A bicycle odometer (which measures distance traveled) is attached near ..2gygbs ty6 xc,tcaht8; 0f ythe wheel hub and is des8t y, cctsfybhxg.a ;0g2. ty6igned for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates z;bozf,h x5esrf, 7+pat constant 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 acceleration? For which cases would the magnitude of either s5e+fb;x zp,zrfo 7h,component of linear acceleration change?

Could a nonrigid body be described by a single value of the angular vj03 +1tshsa r h;q mhg3ruk17velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Explain.75 eh. atdgpg. -ky-mm24kq yl

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 jn)l24yy * z06z)tj/j kjckbh a)a 6owyour hands behind your head than when your arms are stretched out in front of you? A diagram may helh0y*tl2 jy)z )aojkjcjk4)z 6na/ 6bwp you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the ped7l tkgv.+coa f6bwi+g6c 33 vwal cranks. In 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,c3.7kblf+v3o 6vc6w agwti+g a small front sprocket or a large front sprocket? Why?

Mammals that depend on being ab+.f ,3ddd-zhx8 qiqecle to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, expx8dq ,dq3 i- ce.dfz+hlain why this distribution of mass is advantageous.

Why do tightrope walkers (32.yx+ ai eckdufv *9nFig. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the net trm mtd yt5 c33, y83h32wljfororque also zero? If the net torque on a system is zero, is the net fortlmwd ,3coh383 yf 2r3yr5tjmce zero?

Two inclines have the same heigh1q +uoojw (0tht but make different angles with the horizontal. The same steel ba0otw1 (q+houj ll is rolled down each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from bmhp/8zn6ng; rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Whichh; ng6zmp bn8/ has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius yz6 l c e0mkbv*-m2u*(da5pykand the same mass. They start from rest at the top of an yp6vd2ukzkma(cm-y e0 b* l* 5incline. 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 rotational KE?

We claim that momentum and v h,wxgcn3b0t.:zy w x mv/j)/angular momentum are conserved. Yet most moving or rotating objects eventually slow down and stop. Explaihyt ,v3z. nv xxgw:c0)wbj/m/ n.

If there were a great migration im3r2x tof:m22 a3c hftaa g+3of people toward the Earth’s equator, how would this affect the length oiartm f3:x3t 2h22cfa+a3 gmof the day?

Can the diver of Fig. 8–29 do a somersault without having any initialgvvv t o98s;dtlt)2 /o rotation when she leaves the board 29g 8;vtdlvo)/ot stv?

The moment of inertia of a rofgd xntwvhjnhj,j855 :5j0 v9tating solid disk about an axis through its center of mass is 585nv f,v : gx jhnw05dhjt9jj$\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2-kg mass in each outstrcl0 qsh9- tsn, j58rxvetched hand. If you suddenly drop the masses, will your angular velocity increasessnc,v85 tqhjl-x0r9 , decrease, or stay the same? Explain.

Two spheres look ide m,s 3,.oloskgntical and have the same mass. However, one is hollow and the other is solid. Descros ,o.skml,3g ibe an experiment to determine which is which.

The angular velocity of a wheel rotating on a of7 3 7mhxa8mlhorizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasi73l87o faxmmhng?

Suppose you are standing on the edge of a large freely rotatingy5 0 /bwdgonfe3 v49o 1m;mbpv turntable. What happw415nv bg; oy9/em o3bpvf0 dmens if you walk toward the center?

A shortstop may leap into the air to catch a ball and thrownsgy/ et w m)p)2k0vm0 it quickly. As he throws the ball, the upper part2)0mwpk)mg t/sne y v0 of his body 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 angus/e gg;j;z r*zxn6. y2pqk7zclar momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopt /xn7 jg;p2ye;gskq zz*r6z.cer stable.

  Express the following v bjjog4fvi sdb+mbd0lg5+w,5ow3 n -8j+: dpangles 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 amazinga,gn )he;lop,a.fwgy:9 u s1s4 w2alc coincidence. Calculate, using the information inside the Front Cover, the angular diameters (in radiansape,n2 1.s:wga9oy;g fl,uh a wc ls)4) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is direcwm8 0p83j .kq hy-tcdmted at the Moon, 380,000 km from Earth. The beam diverges at an anglet3k .py 8wcq-8djm m0h $\theta$ (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 n2v //n4ytq73 z99 ckmoyofiemotor is turned off during operation, the blades slow to rest99 mczo3 qvy/einkny 4/72fot 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 am9:yjtbx/s6pa 8+n6 nsaxyu: 0hdn4pnother child. If the ball makes 15.0 revolutions, what is its diameth6 pt 8+b: x:yyspnx aam49nn 06u/sjder?    m

  A bicycle with tires 68 cm in diameterhv3co.u7e ed7 0i a(jbjdr:1l vdsp)e6pq24u travels 8.0 km. How many revolutions do t 2 dee0ud l7ebquija6(p41 )vd.3 :7p rocshvjhe wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotateanrz4 t rgr )m:l+olwv 8t65ql z;k06rs at 2500 rpm. Calculate its angular velo:l6mz8 )okq4rr0 zw 6tlgnvr5 tl;a+rcity 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). (a.dhayh g l04z)) What is thay.04h g)l hdze 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 Ea 2h m5+peyi.lt ppvugi,d9;tqs 9(nj 9rth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speek oglrr 5,m(ryjq k*56d 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 )nh pl-j9z8 9pyquz .lp:k nh-particle 7.0 cm from the axis of r-) 9h.h89nnklqzpy: jpulp -zotation is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpm th vm-v;*xcs yvl-5pa;o 280 rpmvca--x *y;; p5s hvlvm 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 x.1qnzsi2o4f t hpc,* $R_1$ 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 spacecraft put thebiui798r6jm ga2kj z xo ulz1/1n ym4a28t*uomselves into a slow rotation to distributey2kjl o6 4oui g27889z zunxm*1t1/a bjruima 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 220 s. Thq/cy*k -, ii0 8;qj(jco u9qo da,xsfyrough how many revolu;qj0j8 cifaq qsi dyy9/,,x(cuoko*-tions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calcu ww ,(i*yij2jklate
(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 t n .wy5mhho128uccxf )he stresses of flying highspeed jets in a whirling “human centrifuge,”2hy.u 8w)1c m xconf5h which takes 1.0 min to turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 360 rpm in 6.5k5 an3, zu gqv(:-uain s. How far will a point on the edge of the wheel have travelea5 i-q(uva,g n:unzk3 d in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It t t2zx a7loy08xv,0ceeju ,6 lturns 1500 revolutt2l8e jxoc x7 vz ,t0uley0a,6ions before it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revou-89azap ;le ulutions as the car reduces its speed uniformly from 95km/h t8l;uaae zup -9o 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed uniformly fm8)di015r h k nygl7zcr 83kjerom 95km/h to 45km/h Tkd70h5yeg8 1lc )8rkz mnri3jhe tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each pedal when cl7vteh*j:3 ( ixnd5tbdimbing a hill. The pedals *dv:t53ejthx i7( dbnrotate 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. What ii+wq21tb y; hhi2p6*/trwvr is the magnitude of the torque if the force i26ywrh*;twir bv/ p21 t+i iqhs 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 thewnvhdhldjp/:f 0w+2k .6vzr y 6+rb os( w(ny6 wheel shown in Fig. 8–39. Assume that a friction torque o6l/6(db w(zwprjy0+fnv sr ohd+2:v 6yw hn k.f 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of ma1csv)1 ma5qb;p.p8yk8le0 h 1lgs aesb/ ep5ess 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 di5;)eake 1c81 epy1h8vp 5gss/ lleab.sp mq0brection of the net torque on this system.

  The bolts on the cylinder head of an e)2,em aa7d)*k rkeruongine require tightening to a torque of 38 7era,a2o eu krmd*k)) $m \cdot N$ 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 zeewkspme.vbh8ds/5e*581gdcj r-0axisge/5 8m c 8.rpsdke *sbd50-e1jzehvw of rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycxb6su+ ,4jbhjle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the huu,+ jbjbhx4 6sb can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, ligh+luv rz2.,iul6cs0y yt rod is rotated in a horizontal circlerz6c .sl0v+yui lu,y2 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 . x)rrbow 7 hiij2w4,oa bowl on a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force betww4b w ,)o7rioi2 r.xhjeen 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 in2.(o5uh,xae3sj-o c uz u62 b(y j8wekmrnlb)ertia of the array of point objects shown in Fig. 8–438rwuxhukan bc2)2( eu zsmbyl.6o-eo( j 3,5j about
(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 wp bzbf 0omy +*1/:jrh;zg qxu)h:2hoh13k sgxhose 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 cylindri7 .shqo oh.z-bn cg85rcal satellite spinning at the correct rate, engineers fire four tangential roo.n.h cqrs 8bo- g5z7hckets 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 unifg z;8brxlg cnd3tmh:c p(ny43t0 3 uc;orm cylinder with a radius of 8.50 cm and a mass of 0.580 kg. Caxn 3 pt4dhc3rc l:yg;0nmb8czt (g;3ulculate
(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, accelerx6g 9c0i fis/qating 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-drive++ ctl zb8on3tgt+1yen 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 two children (each with a mass tgy n+3ltzc8+te +o1b of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is ;d wnjc- td6 5;hik5daeventually brought uniformly to rest by a fricck5ad-6 t jnd;h 5w;ditional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates2j:nmtwcgl.n 95 7nug5w gm/jonk)e5lo 0kn4 a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown sosgye:i e 2j,m0sa+i*slely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. Thesei*asem,y2 :i + g0js 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 fdfhqz4c. +n7rod, as shown in Fig. 8–46. +z.qhdnc7 4ff
(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 cf u3qzqyv48m aj( 9t,xonsists 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 hd9yf w(;i sw2eammer from rest within four full turns (revolutions)ys(ie2 f 9w;dw and releases 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 mo.pc1a eabf; /k si5qd;ment 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 ofwlx0 tp0u3/nuc2k)9ce-wqrv 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls without slbk s9g)fcz *e6ipping down a lane a ef)b9ckzgs*6 t 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy +tu; gzqn3bfhr5z9xrrj/9 w4 of the Earth with respect to the Sun as the sum of two termsfnujxt9r4z/q3hrw + z95 r;bg ,
(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. How much+k3f8 5coph4ycaozp:x 5ig:/gq g )fs net work is required to accelerate it from rest tooqopg pgh) kg:3z8ifsxyfa +/4:c 5 5c 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 9 + tgr3w,f5zs;po7 xglvfe d8ivrp-58sq3 ui mass 1.80 kg starts from rest and rolls without slipping down a 30z po ds;+88e 7 rs5qwi93g-i fv,ufr5lpt3vgx.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 vertical 0pl:ayytjb*-w(e dw)ly on its tip. It starts to fall and its lower end does noy*0wjtd bl a)-e:p (ywt 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-kg ball rotating on the end of aiwvi s1 g yhm 8fh0iins.8(-9c thin string in a circle of radius 1.10 m at an angular speed wi i-ih1i v9mycn0s8.s8 g(f hof 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular i; z5f+daa8 pbmomentum of a 2.8-kg uniform cylindrical grinding wheel of rad+ zpd8i a5fba;ius 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 side, on a 7jwd5r9vv+wbge ,9dfplatform 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 rotate7dwvg5 9jr dvf b,9w+ion 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 inert)dkh0gkr2m 7 fia by a factor of about 3.5 when changing from the straight position to the tuc 20) hd7gkfrkmk 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 incwj itpi 991ry;dl;-ar rease her spin rotation rate from an initial rate of 1.0 rev every i r -pyiw;;lajrd1 9t92.0 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 rotatf/bl kn++vo2(xw4 dseing 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 rotating wheel. What is the frequency of the wheel after the clay sticks o+xf(nwv +s4kb2/del to it?    $rev/s$

  (a) What is the angular moment:lq79 (zy iqimum 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 spn 7dc4qwo):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 dropped onto an idenb8 xe4 -5,mxyrp0f wvbtical disk rotatm0re,b fb8- vwxxy5p 4ing at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns/afcctb* hwjiw*ugc9)dx +72 at 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 standf1ug1.m t*: +z* rffqshf:rr-o/g lhbs at the center of a rotating merry-go-round platform of radius 3.0 m t: +1hs u.1qr/h:rgf l-mff bg*rof *zand 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 angularbwdmw ox-a p2;o02zp/ velocity of 0.8 w/z; w xbp-m2 apo20do$rad/s$ 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 a+fn;bf.e7mf g mp y4d7bout half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new r7p+ 4edfyg bmf f7nm;.otation 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 excess of 1:o5lo17nai-lkv 4 g2q 8egkrn20 $km/h$ 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 18m.q96es e h(uk:l7r a zc0 ywm1caks$ 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, init+tamvmer rd88+( 1e/qng k52oop9 yowially at rest, that can rotate freely without friction. The moment of inertia of the person plus the plo8p9 remq ad (+woy8k+/r oegtn5m v12atform 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-roundz ybfys5 k /ix-nm-h6, turntable that is mountedmz-hib /x6syf5y ,n-k 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 rolls on the ground with the end of the rope lyinjn) e fqz(8g5ig on the top edge of the spool)qn(e8fzg ij5 . A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that tzi cb ax14l/k(he same side always faces the Earth. Determine the ratio of the Moon’s spin ang(b /i4xklazc1ular momentum (about its own axis) to its orbital 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 otr67e,jm6mxcr3 m3h8obbc p1f 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 0.20d a3r4mvf u:8btwv i81 m acquires a rotational ratvd4r1mviu w a8:83fbte 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 tnq7la ua )6mvag2 gm68wo 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 linear speed of the yo-l7v 6gq aamn)g2am68uyo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of the r w. .bysz9j3oqtm7,vcear 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, werr;s .u+rb3 kale 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, losinkr ;a +u.rs3lbg 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-pollutionbm/1gst zz .aca5+d+l automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of +/z atadsm+b zgl1c 5.1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates an 4nmhw859mmzb $H_2O$ 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 h;/f6v *zx sw-ug8q7n veibk ,forizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e.o,;etltmz vk;; 9fvo *, 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 force of gravity acts at the centerm;9eotvoltf;*v , k;z of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is stk;1- 9jsno gi*hfwv. oanding 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 heis ;w-i.v 1nfk9hg*oojght h, where h

  A bicyclist traveling with/n8ax m3nr;geb l m jgb4m+)a, speed v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist and cycle areea4mgbj)/nr8+;nml3 mxg ,ab 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 iov efbsn12d159al3ne3xm4 d fce9 1lsling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his hd 99fbc41fx2e 5e elsvmi3l1a3 no1 dnead, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a soli67bkl-0 t9ybx ,lh pexd cylinder and her arms as thin rods, making reasonable estimates tekph b 9y -7xxb,l6l0for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assemg4be0 ptb0z/jbly which consists of two cylindrical plates, of mass etb0pzgb/j04$M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 track of Fig. 8pn 3 .)korvsaop+f) v d34hz)j–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highest point of pa sk)pr3vh3z4o f j)n .o)+dvthe loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not it9s8du(8v.9o zkl t 9 epow1+u8mcfxassume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as t8 tj :k,wnc pnt5vsr(;he car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) Wn v8 s k5t(;cptj:wn,rhat 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