A bicycle odometer (whg ejvpw dg0,.,ich measures distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on vg,dg.jew , p0a bicycle with 24-inch wheels?

Suppose a disk rotates at c bx3 .:czdnqs(tew:)li/mhu 1)mjb:h onstant 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 accl:.hubm i )sq: ctehd:n3 (1z)/jbw mxeleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a singljtheys, .sbu-f (ch 05e value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than y(gvd9z 9xs/bino,a- a larger force? Explain.

If a force $\vec{F}$ acts on an object such that its lever arm is zero, does it have any effect on the object’s motion? Explain.

Why is it more difficult to do a sit-up with your hands behind your head than r zeqn;++n7-dci1 bojwhen your arms are stretched out in front of yo1nor+;jzq7inbc +e- du? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at;:iu5t rjz w:v vow/ z5zef+6r the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why?vrjw;z5+f5/ o e:vwr:tiz6zu In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lowtccywhtwuoamx j 1c9yi-,n2 6:m0i22er legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics,2x in: uyi2tm- cjc2ca 91o,y mtww6h0 explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narr /i,nddb 3hhj(ow beam?

If the net force on a system is zero, is the net torque also zero? 0.ja 0p0 xnoo6gdk-7z avra /vIf the net torque on a system xrzapkj ogavd- vo070.6a /0nis zero, is the net force zero?

Two inclines have the samej0h,dvddzp -7 height but make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline w0,h -7 vjzddpdill the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneouqz27:4aumv pd 8+ ztzgdt1a0usly 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 incld1v: 8gautaz zz4d+2 tmq7 0upine first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder s (nqau07cyl+7 d vm+vhave the same radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater roqm 0daslvu(7 y7+cn+ vtational KE?

We claim that momentum and angu+jb6s e8qi/,up7t du0e pece)lar momentum are conserved. Yet most moving or rotating objects eventually slow down and stop. Expla ieed0+j7c, 8u ete b)/6upsqpin.

If there were a greatgmkfp -6dv8 m7 migration of people toward the Earth’s equator, how would thm pm6-gkd 87vfis affect the length of the day?

Can the diver of Fig. 8–29 do a somersault without having any initial rotation z9y3z7clh a9nme-yt:when she leaves the board?79n aye93zhy tm-:z cl

The moment of inertia of a rotating fwtuhl ,lq7f+j32 + cpuw*6)ogiia)od ew0 4vsolid disk about an axis through its center of mass il+3 ojhfw 62cu74wiw)ep0 q *,g+ )oitlduvafs $\frac{1}{2}WR^2$ (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 mas9 cvq.e.v6zqis in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, de9iezv.. q c6qvcrease, or stay the same? Explain.

Two spheres look identical a fpzpgo1bx7d 3nu7 s:z(i y69qjxnt4 8nd have the same mass. However, one is hollow and the other is solid. Describe an experim 7nt u(op834sydfj7i1x9x zg:qb6npz ent to determine which is which.

The angular velocity/ 3f 6/ub3t7tn-rwghzf8qstz of a wheel rotating on a horizontal axle points west. In what direction is the lineas8 z tq /tr376wf-th/g3znfbur 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 freely rotating turntablk0y),thupynq x, oy* 4e. What happens if you walk toward u,oy *ynk h,4t y)pqx0the center?

A shortstop may leap into the air to catch a ball and throw i4 n:uh 5 *i/jx4v+clyheosl.rt quickly. As he throws the ball, the upper part of his body ryc4 +*r hxli.:/5vn4 lsehuoj otates. 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, discd 0ie*)3 dgica t1srp,uss why a helicopter must have more than one rotor0decrg1s3 ia *dt,p )i (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following a2-g7oze5 okrxjy2 l6gngles 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 coincidence. Calculate, u;7a rr9i ssq)kb0e;lwsing the information inside the Front Cover, the angular diameters (in radians) of theaswqr e ;7k90r; ibs)l Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at9r )hq,q dclbqp9j)g5.x afkv8 :c:fe the Moon, 380,000 km from Earth. The beam diverges at an a)b)58eqfxg jkf : 9pchl. : ,vqd9qacrngle $\theta$ (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 ,: orylhdxodmjf*th iqfn/qm* ,vq1j+; +;(k rate of 6500 rpm. When the motor is turned off during operat( y hj; x1+vn q:dlr+qtko,/m,o*qj*d;fm i hfion, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to anore l8p3v71 gsither child. If the ball makes 15.0 revoluti78g 1vrsepl 3ions, what is its diameter?    m

  A bicycle with tires 68 cm in dw qexdq1;b+n4 iameter travels 8.0 km. How many revolutions do the wheel1qn+ 4xd ;ewbqs make?    $rev$

  (a) A grinding wheel 0.35 m in diamet3d/nn/h t lqx5gi,r7n*dwr b5er rotates at 2500 rpm. Calculate its angular velocr7*xb n/g,i hlrwn/td 35dqn5ity 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). j(+xrw v5xz,9r. iozcub 2 9kt(a) Whato5. vxjzrxk 9cr+9u(iztw, b2 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 (a) in xhcjf2w (g96h) c0 c u,au4elaits orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear spe -kt,ud xiey 3jp+af y-r5;,amed 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 centrifuge rotate if a particle 7.0 cm from the axis ofuq qj0/ryl(z. gnb 9(k rotation is to expzbr (/(u jlqg.0q 9nkyerience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates unif3nus lq6me drjt4a i/l t20y;;ormly about its center from 130 rpm to 28jmel;;irn4q 3tts2 / u ld6a0y0 rpm in 4.0 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 radius rm,yfz w ok82cy6uyq(th 6p/0 $R_1$ 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, astronats ,btg7 c33dzuts aboard the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated 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 diame g cbts,7d3tz3ter 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 6e ak-(f;zmx cz2l-g us2vxq0uniformly from rest to 15,000 rpm in 220 s. Through how many revolutions did it turn in this tk0-zzl2(6x cvu2-efg x;mq as ime?    $rev$

  An automobile engine slo;h9nk6/ acye - dmp3 zxx+u4huws down from 4500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying high1t(; 42 (gm ;nhyvmxcdf/orij speed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 cooj4t i( g;hv/12;cd(f mxnyrmmplete 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 frow j/fq/b c*j(x ewsa45m 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have travejej/a/ 4w*5sx(b cqfwled in this time?    m

  A cooling fan is turned off when it is running at 850reelko3. i qr*x;qnc1o: v/min It turns 1500 revoluti1o ik*re3qlc.; nq xo:ons 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 makeccu8zl b6m;l,ybcvind x 5g20a6 e4x( 65 revolutions as the car reduces its speed uniformly from 95kmcc(dg 8 lc,u mvaz60iyln5 e 2b4x;xb6/h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces19uhpb a0l5xs its speed uniformly from 95km/h to 45km/h The tires have a diameter o blh1sxupa95 0f 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 ea jg,tg4cr,eg 6ch pedal when climbing a hill. The pe4r,g,j tg g6ecdals rotate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cm wide. Whj hq+ i2bd*lqqq: ynhphugd1(2/6 v5kat is the magnitude of the torque if the force is exerted qu*g2n5 (:j6pqdhh k/2+vqhiy1d blq
(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 5b3mryngt00k1erk f-axle of the wheel shown in Fig. 8–39. Assume that a friction torque o0kbn1 3tg0frk- erym5f 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a massless rod 81p,. saog fc3wt:lflwhich pivots as shown in Fig. 8–40. Iniwlf8 cof 3tsp:g1,a l.tially 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 req eg1je,h9o0t huire tightening to a torque of 38ge,hjh9eto 1 0 $m \cdot N$ 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 ,+fc w*gelyl6radius 0.648 m when the a6ecfllwgy ,* +xis of rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertiaw ,wmxn75ls6c of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The wm7 s,5xc6wl nmass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod iyvj5:f4g us8is rotated in a horizontal circle of radius 1:yfg i5jv 84su.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rou 63/d0s lxnuktating at constant angular speed (Fig. 8–4ud6 s0x/3nlk u2). The friction force between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of point o.qlu fo cpxp0ajjpl)7 h.q(paf0*+/ ubjects shown in Fig. 8–43 a fap pup*. ajp0f(huqqjcxl+o ).07/l bout
(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 oha3 r by )z+ki3 f+rhs-)vr-scf two oxygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the correct r;24 0gpgpgyla 5 l)uraate, engineerrp gl5a g 0a4gu2pl;)ys 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 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 radi9 uv40qmy+8 ies8ii s,pppre *uu0g8i us of 8.50 cm and a mass of 0.580ypv*80i+ipe 0 4u uge8imu9q8,rpis s 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, accelerating it from rest zp+9 c magg8*oto 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 and is abwr as k)r+u5mgr 0rlihx-382a le to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume +xim3r0rwau5 h2s )-krlg8 ra the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and ivs+5t *ufz u86cdqovk(xf4 j+s eventually brought uniformly to rest by a frictional torque 6ofkd+jf8 q+(zc*tu4us v x 5vof 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 a9-1 9mb syx6hpgvz1sw 0(6gryl 2x eyft 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the ac; (4 *knu/otlc eqc9m/hp,+1gxqwunction of the forearm, which rotates about the elbow joint under thpqc(xu kn+g;h1nlq,e 4w/tmu cco 9*/ e 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 considered a long thin rod, n 2hcste /.cu 2dmaty 2.,qhb)3g6fbdas shown in Fig. 8–46. )g226 ,. q2she3ctbdd/ m.fu chba tyn
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine consists of two mauh;hj3 nf o)gx4,os w hzl52p7 6qvu;usses, $m_1$ 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 thnfb*ehe*ysr8+ba5 b pf+ eygr) p3.y8e hammer from rest within four full turns (revolutions) and releases it at a speed gypn*+ )b+5e8er ffsb83y py rh*bae.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 ::crt-y n 7araof 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 40hl ;akih ey73uot/z$m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and rpvj2sm2-pi:n-9my yiadius 9.0 cm rolls without slipping down a lanem2ii pp:- 2m- yyvn9sj at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the sum o l9 vwa7/*5r fojg7 k5jj*+itq*t0vsux w2dfvf two terms,tkrt+i*09f*/ d 5s*v52uvg o7qwjlxj 7vjwaf
(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 kkz9bczza:s mfc- a(9+1 z8to 1640 kg and a radius of 7.50 m. How much net work is required to accelerate it f k 9z1 o-cc+9az8 ftms:z(kzabrom 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 1.8 b,ro-7 qfduqb+jx 5o0g;jzo 9 itwp;40 kg starts from rest and rolls without slippingdrtqqo;ujo0- g5iwbfzjx 4,+7 po9; b 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 vejcb 3c 5wa0/ 3wxpit;nrtically on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation o b w033;p5accxj wt/inf energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotatvxs(cj8,7n z i x3g6dzing on the end of a thin string in a circle sgzdviz( x jcn683,x7of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular mof1 sf7k0na(xyfr.:81. vn k+vxokb p lmentum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm whenk0+po8l fxx(b17n. k sr .any1 fvfv:k 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 rotating5gd)6ciq2 5rp yrql9 pj ez,i5 at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotp,ipijg6) 5cd2 lq rr y955eqzation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one s* dy-i 2g)1;yrtdy, jlzsp8 *ckgqrs 6hown in Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck pos yzr*kjs,pt*s16 yq dcyg;2i8 g)r dl-ition. 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 izaq9nec8 ,b a*nitial rate of 1.0 rev every 2bzacna*,9 eq8 .0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rj dclti: -2te7,3vlvpotating around a vertical axis 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 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 rotc j72lvl:tei ,t- 3pvdating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figurerb 0h6diths 1.1dyyn41m5 ezw 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 angula fx5)::jkzf nsr 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 I is dropp,.x 6f *u 6n9ltp:sxwlkj+pdf p8l4 clh at:6hed onto an identical disk rotating at angular speed .tsl wad hk, :pf+6 x*plntfjll :4u9p6 cxh68$\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns upe5icd ko5f-2w t p-ye:nn c+o* x7pvcv41n4at 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 standgap fa4usdzrz 69f;p,u+ogv n/k) n,2s at the center of a rotating merry-go-round platform of radius 3.0 m a26u/;npdup g v)s,ao ag4fkzrz,f9+ nnd 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 angugb9bi2bw yz3y +y xt 9m/pc/9uf3a v)vlar velocity of 0.8 myy9z +pvbi 9)/f3xuwgay btcv 39b2/$rad/s$ 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, losing about half its0o ctt3:py*qnfrri66 sr3i q3 mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the l36iqy pn3c q3rr:t6ifr*0t soost 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 exceswez13;o7. mxftr - ztds 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 nflzn *2.q/b h$ 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, t o4 a+su(.y5hx osylt:j 6gip(hat can rotate freely without friction. The moment of inertia of the p: y xo 4y( h(s+s5l.tuigo6ajperson 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- /asvt 9-adv(fgo-round turntable that is mounted on frictionless bearings and haa9vsdt f( -/vas 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 rg o*ylmub6).u -;dxd (*mdbpground 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 spool rolls behind the person without slipping. What length of rope u .doxd-;l p*gdyr6u*mu()bb m nwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same sidenc+lt k9sc.s( 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 caskc+ n9(scst .le, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 1 c.)j4lgfo o d) 5s*tb::rfeq *cuvv)0w -j lcbm/$s^2$ 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 cylixgv5 -qix5.us zh7(jynder of radius 0.20 m acquires a rotationa.yi7-szuxg q v5x(hj 5l rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, ea joz5,ard: jmdf3d*i0;rju9etf b- 5i ch 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 linear speed of the yo-yo whtfe omzd*55ia:j 9 rbd; 3fru- di,0jjen 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 th*tm y:gfj ia au99)l;ce angular 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 timmof rv,68bnrf viv5/ 2es as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carrie 6vnb5 m /vf2,rvr8ifos off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollutiont+au6 o*)j7wy ;cc bym automobile 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 b+* c )a6tj7y;ywmcoukg, 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 illustratean .8(z w nqfm+.mlx(nfm,l2is 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 zdgjfa65/ appyx36 .irolling 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 pivo6 ccklsqc n (/8nv84fqt freely (i.e., 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 tcnfc(488cvqkl n6s /qhe 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 sr oh*+ ,ey)6yom.iro:e h exj.tanding vertically on the floor, and we want to exert a horizontal eh) jr oyxyeo e+r..h,*:mo 6iforce 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 travelinlzbtl3x8 5x g10iavy2 g with speed v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclyi5l1a8v2 x0tl z gxb3ist 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 av-fq72zx (mk oesje ewf7s27: 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 vk2f7z -e:fo s7jqm( w2 sexe7feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms bhgs kq on,p-8) vyk.) 3t..)v+ljko1tksjy b as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arm,vp )1jtkj-tbh vkskq s. +kbo )yog3)y8ln..s, and with arms held tightly against her body.   

  You are designing a clutch assembly which jv gyaeew8;*x)8u kra3+g3ma consists of two cylindrical plates, of mas3 j3 g8a+k aue)*rm a;8gvwxyes $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 t4(i;p dyq9 i6xe- wgy7qkul d)rack of Fig. 8–58. Whaqwyy ld7u49gd q)ip ex(-ik6; t is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not owz7xcvy.z4e*p g o,,mru9m awv g-)0assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the a2xv hsnpn85ws hr :zoiy8))8car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular accelerath sspa8o:58y in zr8xvw)) hn2ion 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