A bicycle odometer (which measures distanczdc0ko4c(3 tle traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 2ldt 0ko(3cz4c4-inch wheels?

Suppose a disk rotates at constantpsr:-bt2 vf*,lsn rq6 ft6t0t angular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential accet,sb6 -tfrst6n*l0pqv2trf :leration? 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 q,1; -0y gtzy2eyqms 3 vy4fcrangular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a lar7qg w/phe 0a3gger 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 afr9t* fx c7o*n sit-up with your hands behind your head than when your arms are stretched out in front of you? A diagram may help you to anx c9f*7ftorn* swer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the p8 xih 6i9nfqch*ci2h4edal cranks. In which gear is it harder to pedal, a small rear sprocketi89inx*cq4fh c2 hi6h or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have a ;p (ur fqi(wga l08fli99xw7slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explainlaralxquf0 7wg(ii;fp w 98(9 why this distribution of mass is advantageous.

Why do tightrope walkers (zfzo 5(7 eayu7oxb8 m0Fig. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the net torque also zerjeqtfot0 ;q52o? If the net torque on a system is zero, is the net force zero?qtf0t;e5 o qj2

Two inclines have the same height but make different angd5ot)o2d-g, rlu lg()i4ft fgles with the horizontal. The same steel ball is rolled g-l5f (tu2io g,og)d4t frl)ddown each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultul+l5s k7c2o2(glb1luv dw:h aneously 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 inclincvu g o5s+1(7:l2w llldkhb 2ue 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 radius and the same mass. They starth 8i5s-*nklpu from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom?lk8p n uhsi*5- Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum avqwu s) -yt3m ;xe)3y2gzlb/v)fj( a tnd angular momentum are conserved. Yet most moving or rotating objects eventually slow down and stop. Expls-3(xyzmq)l3/w)evt) tf2u ayjgv ; bain.

If there were a great migration of people toward the Earth’s equaw58vj7 ra/fvfo :05me e s:ojgtor, how would toe5wj m5fa:0 e j8vsofrvg/7 :his affect the length of the day?

Can the diver of Fig. 8–29 do;1n c0tzgvfw.cr/.j7n y,z usu.8nf v a somersault without having any initial rotation wh, .tn jwuv..n8 uczg107cznfsrfy;v/en she leaves the board?

The moment of inertia of a v+l.,af(j llkk(6rwg rotating solid disk about an axis through its center of mass i kwrfja 6gk(.v( l,l+ls $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sittin+ 8bo6uylkx+l m -e9vcg on a rotating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, deym9 v x-+ kcob8+ell6ucrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is holl8jso/fu+f w gey:/g7jx9b7g l *:hzzi ow and the other is solid. Describe an eeyg: w/ sg 7iz:jjo89/ug7 h lbxf*+zfxperiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points wes/lxpgh*jaxw 3 t( 3a6dt. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangenp3/ ja( h *xtl3wxa6dgtial 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 omm pmfi: fc5r3v z5v9.f a large freely rotating turntable. What happens if you walk toward the centervc5m z3m 9.f:r f5imvp?

A shortstop may leap into the air to catch a ball and throw1c woyk p-o4 e+t45rjk it quickly. 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 in the o+p 4kw5rkeyc 1 tjo-4opposite direction (Fig. 8–36). Explain.

On the basis of the law of con0je .y dtzlm.:servation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the se 0dz.elm yj:.tcond propeller can operate to keep the helicopter stable.

  Express the following anggp 0ztk qy.u0u83rip+ les in radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coin7q+xyh. +w xu vf.1aencidence. Calculate, using the information inside tf+hx ya7.uwxnevq .+1he Front 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 thel+vn34i2hmi ,t5n -hb b.gg jk Moon, 380,000 km from Earth. The beam diverges at an ai k.tvgj2i 4h+g5n,b hb nml3-ngle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blendeml2jkqziy y8aff1z 825t.fz0r rotate at 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 acceleration as the ff y lzzk 0m8tf1aqi2j5y8.2zblades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to anot ;j1znhv4v4f)fjs+o)n rwm m. her child. If the ball makes 15.0 revolutions, what is ;j zfm+v1)f4)sowmj .4nvnrhits diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How mb79wwgy8v n2 1o(fthp any revolutions do t2 t7vhop fy8 n9(b1wwghe wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 ru 9m :it(jci+opm. Calculate its angular velociot+i cu:j im(9ty in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.0 s (Fig. 8–38)7 l6g/ ym(plup ,e3yzd. (a) What is t,6gp mpy/y du7z(ll3e he 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 (goe4 f:sd:8 dja) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a podc+ wx/f9g)-3nouq,ot v, bz7lru 3ugint
(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 centl.v/ex )ugf kn abx37.rifuge rotate if a particle 7.0 cm from the axis of rotation is to experience an accel n)7u.e.g/ 3av lxfkbxeration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its6bjlh f rst-. si)n5y6k*7fke ++laig center from 130 rpm to 280 rpm in 4.0 s. D+ ts hlbfk +yk-6si)*7e 5fai6ln .rjgetermine
(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 radiusq0cbh ; e)sad3 $R_1$ 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 spacecraywlr1r:p k ,380fb vee o*dp;rft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the starw, l rrv;ybo30ed: 1 kepp8*frt 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 indqksg+1(9gf v b7 l/ :c2ehbic-w*; gkossny/ 220 s. Through how many revols+oy 7; w bgg: /1kqks9sl/bch(ei-gvc*fn2 dutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calculatzykakli p)- *q1ym),oe
(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 yb 2fju, ;/ 8lq3eqcvdfor the stresses of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before rf u8qbc,q2j/3ld ey ;veaching 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 insbw2*a,n2sonp1mf4p *yn k* e 6.5 s. Howfepk 22 ,w1sboymnpns**a *4n far will a point 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 turns 1500 revolcd2(n g.oc9 v bt8ywe3utions before it comes to a 3(8. gdnvyw9 2 ccotebstop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces *yo nz:dh(3 0a0dbaol)+aqryjx8ux- its speed uniformly from 95km/h to 45km/h The tires have a diameter of 0oa *x 0l0+quyyraj a8b-d(xho) dnz:3.80 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 carfife xu2sg/*rux7 2.p0tovr (nm:4n d reduces its speed uniformly from 95km/h to 45km/h The tin4d(moi/r:*egrxu suv 2np0.tf 7 2 fxres have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each pedal when szv.lss++i h n+z--g mclimbing a hill. The pedals rotv-n.hmsgslzi++ s -+zate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cm h+8v2c1iuf uv hbw8 g-wide. What is the magnitude of the torque if th8 fhvv ui+gcb1wuh2 8-e 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 wheelnvqoqikuv8(h va 6e*:8:v5u h shown in Fig. 8–39. Assume that a fric*k:u6h v vqq8 e(:hua5 niov8vtion torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attae rnnr1*q kk+s1 -vsmvlp28w 0ched to the ends of a massless rod which pivots as shown in F2wn1mvq-01 * srreslkv k8p n+ig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening to a torq an5*x)1x jrrkue of 38 *jr xa1 )rnk5x$m \cdot N$ 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 ofzyo au4wbdfr a/ld0j8h.;7(n inertia of a 10.8-kg sphere of radius 0.648 m when the axis of rotation is thrd;( 7hufrba4yo/ w0 a8ndz. ljough its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 667r 8,oc2tx oq -hnc.kp.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub caht27q8r o.n o,pcck -xn be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball eyk:ekr9 *l jb*jjs(- on the end of a thin, light rod is rotated in a horizontal circle jy(-rjk*esj9bl*ke: of radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a fogxu)-h5 e3p potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and the clo gpuh35ex )f-ay 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,7h:cs 8p*zeqk 6ax 9zgxdp z( array of point objects shown in Fig. 8–43 aboh k *6z(qp:xed9zcx8a,7s p zgut
(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 isfy; : *qwrvba+ $5.3 \times10^{ -26}$ 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 sj*8yr-lb ggnmo:;4 lcorrect rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a ra-ygb4 sg rjlml: *n;o8dius 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 with a radcu* 1qbejy1h+ .fcs02 ,tou piius of 8.50 cm and a mass of 0.580 kg. Calpuy2 i0oh1s jc,f*.1ctbu q+eculate
(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 u3/igk y xyhf yf.d-*7(h*j dta 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 small hand-driven merry-o2hd4v(lwh 3f6 , efoego-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 lh (h v2ef6e4wodo,3fmass 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 r1.n9fqfp ij 1+ oed9kspm is shut off and is eventually brought uniformly to rest by a frictional torqj fnk11+99.siefdp oque 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 7 hg jo:jx7qh.)d/1etd5su.x u + mi9z q$m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the action of the f8xm e i7.rqq2xorearm, which rotates about the elbow joint und7xe .mqqxr8i2er 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 03saoaxvo 7q/ be considered a long thin rod, as shown in Fig. 8–46.7xv03o aq/ osa
(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 8;)c4j4 eyt q1cp zofnconsists of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest within four xa p6y(,bw,umfnua;4full turns (revolutions) and releases it at 4nupw ay(b6 m,xafu;, 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 momh3me g+x3fv6asqn4 i0:mh m8wbs/j7 pent of inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque of 280 5tr,c,/ vw yffawj v0(my 3he-$m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rollnl;e9d d b8t4 t/fzk2.c 6nfzhs without slipping down a lancd6bh k.8z lnf4;9/2et fn tzde at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of+us- 5zb1monov tkh8049mleb the Earth with respect to the Sun as the sum of two term0 8-ok1m bn+5o4h usvlze9mbts,
(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 gmh5pwj:oe -37-)i ech-xhc smass of 1640 kg and a radius of 7.50 m. How 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 a s -)siepho ecjxw-m7 gh3c-:5holid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.nwe6grlb (p-i i/v, wjw. vg1.80 kg starts from rest and rolls without slippiei lw(/w b.v,j-rg.n gv i61pwng 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 to fall and its op),fyjj gg3mfx;4rl *7p6il low43 gy, 67pf*l;) imogljfpxjr er end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-z4gu jm qg2)+ 2q3kgx6yc hzb7kg ball rotating on the end of a thin string in 2cqm2hxu4gzz3 gg7j6yk b+ q )a circle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 6s 5k a5 n )6h46bgimgmva pey c8ghe9*+4ying2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotating at 150 sc 4an9yg6ga+hm4i hkbe568gv yimgpe 6)5n *0 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, sgj.s* ts su /sm58au(6o1gh/p w7tz mon a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal posiswgzhus* g/ao js 86s5tt sm.7 1mp(u/tion, 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 moment of ie1 qf.g9zj n0zp(cy4sm zk:2b nertia by a factor of about 3.5 when changing from the straight position to the tuck0 2 z4.qzn:eg9bmfys 1jpck( z 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 l7iayl+ sn7c5 rate from an initial rate of 1.0 rev every 2.lyn7i l 5as+c70 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis w0x e:..fznvuvvr( )y through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 r0v.fw v.ezx:)( ynvum. 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 3 zfo-;vzeqc3 ke+gh;momentum 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 angular momentum of thebl+qbin q r/aa /;r-,wq- *iin 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 inertim3oa2c sx hx 6+zft/8 0fxb8nnzv8yu7 a I is dropped onto an identical disk rotaft/o6 hy8sf8vz8m037u nzxna +cxb2 xting at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns ld5wdwqq:8a u2 o6*yoc/ 4hw eat 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 centpm,l y3 v :d9 a*:li;wvttho6ier of a rotating merry-go-round platform of radius 3.0 m and moment of iner :v;6awo llhvi,tm p:t id3y9*tia 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 rotatw 8 z7j34gqh7sycdnb0ing freely with an angular velocity of 0. sgwhy 748cqjd0zb37n 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, lrb2/aj1q* wtx jx9o 2oosing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass cab1tj wxjo* 9qx/2 o2arrries 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 win63muj tx9/(fchiv(r paujm ua; y--; zds 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 5 ui62fz 6bgfwmbz;df7v4qdy(ti z c6lj6o ;($ 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, initi +avo9 ae ,)9wa5+d0czndyckwally at rest, that can rotate freely without friction. The moment of inertia of the person plus the platfor+ww 5ydzdo)+cank,vaa 9c09em 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 div)w*dagb;j5 icb jcxn.lxr4ow.-e; * ameter merry-go-round turntable ) l xon*eg5vbjd cc;r;b4.j-iw*.xwathat is mounted on frictionless bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope ro 0tb v65-ccr3ux odi*plls on the ground with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The sv03t- ir coduc b5*xp6pool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces the Earth. Det39d0xoh2ecqn qu j8 mig(21nh . ibu+nermine 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 Earth.)qmh2 qg1 nu9x2u(+iie0co ndh8n .bj 3    $\times10^{ -6}$

  A cyclist accelerates from rest at a ratev,bkp vit;s2y5mm sl9m g0urjcm7; l ,j3,d0x of 1 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 radius 0.20 : g 3 2p s58dudci-gaxelxa,l3m acquires a rotational rate of from rest over a 6.0-s interval aigaexd8x32spl cal5:g3 ,d u-t constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of twgn r;,.qk; vbro solid cylindrical 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 energy to calculate the v; g.,bk q;rnrlinear 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 angul6coxog tk4s)7 zw:+jv ar speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times as great, were rotp4me.z -;x ib rpnd:+hating at a speed of 1.0 re.-m: e;+ibzp p d4xnhrvolution 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 qfaxb)33 g6wa hgo*s9s9 xwt, it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindricaltxsax 3qh 6wba*w f9g3 o,s)g9 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 +ol , (8tenfqsan $H_2O$ 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 at76*kh k7oxrs e speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., we ignore fec( 7j 9wy47hcy aci0z,l-tfyriction) 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 (z7yf-cilt0hcy4j wa e c7 9y,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 rad1j36jzs ,mkdm m*0 p7rhfqj,+ltm2yq ius R. It is standing vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has he,zm7m jm f ts 3pq,hj6+rk2mjql0*dy1ight h, where h

  A bicyclist traveling with speed v= f+mxohd *98hpna35qe4.2m/s on a flat road is making a turn with a radius The forces a me38hn 5fha +*xdoq9pcting 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 length 1.5 m and begins ,vx)agavatu6eo6j1 q9rz+k* g+y x-z whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate o yo*6 6j1au+x zqvvxr ,zt)ka 9ge-+gaf 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solxh m4.2ud- zy-p8t . el 5ivyce3ivt.yi.: hplid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a s - .8ivil:-c4x.5tdutzy2p .ehe.hlym vpyi3pinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consist hl 7ea7e5x t);;6bdyss .rbcks of two cylindrical plates, of mass)r;s7x bbt s7.;ca h6l d5ekye $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 16pnr7 cy)k:b r/ / psks8fgxyr rolls along the looped rough track of Fig. 8–58. What is the minimum value of the vertical height/ygr6f) n7s 8k cpbrks xp/y1: 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, but dbg3bl nxwvd (h :8v. / 78-cjzlepwwj.o not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutiz19kxbvg 8 q9yons as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diamet 8qg91yzxbv9k er 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