A bicycle odometer (which measures disofyvs:z)7 a9ttance traveled) is attached near the wheel huy)a :otfz7 sv9b and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a porky-s 0k3*au3z ni9njy r0i,l int 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 mani0yr-a3k ,z9s ljnk3uy*ir 0gnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the ang*ty 69bgwu18rsb c0h jfaua 60ular velocity $\omega$ Explain.

Can a small force evek29 j w z9zu3muhlbw3rqq5)a9r exert 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 difficp6 6 x:mkykzl- *n 5g*clz1nlault to do a sit-up with your hands behind your head than when your arms are stretched out in front of you?c:kxpm gz5nnl *a *6z6l yl1-k A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the pedal px vs tnm:ebr7e:9 38dkk+7 wocranks. In which gear is it harder to pedal, a small rear sprocket or a large rotkk edbw xn79p 83re+ s:7m:vear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on be(;kja0(opm vfing able to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of p0(jak;o fvm(rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fzrzhr.. jtfd :i8)x ;sig. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the xex.t0btw r5hh(w3b. net torque also zero? If the net torque on a systb xw0tr. xhbewht(3 .5em is zero, is the net force zero?

Two inclines have the same height but make different angles with the ho nx1r) gzq5a qol0zf4,rizontal. The same steel ball is rolled down each incline. On whz1g l ,qn)q 0aozxf4r5ich incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneo as z5u mv3kwi.j(ws951re,9mi7 ukxrusly start rolling (from rest) down an 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 ez(am jsi17ukw mevi9.usxr5 ,5kr w39 nergy at the bottom?

A sphere and a cylinder have the same radius and the same mas fbmflv ,,+0bvi++eyns. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the m+yb i,fl+bfe+v,v0nbottom? 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 moving or rux45qgd * xjq(otating objects eventually slow down au 4 qx*qx(5jdgnd stop. Explain.

If there were a great migration of people towaav h.ls-cu)v/ arf-q( rd the Earth’s equator, how would this affect the length (cv-h q./- uarfal sv)of the day?

Can the diver of Fig. 8–29 do a somersault without having any initial rotatilu 1-95mcbg cion when she leaves the boar- lc59ug1 ibcmd?

The moment of inertia of a rotating solid disq/po3g/ iz/ ozua00a ek about an axis through its center of mass iso3 /goe/aizap/0 zq0u $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2-kg mass in each/kav4 09yhsamuw 7n/yz .)eiq outstretched hand. If you suddenly drop the/wmsqa vn0/e z4yi) yua.9hk7 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 xtpbt+,7td9i f4k,uy and the other is solid. Describe an experiment to determine which is which. f,ti+upkbdt 7y,x94t

The angular velocity of a wheel rotating on a horxer0l +g/fdt 9m*3 kfnizontal 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 a ntk90e3fr*d fl/xgm+t the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating turntablxp gr- oa3lbo51 , 8ohq0wq7mwe. What happens if you walk toward the cen ql, -ao0wg8r3oh pxbq5 17wmoter?

A shortstop may leap into the air to ck2z 8czin3n ,oatch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his 2czn ki3n,oz8hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of anaqyf;i5c mq 3nk30nyf78t;i u gular momentum, discuss why a helicopter must have more than one rotor (or propeller)qq8y5 ;unkf i0mnit;y7fc 33a . Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in ra7f2alfrwp l,t( 4 pk/tao7+dsdians: (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. C ;v2v0 c7(0c4t5bjjueaskehatwa/(kalculate, using the information inside the Front Cover, the angular diameters (in radiaja 7uc(j tb2va0 aws0;/5ck4te (vk ehns) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at thm g/zey4gjs9 6 7+fzhie Moon, 380,000 km from Earth. The beam diverges at an angle i gm zfjs+79yzh6g/e 4$\theta$ (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 i;dc k((bk m11i mwoso9 5uv cxttf1h b666py0cs turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration sw196vi(01bp;th1 tkxkc mo6y( mdu6 5f cocbas the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the balldcj:n cr04 (tn makes 15.0 revolutions, what is it(jncr t0:cd n4s diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. Ho,hltd (n/v.qaw many revolutions do the wheels make? q/,nt.h(vl d a   $rev$

  (a) A grinding wheel 0.35 m9 ftpbn ttj)14 kg u4e:wg/bv3g.zq x. in diameter rotates at 2500 rpm. Calculate its angular n 1 bb :/wvzg.t)3pjqufgx44gtek9t.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 c(i5 (sd+ stm:p0rnjruomplete revolution in 4.0 s (Fig. 8–38). (a5u:nt s( r+ijm(rps0d ) 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 velocity of the Earth ( 0 ,q.fs)l pxn+ kmt)k:fkc7rda) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speezc)-ips y zbxheteos-ni;*m ,48 s1-od of a 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 c+c +avxlj:w 3kentrifuge rotate if a particle 7.0 cm from the axis of rotation is tv x+cw:3a+kj lo experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 a u9 yacp ;nr+euf(-/xalp .0v:v6ftr rpm to 280 rpm in 4. 6 9xaapyf0/(a+rv;u-pe rtlcfn:.v u0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radiu1;pq 8i5( ztp8iwzan- adx9zis $R_1$ 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, astronao llq 6zjwq iw1 0cv33jrk6,-zuts aboard the Apollo spacecraft put themselves into a slow rotation to divw1wqz 63-l ,r0 kzi3jlq6ojcstribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can 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 22:7nynx .lev j u8,z msm7,k jk*)6rnfw0 s. Through how many revolutions did it t8vj,nmu7w6lz xjen 7:.nr k*)fskm ,y urn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in n -3bnzix3f24g 1pwyh2.5 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 whir kqsws/0ty,t 4ling “human centrifuge,” which takes 1.0 min to turn through 20 compsy kst4,0q/wt lete 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 f y0w , rsbf9purtvv eu,f9l;(m0 6pf3drom 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have tra( t;u9,6rwpsbm 3ld,e u00rpfvff yv9veled in this time?    m

  A cooling fan is turned off when it is ru8i084 6 vw9re:olfgaodpy g-g nning at 850rev/min It turns 1500 revolei-pggw4og 6df a0vy8: r8 9loutions before it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolution/+4z*sri df9i7kdu+7a rq k gxs as the car reduces its speed uniformly from 95km/h to 45km/h k s4xqi7 add /+fur+ *grz7ki9The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed ogy8525b bvry uniformly from 95km/h to 45km/h The tires have a diameter 2ygbrbo8v y5 5of 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 8tuqi-,ay;kvnp vthh + 8(zd. her weight on each pedal when climbing a hill. The pedals rotate in a circle+ty8kp qdh;v a,nhti- zu(8.v 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. What is the madug-,0k1j;t svq o3 yzgnitude of the torque if the force ;o0udzgvqsk,31y t j-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 t n1 u/t8f4zbjzhe wheel shown in Fig. 8–39. Assume that a friction torq4f jz1ub /ztn8ue 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 masslesg;p2qp.4 ;o9cgzwlmoy4k + c qs rod which pivots as shown+;y 4 4.popwzk;ggoq m2l9 cqc 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 engibcl+fo,1by z7k9.mem ne require tightening to a torque of 38 c+f7em k lm b.y,b9zo1$m \cdot N$ 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 ineryg)j fg0+ wq oa;b x(t-*3rlnrtia of a 10.8-kg sphere of radius 0.648 m when the axis of rotation 0ft)jr;q gg o*3(larx-+ynb wis through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm +y 1ph mk6f0hvin diameter. The rim and tire have a combined mass of 1.25 kg. The mass of thep6+vmf y0h1hk hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is dpo6wqijf*+2( epdh *ieh 0 m/rotated in a horizontal circle of radius 1.2 m. e6pq *0mi2dih/*hf p ojw(+e dCalculate
(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 w4ln65mapcl7a bowl on a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her 6alm 5p 47wclnhands 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 inerti58 (7l(bp l toer yii2jbmv 94opp2u0ja of the array of point objects shown in Fig. 8–43 about0p2opm2i 5lepv 7(yob( i4btjujrl8 9
(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 ox.w1d vcr vt7+ nux1az7a.b5ja ygen 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 gah -57ryrj4v 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 rocket if thejag5vyr7 -4rh satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cyz(ylii jkyx,c ;w(j9 +linder with a radius of 8.50 cm and a mass of 0.59cj+; jyy lizxiw,k((80 kg. 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, acceleratijlola o1oq8*dxl* 9 )vng 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-round an3ewjqho w7 926aryf d2d is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the7jfww32oay2 d hr6 e9q 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 16 vyb64cijc( q0,300 rpm is shut off and is eventually brought unif b y64i(v6qccjormly 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 accelerates a 3.6-kg ball at /q;4vxvso.0fm ,;v- ahqm)s5vmdy isg7b 1cg 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 ballk5 7j(4az)lxur ry l)b is thrown solely by the action of the forearm, which ror)zlkb(j5x4)a u7l rytates about the elbow joint under 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 con62eo ny+hdzz-p1 6-e o0ol5wjmt ,sgbsidered a long thin rod, as shown in Fig. 8–4d6eoy sewgmzph t-bol,5j2n1 oz-+ 606.
(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 tg/m th, v3j-oswo 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 tn 1l9qdza4+e **zp )9q:fz 4 cknlnqsqhe hammer from rest within four full turns (revolutions) and releases it at a speed of 28q)* dl:an9 q9+nz*nsz peqkl44cqfz1 $m/s$ 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 ha3y( y 23s7zkjdsqtf1ps a moment of inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develop -7u2z nxyebiswby0 05s 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;o9kb ;w,che mpcik56yy6g* u radius 9.0 cm rolls without slipping down a lane at 3.g6yh;cp9k 6u 5*eoc ,wy;bkim3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respectnht qjh4() c;f to the Sun as the sum of t fn4thhj(q;c) wo terms,
(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 1(a3vn-0jnkyc 0n;jhe 640 kg and a radius of 7.50 m. How much net work is reqh-nnyc;va ( je0nkj30uired 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 kg starts from rest and rolls wfq r7ij nm)(gxf* 4j3oithout slipp fj 7f3iqn)(ro4 mgxj*ing down a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

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

  A 2.30-m-long pole is balanced vertically on its tip. It starts tpz s+)u;p8+8akljwjz o fall and its lower end does not slip. What will be the spe)zjl+8;+ uwaps j8pzk ed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on th9up(pvopf xde/4q+ d(e end of a thin string in a circle of radius 1.10 m at an ap o(v(p4dxu+e9pf/qd ngular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum oeb84omp.u0x zf a 2.8-kg uniform cylindrical grinding wheel of radius 18mxu pz08 e4ob. cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that is rotating at 1u 0cxd1io a;96-fdlt 9damw+oa e5sba rate of 1.3rev/s If he rou xflc1 +ew9i16ts-;m0odadb5a9a d aises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce her mceve*7m /m v2poment 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 tucpecmv/e 2vm*7 k position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotatio a8m2 4qdbzlv2n rate from an initial rate of 1.0 rev every 2.0 s to a final rate of 8a dbq42 lmv2z3 $rev/s$ 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 through its center a2yz85lgqtl/ r0lk2h e g pl.cze9f86lt a frequency of 1.5rev/s The wheel can be conlprlg0yclezl5h8 9/ 8 ke z2.2g 6qfltsidered 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 after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinning at 3 f,qho xdw*vakxs;ji5r7 ky9 *7ds e4).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 2rao7fn rhavh x; /vxj(h -;2z 1d3v2kmgsb,gthe 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 im,i8 g5*ob)x2n k0x;somtzx moment of inertia I is dropped onto an identical disk rotating at angular spee)8sox mt* iz knbg, om0x;ix52d $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at sex/ke8 d0rz 12.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a rotating mer.dwjyos m 8x4d9 oc20su3x .pfry-go-round platform of radius 3.0 m mo cw 9pu.3j.y2xdsox4s0f8d 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 vegq .nme*2jkr 9qy ::wzlocity of 0*r zge9:m jwyn 2.kq:q.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 even*utky js3yxkzb 3z220 tually collapses into 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 at zy 3u3sjxbk*0kz22y ngular 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 in exniu dch5r q6 n0 k9;a-jga40fqcess 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 *unix ovzi5wcvi )1d j -f6:mx) de84f$ 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 rf-ac7th; )/koe hx8y)iq t6a( wc f.swotate freely without friction. The moment of inertia of the person plus the platform i. c;hfs6h8 t7/()w q-wx)fi acote ykas $I_P$ 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 aptbm7v np* vdu: g8+i4t the edge of a 6.5-m diameter merry-go-round turntable that is mounted on fricvng pbupmtv *74i:d8 +tionless 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 rolls on the ground with the end of the rope lying on the vb /l9dhfkq:9ipr+ ev+l a7 tb: o/7uvtop 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 lle9+a q tv7/fbo7::/ul+b dk9p iv rhvength 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 sh a-f,r999u bs5vo wuaide 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 casw s5ua9a99rf,h -buvoe, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 1 m/+hnkbwuaf2 qwe hxbs . ../w)($s^2$ 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 sha- d* tbng;6eogpe of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at conse;o*6 bt-gdngtant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindri,e 1*bihi*s iqcal disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of eq1,s* *i bhiienergy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of the rear vr bi* h,23v7a znhp5o5zgs.5csnir7wheel ($\omega_R$) 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.- anbcb4w+p of88;byd*osm +i0 times as great, were rotating at a speed +on swyi;m-8dboa*b p +c4b8 fof 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, 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 h,s8vkv *,g 3jwbr1u muse energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mas3 kh8uv, jvw*s1g,bmrs 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 illustrate w.bsm4l9 cgr-qmz 0f3s 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 rollr69mkkejv35 y ing on a horizontal 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 mass M and length L can pivot freely f:j.xipto(1z5.ow9 ca )j b lw(i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then (5ol otjx1c i.w.9wbpja : fz)released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on the s a xb75xo2fl*wzn:i+f;w:kafloor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The sta ax:7x2 fwsflkin+zwo:;b*5ep has height h, where h

  A bicyclist traveling with speed v=4.2m/s on(6 b kwe22:kooudjvkt7lq k:,zno9 t 4 a flat road is making a turn with a radius The forces ac49u2vwjd oo: kk6bt(:ktlo, kezq7 2 nting on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of le /19pya3 c5(wyv4cc9 mmpi3vzq s-dwdngth 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come fyd wcamwy -ic5qdv/ ms31 ( 4v9z39ppcrom?    $m \cdot N$

  Model a figure skater’s body aufz4gtb; )ob7 ls)6ho( nw8yas a solid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with a7u8hf wn g ;la(bb4sot)6zoy)rms held tightly against her body.   

  You are designing a clutch assemb -n* y3*qwa+zaqt,r ekw5 4rgd zjc/y-ly which consists of two cylindrical plates, of zztc,q-ey*d -q5kawrn*/gw4rjy+3 a 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 radius r rolls along the looped rough tri;0snao, eqc3s0j,zy wa, ,m gack of Fig. 8–58. What is taico,zman, ,ws g0,q;j3s y0 ehe 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, bnagtdcvb .i8 6m0bb3 +ut do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car -d- rv ulkk7b/reduces its speed uniformly from 90km/h to 60km/h Thebkru lk -7v-/d 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