A bicycle odometer (which measures distance travel:lgak rr57r9 t 7pg1wccvye8(ed) is attached near the wheel hub and is designed for 27-inch wheels. What happens if y t57p8y wc:ar9gv(clek1r7r g ou use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocitmfd5-r9 p v*czy. Does a point on the rim have radial and/or tange-zr*c9v d5fp mntial acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the angular velocity 9q,gl92x ueu.m1fwr3tu.z lj $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Expla 6f4/*s by6oinjhu +tmin.

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 uew xs31/,he 8khya9jwith your hands behind your head than when your arms are st8j x/k,ywe he 391shauretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle hau9qq5dqby;zm0lgs; w 5eo9xz x; ;h(is seven sprockets at the rear wheel and three at the pedal cranks. Ini qqo;xy z; ;bgqx95ul0dh9 m(5s;ewz which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fask uple1/ej70*i rqed0 t have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotatd0pq0 li71/e ek rj*euional dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig.+qq7qb xqx5 8g 8–35) carry a long, narrow beam?

If the net force on a n * ic5q4xsl+0udc:z5yp trr 8m;v4ehsystem is zero, is the net torque also zero? If the net torque on a system is zero, is the net fxre zcvn0dc5;:4lp y*uq5 8r4hi+ tsm orce zero?

Two inclines have the same heigc*7 le 4zsosy4;k9 ieyht but make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the b 4kc;sleo7ysy49 zei *all at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an incline. nh/mj8gey,o:xddn)c z)u vf2v+e w: 6One sphere has twice the radius 8vhy,vwdu /xc)2jo:gn:6 z+een d f)mand twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius3 7d8twh :ucj6:uxyaq 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 gxctayhqu :6dw3u 7: 8jreater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet mostb aatm/mb +7)j moving or rotating a) mjm/7tbba +objects eventually slow down and stop. Explain.

If there were a great migration of people toward the ;emitrufck 8d/dx*4 p r91p)k Earth’s equator, how would this affecurfp1mcd ei9p t48r/x*kd ; )kt the length of the day?

Can the diver of Fig. 8r 0smjl.po1vw8j/br3* -lztu hfs; 6r–29 do a somersault without having any initial rotation when she leav0; 6tlpls f/zv*8sh rurj 3bj.mw1o-res the board?

The moment of inertia of a rotating solid disk about an axis through its cen b 0oxum2 kvl4u-ezh3u7/g-rb ter of mkxu4h - rv u2b0lm7bo z-3gu/eass 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 stoo:uj1 xzv bi55hj t2fr3l holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will yof2b hi5v5jrt :zuj x13ur angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and u.nmxtk5muv j3 0 l5;zhave the same mass. However, one is hollow and the other is solid. Describe an experiment 3 zvlj;mm5t0k.unu5 x to determine which is which.

The angular velocity of a wheel rotat2lpe0/ 1dsabyhhlx, 2ing on a horizontal axle points west. In what direction x /1h,b02y llepds2hais 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 increasing or decreasing?

Suppose you are standing on the edge of a large fq8i-ed eembld )twht/zb10 57 reely rotating turntable. What happens if you walk toward the centerhd 5b7 )wqi 8teme0bl d-/1zet?

A shortstop may leap into thedd jrd1 erz1ti/1h y8* air to catch a ball and throw it quickly. As he throws the ball, the upper part of his body e* ijr1dr h8d1z/ty1d rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discust :y,u.vgz km b.le(5fgw* sg1s why a helicopter must have more than one rotor (or propeller). Discuss on fvy1. ,us5eg(.l* bgmt: kgwze or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (a) 6cee3p:cu 3* hw+bdvg30 $^{\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 co ny ulgzz 0cbs4n3lj9q;*kx62 incidence. Calculate, using the information inside the 3 syl*xnb2uq9j0kgl 4 ;z6nczFront Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Eh+2wqjd01xglevx k* , 5my8bcarth. The beam diverges at an anglgve5xy1b8 k*x2 c +qh jwld0m,e $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. When the motor is turned 7 qj/): drwogypgydw u))bu3: off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow down? gyur) dd gb3:)wq p/)yuw7:jo    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the ballb djmm0wp 2f;1 makes 15.0 revolutions;jm2m0bfd pw1, what is its diameter?    m

  A bicycle with tires 68 cm ins 9dksi:bp,6. jbluq7 diameter travels 8.0 km. How many revolutions do the wheek db qlpu:si6sbj.9,7ls make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate itsg4d*f r cl+mi0by7l c9be l5;b angular veb0icy *5c+fel g ;dbb97 rml4llocity 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 makesa u1nf pf4n:k/ one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child sf 4pnfu:na1/ keated 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 (a) in its orbit around the Sun/hm0u 88 c.wggnavff5    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed ofa9ktnmkz .(ri /k9)fk 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 centrifuge rotate if a . je6wt(dflx*7mk.ta particle 7.0 cm from the axis of rowm e.l(6tkx*fjdta.7 tation is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center frg11/:ya rlzxk5r yzdzp6 0)g wom 130 rpm to 280 rpm in 4.0 s. Deter 1krp/0xwgryg5zdalz :y)z1 6mine
(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;hzhtc( gd7g)d 0jvz5s2cq1 ks $R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboard the Apollo sskvkb328x rm (fm40ut:dzza-pacecraft put themselves into a slow rotation t -tu0svak83 2z4k:xzbdfm( mro distribute 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 2 f 7x0f985)+ ppasv wcmw.odvg20 s. Through how many revolutions did it turn in thisv+xwv0spc 7pd 9gf5f om.aw)8 time?    $rev$

  An automobile engine slows down from 4500 8iu4ugrgv zyt 75 h*d+:mh7zh2h d.fb 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 the stresses of flying highspeed jets(o66*,xdl keqyb ul9 m in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complqk d9(uy*o e 6,lmlbx6ete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 360 rpm in 6.5p-ganh (wd* )q s. How far will a point on the edge of the whee*d(pgn a)-wh ql have traveled in this time?    m

  A cooling fan is turned ofizc::8te (d 8m2yajj bf when it is running at 850rev/min It turns 1500 revolutions before it comes to a stop8e:jy 28a:zicbtm(j d.
(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 revolutionsf2 tbs *voah w. 9s)2-;vjuxfz as the car reduces its speed uniformly from 95km/h to 45km/h The tires;xs f9z )osvhb2vj -fu*ta w.2 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 reducfx ngmq 2t4:cg+7oy 6ges its speed uniformly from 95km/h to 45km/h The tires hagnfcgogm6 7:24q xt+yve 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 wh cz aodwt6mn*9nel, 34en climbing a hill. The pedals rotate in a circle of radwmn cl 6oda4, 39e*tznius 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 th5 tdhe cu(xpb g,z -/)7dlq7gfe magnitudxl7 /)7b-cdg(ef z5thgp uq ,de of the torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of0e;:66dw dr(lyymuw9v sh f4q the wheel shown in Fig. 8–39. Assume that ayq ueh0(wd 6; dvf :m9rsw6yl4 friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mhgviy, 1pl v*(ass m, are attached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque l,vv pyhg(*1ion this system.

  The bolts on the cylinder head of an ei,3 focly:uq2z/d nx+;fj y;ebf.b:e ngine require tightening to a torque of 38 / xb3fo:fncy; .el q:2d+f;y,bie ujz$m \cdot N$ 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 when the a g wz+l-6ie.t.7tkrct xis of rotation is through its ctw. 6k7lte g+z.ic rt-enter.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. The rtolkgb n+o5i7g+ )2f) b-e jbqim and tire have a combined mass of 1.25 kg. g bjf)o 2+5on b7+t leq-k)igbThe 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 horizontq/m*c 1a4teat a ukm33al circle of radius 1.2 4c km te/aqat1a*um33m. 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 rotat5ia0: zn2afv1 .ewf skkaj7c *ing at constant angular speed (Fewvjnz 2kfak.a5f* 7s1c 0i a:ig. 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 a6r7cgumi /e:33xtqbf rray of point objects shown in Fig. 8–43 ger 37u t qb6cx:3/mifabout
(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 maq q:6n7q(di fn:ph/bnss is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the correct rate, engind7vgfi7.6ts k j7 c2vseers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass djv . 76s7gctf72s vikof 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 withs/rhcl .8d3zbmtk *eqrf.,0 b a radius of 8.50 cm and a mass of 0.583q/.mdth8r.zecs, lr0bk * bf0 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 sw.uis28(t,vlh 3tb mdwz+ vw5mings a bat, accelerating 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 smal* jbyp8b3wn)t*w q zm7l hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and tw*p7 wjb)wny38 b* mqzto 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 iped3* suq ns-:.sf(42o8 nu l( qckkpms eventually brought uniformly to rest by.du 3 q(eokp:slu s4fqmcp2n8 *s-nk( 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. 8–4//8u )l:5vbm*fyke .ce vmchz5 accelerates 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 ve gb 24f i*tibj2 2)f)mcnqo5thrown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelefgme2bvi)2 2*ot q4j5 ci)bf nrated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, as show f+6loim ,x;uzn in Fig. 8–46.m, ofx;+l6 zui
(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 osgf;md lb(2 h*3ud3 cdf 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 acceleraa 6v o;g8ml1httes the hammer from rest within four full turns (revolutions) and 1ov;gaml8h t 6releases it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a moment ol wnjyf93 (nz*ngtid:7im .0zf 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 torque of 28+gz 0kxz7 cu8m0 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 pj3dr+o sv:l )ly-i8icm rolls without slipping down a lane ati-jp3is)+:lrd y ol8v 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the yqj*fn -2py6++mn4kk vrohf .Sun as the sum of two termrqpn 6vh2fynkk+*mj4f. yo +-s,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius of 7.50 m. + yqwcr 4dwk p6c:t)*jHow much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is ar jc d+)w k46wy:tq*pc solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1. 2na 61sgdp5xsar-;n /ncio-g80 kg starts from rest and rolls without sli dr6ap-2o5sg ;ngan i c1x-/nspping 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. I-k0ymyeuk 3)jm s 1ebu)ea r20t starts to fall and its lower end does not slip. What will be the speed omb2eayres-00eu1 y ) k 3)ukmjf 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 the enf eeu9 ude(+7lzc:xd0d of a thin string in a circle of radius 1.10 m at an anguc:dx0lee f+ e9u 7(zudlar speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform -no*z:4sgpazs+c . x;4pfxxkcylindrical grinding wheel of radis+.gpxf z4;ocs*4ap xkz n:-xus 18 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 -o ;qo: gpv)oex zusa,2+ca g/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, the speed of rotation coo)esaau pg:/2qxv ;g,z o-+ 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 of icf*-on s /3wqdoa* q)rnertia by a factor of about 3.5 when choodf rsq)* cn/-a*qw3anging from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initig:f34.aapol wwtz. w p(9 tmk7sd vrs.k91hy6al rate of 1.0 rev every 2.0 smtrf1(z gkv l.w9: ho.asdpypswk64 t 3 .a79w 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 rotatingg.t fuv: eqg (e6to:r2 around a vertical axis through its center at a freque qfe:trg2e:g.v(t uo6ncy 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 after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum f nqa0.qilw 9+of a figure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angularabtaom t 2-9195jz iga momentum of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia m oq+7ny(hs(5yko kr78isk dhfr66t9 I is dropped onto an identical disk rotating at angular speedqr6mshyyntk ( 876ishoof9+5(k rkd 7 $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns 4bnuavp65e+6 a*a hrrat 2.4 $rev/s$ 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 merryrr71w 9-n hvry-go-round platform of raw1r7h vr9- nyrdius 3.0 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 rotating freely with ayrbz(c l* j-f;n angular velocity of 0zlrcfj-*y (b ;.8 $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, losingj1e,f 2 /qa() acmqaug about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assumcj/,g a( 12qmaaquf e)ing the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in qa ri3f,0e92rf ny0lo 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 j/vg;m l1 fmrp .w7ov(/ef,b.gdakv9$ 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 atl o:us.gmu j( a4 ou8zbi6j*)o rest, that can rotate freely without fro.iu: jlmaguu46(8) j*b zosoiction. The moment of inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diameter merry-go-round tj9xiuy 7s:t9 mj v4r1tw rdk31urntable that is mounted on frictionless bearings and hawt9s1tu mikr yd:47xr j3j19vs 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 en e/bo1.m /eiv(eyio) 5;p zrxqd of the rope lying on 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 doe.1 io;ovieq(e y /)bp5zm/xers the spool’s center of mass move?

  The Moon orbits the Earth such that the same fx2.z 4( ) agakgmzr5lside always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orb4 fzzl2kx)g5ma( rag.ital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 1 m/ ,tfhs8x mra 9hk1-zg67im. gy$s^2$ 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 radius 0.inv/183faj ks9h. ;vum)muzh 20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angularnuu.h himv1j vf9)3/; 8 zasmk acceleration. Calculate 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 k jyz.,pkg -/m xrp5uluj-08ykg and diameter 0.075 m, joined by ul mpj0 y.pr,5 zxy/k-g8 kju-a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo 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 spea ,2j 20j0.x v.mkxmlt( yqreled of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 timesgli9p20v 1sakd )-atp as great, were rotating at a-12g p )0 iksaldtpa9v speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution auto6-r sf0gkfk jyx7:(vegcu*p / mobile is 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 flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km f 7*exk0jyc vr pf 6k-ggu/s(: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 an5a 0v94bvaglx $H_2O$ 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 spemd (2n-;ic blwed 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.252frcpze0:hqfy, uz , we 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 tip of the rod. Assume that the f, 25 yfuec2 zqfrzh0p:orce 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 stapu7s7d 5 h5j+x:;s(uebo jh d02xsoid nding vertically on the floor, and we want to exert a horizontal (e5h j7sh0;2bxjdoi s +7sodduxu5p:force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is makevnc l 3 80 *flt/)qj-zqykko7ing a turn with a radius The foel3l)of/qk yv7 8t0c *nqjk- zrces acting 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 aj*95 fqr lfnp2 0.50-kg rock into a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque c 9fr2jfq5nl p*ome from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her ara (e;- )qv qpikap)s .v.myhbt61cs2ems as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstrebpk(6s p .e.th aa))s-yqqe1 v2ci; mvtched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindricalao*9g :y+eqrd plates, of +grdo9yq :e*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 rou(.8b/fbj 7ll nkm s+sugh track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must dr s( 8k+nl7fbbjl/smu .op 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, but7b,tmj fy22* mljlr8z6exp u/-c1 ymd do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces k74 hfhqkxb1 r ess.b8j+h -vbhvl1*2its speed uniformly from 90km/h to 60km/h The tvhb71eh4kl1 2-xhhvb rjs f8+ksb. *q ires 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