume -: w qtkhf2.o /7zply-4y+t;ubihm ql that a frictionut+tiqybhof/h7w. pz - lm4-y:q k;l2 torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a massless rod whichxr s2xa-aykj1 7(z)jh;cr(z w/:sz kd pivots as shown in Fig. 8–40. Initially the rod is held in the horizxka :-/7jzhza j1rc s d)k(2 ;rsw(xyzontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head g:2 kd ;a*kuf6j/kmirof an engine require tightening to a torque of 38jr6kd:ukk2a;mfig/ * $m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of radius 0.648 m wh7y.n poq;uli , u2l7alen the axis of rotation is through its ce7lpq l2il7 no;u. yau,nter.    $kg \cdot m^2$

  Calculate the moment of in d.sxt6m3ko:7 .cm+kyimnl 9 9 tyvys6ertia of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The k6.3tmy d+k9v:s 9m7.txmcyosy6n li 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.diuea70/8i(h j l jgc horizontal circle of radigje07 8h/al(ijudi .c us 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 wheel6 yszdjq t3/-x rotating at constant angular speed (Fig. 8–42). The friction force bej3s/ -qt xz6ydtween 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 in5df:dtl 9;,pvaqqq0 dertia of the array of point objects shown in Fig. 8–43 aboutf5p d;tvq9ad :0 qqdl,
(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 iyv7ek;a0v/qdut d /1 35er7e tvzpvz4s $5.3 \times10^{ -26}$ 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 cylin 2oe(t00qkciu k:nn8tdrical satellite spinning at the correct 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 8ockt0n (k0ne ti:q2urocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.50 cm and a mass zo14mw0v +czfspcweg0p 9 1h,tyu7w4 of 0.580 kg.fheyup 4p gw 1t +v7 w0z1c9sowz40,cm 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;0yycc6lxr6oebjm:mns 7* n* 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-rou7vdp 2 /+kb v yh(cmaf:zhx8n9nd 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 rah+nm7( vczhya/p98: fxvd2bkdius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is eventd(oz*wvm 7im6ually brought uniformly to rest by a frictional torqwmzvo* m7( id6ue 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 a5w6cnai 0oz *;shek/ rw ;umr8ccelerates 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 solelyex6pt/ 5pax8;jzb 1hn by the action of the forearm, which rotates about the elbow joint undepb5z j n6/1tp h8xx;aer the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considere37/izyfz) t)raah 2vo2wm q4y d a long thin rod, as shown in Fig. 8–46.a)y3 i2 )t 4armvoy/2w fzhq7z
(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 gb6n6 -1gd 04vuk2a 8;b/q k1qxvcuj l(wgrx pof 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 accelerates6x3/zb: b,ps sf9wojw the hammer from rest within four full turns (revolutions) and releases itw9 xfs6z3bw/ b,:pjso 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 moment of inertia ave pf(xu; , x.6rdw/xof $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops6x24ua xql.kz 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 ar4)jz)yst3ax 43w b2k5 xtc jrnd radius 9.0 cm rolls without slipping down a lane at 3.24 xc3 3zjrt)js5a4rxkt) wby 3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the suqi zrsz m1j 8ur+0 mr/dcu.2+gm of two term 2/crrz+z+usdrm iq 18. u0jgms,
(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 radil i1v3o55dqt 6ukr3ltus of 7.50 m. How much net work is required t6t d13k 5tluirlqv 3o5o 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 kg stary *-ug/e :w,6hdl gny hfcsr)/ts from rest and rolls without slipping down ah6,)/ed ul - :rgny /w*ygcshf 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

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

  A 2.30-m-long pole is balanced vertically on its tip. It starc*vkr lb,4h hbzo e)5-ts 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? [Hint: U4-)bcoz *r5 hlehbv, kse conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kgqiyga9ks tj);)ww vv3/tfa)4lms0o 4 ball rotating on the end of a thin string in a circle of a /iqo4swa))m tl k094v)s;f wgt y3jvradius 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 unb* aibhxx , k/k7w( 9gc+nh3 wp92zqrkiform cylindrical grinding wheel of radius 18 cm when rotating at 1500 rpmakqbx,/ngcx+( h 9b7w9z3rp *ki2khw?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands ato2c 7iuo x.b ;:n/tnssl(zlt: his side, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, .(; :so7u:2l ilbo /ntxs zntcthe 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 moment o qrv/ymb/7;u pf 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 tr;/vpmb yu7/ qhe 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 5i50+;i tcb c9jb h29c syzcyeinitial rate of 1.0 rev every 2.0 s t5 + 0bh;5cjsb ieyc9 tci2zyc9o 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 thstd 69c,ulok y40) gbxrough 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 chun94yokuc6 )gtlb dxs0,k 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 figuy90e;z1 h fvqrre 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 momentum of th5 lrhy94 4o8jmom vq6ne 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 iw-55mzf 7gqwy* hjv ,ddentic 7wyd5j,m*qwf g-5hvzal 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 i hb 5i2c2/kua2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of 29 bll izuv)p4a rotating merry-go-round platform of radius 3. )i9pv2uzb l4l0 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 rotating freely with an angular velocity ys8tj q;9aul2of 0tsq9a;ul82 y j.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 w+jr ljy/ 34z -rhu6okqhite 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 nearest +q ozrjjrkyl36h/ -u4 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 wih0lg6 pgi.5 pvnds 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 rob7-)bmb1 d sdt9sp -$ 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 rotate freely without/gvxe1 w/ gog5 friction. The moment of inertia of the person plus /g evwg/xgo5 1the 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 ofjp(bgaa 1 r,2ed+t 7wd a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of ig2rjap7wd btad,(e1+ nertia 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 rolls on the ground with the end of the rope lying onj* :kr 23v 2kied(amng the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s cenv:k*i2 jrmdga 3( ek2nter of mass move?

  The Moon orbits the gcw ff 3g24cp i,t g;-:giphh8Earth such that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Ea wpfgg g-8;cg 2hcpf3t,iih4:rth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 1 eeerbknw +w(k ;jxq2p.)kg79 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of d;ff wer 8lg76radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interve 8fdw67l gfr;al at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disksb :g/e7drwl 4 sy3cx5e, 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 calculab 7e3x5el:s 4yrcw gd/te the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of the r,g.,z r8x .lvafgzic 971 wnsxear 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,i ,bk/wa17woz 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 spherb7wi o,za/w 1ke at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-po8ga nv (3r mzm +khy(xdw1me *-u8nm2h8p 8lcfllution automobile is for it to use energy stored in a heavym2f3 ry cmex+a8wn-1d n8 l(gh *8zu(p8 khmvm rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates wej 90oj3v /3p f3-a kg/hjusxan $H_2O$ 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 surface at e4fb-w r) ea 7v7tab3hspeed 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.,r8 *z5nxh /;prhokfa ftma);+ we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is hear)z kn fhhtamf*o5;x 8r ;+/pld horizontally 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 radius R. It is standinnz 77e+ 4pwhd1tgz1xjg 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 (Fpzhw 14gd x1+nzj 7t7eig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s kqqn. ypb.x7 7on a flat road is making a turn with a radius The forces acting on the cyc q.ypb7x q.kn7list and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.5 m and begiu fj:uznay933ns 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 re:u 9za nj3y3ufquired to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid9tl+ lctp8gq ,ja5y-q pfh:c/ cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinnla hqctp5 t/y- jl,p c8f+qg:9ing skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindrical plao 2h:kh,j 4stuif,dla-4ih,a tes, of ma itah,-djus,i 44,okfhal 2h:ss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 radiuss5 xdk*c p 6e)oy)k lblr1l9b+ r rolls along the looped rough track of Fig. 8–58. What ek l5 61o* r)yc xb9lbsp+)dklis 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, b*grb .qygm,v .q1 3jh7gb;xjgut do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces 9i da:gv**c,.)emq hjpt-e us its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m.sqc*d: e)t j-,pmg 9*e.uv hia (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