A bicycle odometer (which measures distance traveled) is attacarg9k3 /bv.3nxt :ec bhed near the wheel hub and is desixg/n9bke .:3vr3ab ct gned for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angul78:0wldjn l 6ss7vidiar 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/or6llsj :0 vdi d77swin8 tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single vaou1 xm)0eo8h p3r.ziqlue of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Expf ;ca7nt -yc)t,o:rm2uei 9xx lain.

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 dm sk prb j,bjy d*0lh) ,w8/umh/46ynuo a sit-up with your hands behind your head than when your arms are stretchj4mnu/ up h60br/y,y lk,*b) mhj wds8ed out in front of you? A diagram may help you to answer this.

A 21-speed bicycle hag2t :*1tcagbkvu mw*,s seven sprockets at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Whc1maktw:vgug 2bt ,**y? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able t8w h2gbfmgegkd, - ;e, +zc6dio run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the f 6e2ikedw,gmcd8h+;gg bz,- basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow beam? ;8fyhib)e 0.gboctx+ijz ;t94i5v: kxmd3mn

If the net force on a system is zerog:r9xy/ 5 kfnu, is the net torque also zero? If the net torque on a system is zero, is the neu:n/ fx 5ykr9gt force zero?

Two inclines have the same height but make different angles with7prk 6dp3lb *r the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the botto7pdl k6rbp3*r m be greater? Explain.

Two solid spheres simultaneously stl 8hv /v*jhtxg(e11op art rolling (from rest) down an incline. One sphev1l*x t/p o1j ev(hh8gre has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have thx (ele6mh1qz : f-k)fp 2x5iske 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:kl2 fp hqx5kes6 x-im1( ezf) the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most movi:or-ck0p3 tg;g8r c kfmgg3*wc *cm79v *dnfn ng or rotating objects eventually slow down and stop. Ex0c 3c*9 rnvdg; :-*3cmmpfr7kfgt *8o gkcnwgplain.

If there were a great mi j*28rbxp, zibzsmz 9 uc(5wky2lb*)m gration of people toward the Earth’s equator, how would this5wyc(i28 s z*9ulk*b j,2mmrbz zx)pb affect the length of the day?

Can the diver of Fig. 8–29 do a somersault without having any initial rotatjst z 55/1h .+oswpzdyion when sh +1. dszowp/ 5tyhsj5ze leaves the board?

The moment of inertia of a r)paxzcr s1d 80am kup7su5)r ;otating solid disk about an axis through its center of mass is8z10xpa5d sr uukr mc;pa)7s) $\frac{1}{2}WR^2$ (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 mas lmkqeded,d3 eg53:z x7.fb y0s in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, dec3d570ymdeg b3keq: ,.f dlzex rease, or stay the same? Explain.

Two spheres look identical and have tihicuxh:.6lx().o g the same mass. However, one is hollow and the other is solid. Describe an experiment to determine which is which6cx.tiog)h( :ihu .lx .

The angular velocity of a wheel rotating on a horw*;*lztj*ljory 9p,nbfq 12xizontal 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 ofj1 q,j*pf xlo9;nry2* l*zwbt the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely r 01icx8ia61 15mbzffi ,ibrfcotating turntable. What happens if you walk toward 1r fafi1,c01 5ib6xcizif 8mb the center?

A shortstop may leap into the air to catch a ball and throw it quickly. As ba;sj.) qe v*t i wuz2.n4p4dphe 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 oppos4)p4wb ns.2. zqejuvi;tap*dite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of ang*av2pqna4j x1 hx.r 7yular momentum, discuss why a helicopter2ry 4hqja1vx x7. an*p must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians qe i;g+kv5lx*: (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 coim cs;mw(m;3w7wfva:x zt4 gg1ncidence. Calculate, using the information inside the Front Cover, the angular diameters (in ra (vgm;f4mas37c;zt:xg1m ww w dians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The bein c 08oef8gy-n7/ 3pl/zssps am diverges at an yp7 sp8szcs3l0o 8gfe-/i /nnangle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. When the motor is turnedga9 tj;t2g+ es off during opera2jt gsgtea;+9 tion, 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 f ei3xko4nxpi.c2zr.t+ /czc, loor 3.5 m to another child. If the ball makes 15.0 revolutions,+o kxn .c3zixtpz/.cri4ec, 2 what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revol9ka qi9*k6 sdy5y(ck6u4cm m mutions do the wheelskkiymmc96 m*y4dc56 ksuq 9(a make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at0uo xdyl*iw2,c7u k.s;-2yekr5 e v+sgqj; qj 2500 rpm. Calculate its angular ygq+;*0;-u ksrd2ov w5jx,7k2yjqeliu . cs evelocity 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 revolu9 w uete,g1rca; 4uh iwg;,wo8tion in 4.0 s (Fig. 8–38). (a) What is the linear speed of a ch;e wgtc; ,w1 89irau 4oweuh,gild 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 z aura7e3;m4/ ;dwxy aEarth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear spee1t*mt 5woqp.6d i dl in-x1+e;xs/ dsed 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 fro xt b78t.m )s0yzt7otj l8,ccbm the axis of rotation is to experience an acceleration of 10xj c07t c)tzttly bo8ms,7.b80,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its centersjau-gn, 9d v; from 130 rpm to 280 rag j ;9,nus-dvpm 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 2v jv o 2t+g3bk4h3esb$R_1$ 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 thebse).5mtu j)z3xf z0e,lni6 umselves into a slow rotatii)u) zt lxnu30s,. 5mee6b fjzon 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 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 uniformleuneo;ry 4;+o t3buie l+55hh y from rest to 15,000 rpm in 220 s. Through how mahuh+55iot+be; nry 4le;ou e3ny revolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500gv xa/tc8-pl0 kb+5cv5htt; y 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 stressesint :;p3d,thefrn)aen obo+1oi 54w7 of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 completo+4p,nehfadnbttn7 eir5:1; ooiw3)e 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 diamu z.o/7a af ;e;wh4w;.pg w5( pfuekjaeter accelerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled in this tim;wapf5u.w(7 uaza; wf 4/o.ekgh;ejpe?    m

  A cooling fan is turned off when wnv0bq (e 2-z005 gzukl cxm*tit is running at 850rev/min It turns 1500 revolutions before it comes to x5g u0t vwzknze*m(2q0cb0l -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 revolutions as the car reduces its speed uniformly f0)gvt1ldu0 l9g d; zy,(twg chrom 95km/h to 45km/h The tires0 ,dgg9v zdu;ghl10)ttc l(wy 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 makfj yx+f:p8y0 mq68ox0p pv(7 mm.pev a6ow th;e 65 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diamete 80x:yv ypffp0o+v(atpp6m; hqw.8me x6o7jmr 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 putf;z j2+ lxd;v-pg7 xcts all her weight on each pedal when climbing a hill. The pedals rotate in a circle o+gz;lc-p;jxdt vxf7 2f 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 dgbugn,.h u 72s0 7yxkyoor 74 cm wide. What is the magnitude of th gux.y7,u0 h gbsn72kye torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle smh7kl -m l2 0bbik w,1.d6aceof the wheel shown in Fig. 8–39. Assume that a friction cm6-7s k0ah,bd kmi2wb.1lel torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a massless rod w kuy duyu3x/,.hich pivots as shown in Fig. 8–40. Initially the rod is held in the xukduy./uy3 ,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 igi4 ;fyji- a)torque of 38 ;4a fij-i) ygi$m \cdot N$ 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 a4cu.mn *y5 xw;vdc uu, 10.8-kg sphere of radius 0.648 m when the axis of rotation is through its cenn4du5u um* xccw.,; yvter.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in d 87 gtyx4m+h 1g9z8vrp q;tpxuiameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignored (why?)g 8tpxvg zh1uq 9py r+m8x;4t7.    $kg \cdot m^2$

  A small 650-gram ball on the end of a upb+)a 6d *a/hp,mf)a0ikdag thin, light rod is rotated in a horizontal circle of radius 1.,b ak+ d6*fdmu/pap0))ahi ga2 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 rotating at constant angular spee;/k*gpds g.pn d (Fig. 8–42). The friction;pgskd p *.g/n 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 m2a/7ylsp r(y (9gv q+shl t7opoint objects shown in Fig. 8–43 abouvhol7my7 stsl(r/ 2qy9+ag( pt
(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 oxnl r +n*mvqf,*ygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the correct rate, e vp ic3s6+v s05lovk4vngineers fire four tan lv0vpco3 ss v+v5ik64gential 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 radius of /2w9yywa b(ou ug9 4ku8.50 cm and a mass of 0.580 kg. Calcu/2uagw yo(w9byk 4uu9late
(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 swie f8 +4bcdxd/gngs a bat, accelerating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a1abmiy5qf;; ad e645c ucr.i i small hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform dis ab1ri 5cyqd;iae5 m6;.fci4uk 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 is eve+oil +9n ml+ kks7.xhmntually brought uniformly to rest by a f9hmxk.7+k ++imlnosl rictional 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. 2dgjd2)gz)kc z5hfd0 8–45 accelerates a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown soleaxmlwrai, )/6fe:7 drly by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest m f: dxl),rea ia/rw76to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, as showr,k 2ftiqx t( p7 10mnr4t qem8/xp3zen in Fig. 8–4mq tik 3t,rrtp0n2/mqxe(ef7 4xzp8 1 6.
(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 cona+kl1qn ocw(;zfil i.f i2vrw .u6o9(sists 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 withi4-e q. nbrf1s8lw;9afg mxo0z n four full turns (revolutions) and releases it at a 9m;szofe8f 4n1-xlg0 w. aqbrspeed 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 of inertia oa /iptq6s n7nyv(b/ vtx2b j+7is5jp,f $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine deb 5 5s18qkb4mmee/l ssvelops a torque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls wia *k. 4. tor)rao;xkcxlb(uv kd;,/ jhthout slipping down a lane at 3.3./kcuo)(jkthvr4ba;o a x*rd .l; ,xk $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as q qzfnfz:ijvtdf7 zw5n,.jn 3i),1 (q the sum of two term7z1.qnjw jzd tf3(i,if q:fvn,nz ) 5qs,
(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 5kl oc7(53t7lh/ruls:qrno pmuch net work is required to accelerate it from ru qo3ncs/7p(lk7or:l l t55hrest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest anf )3 hgs tvnm 6a, /hopy+onvju.q3o8.d rolls without slipping down a 30.0 h o/j yoha .,oq3f.)tu m6gpnv+83vn s$^{\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 ti*of:8*q5upjx 1ecrwl59 gotz p. 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 of energljq:1 5oc8u5t9f pg*x*wzeo r y.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end of; 5jyminav/o (ckph7l tal2*8 a thin string in a circle of radius 1.10 m at an angu;ai (plkcjal 75t/homn 82yv*lar speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cxt s8u)m0mm*fylindrical grinding wheel of radius 18 cm*ut8m f)m0smx 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 * ti,tn ijou10rmu 90hk;+3acfk4aek side, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a h,ha9;o0uki tca4um1ikjnkr *t0f +3 e orizontal position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce her mozf 5 pyxrg8k;0ment of inertia by a factor of about 8f0g;y r5zx kp3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial jnac 6)9k9l5ud2m4ly+i o2crcezpr 6 rate of 1.0 rev every 2.0 s to a final rate 9 r a6ud69zi2 rlkc4y) onj2 ce+c5mplof 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 rota2 ;r1ayu *tbkk,8zy a ta4cue3ting 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 wheelk;r3a b8 a1t,aucy 24*keuyzt after the clay sticks to it?    $rev/s$

  (a) What is the angular mom9knj,ew4z s9g entum 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 thh7f k12g s: vfhizf1ehxv *vg0 8n:ub/e 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 inerti,afa j(kpjb-xp4q 7q,a I is dropped onto an identical disk rotating at angular spj7b, p4ja qka,-(f pxqeed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.os8xpmkj6 9-5rz qs)g 4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of a 4rtq 8qukjo ;/rotating merry-go-round platform of radius 3.0 m and moment of inert;/orjt qk48quia 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 angularffh6 xajlrjdf .b;k9 n- .j.q. velocity of 0.8jnj lrj.haf-fb...; 9dqkxf6 $rad/s$ 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 7p9i4k 1u4tf ud0j gtkinto a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round tout k44tudj01p97fg ki 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 e ;udk rb5+;meiak;oh;xcess 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 asmb1 (b 5psfffbk; .1$ 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, zf ;;z))f0 rlz cpy(yethat can rotate freely without frictpfz0 )crl fe;y y;(zz)ion. The moment of inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg personnt+ o3hoo) )pa stands at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inerti3)nhp+ ) toaooa of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of the rope lying on 72hunah( zasi (-aqu ,5u( nupthe top edge of the spool. A person grabs the end of the rope and walks a disnsauaah2uin hu-p7q ((z5u (,tance 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 the same side always faces the Eart;txvwr y6y ab(6 md+a:h. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latte6w: v+x6;dmy (brta ayr case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a b p i/ 8uux,q:o91cclirate 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 ua8tfi38en a x8s5watd(-)ey cniform cylinder of radius 0.20 m acquires a roxsf tedw8)5t ciy( a8n8aa e-3tational 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, each of mass 0.0502irj qifz m,;.i.itoef( z+j: kg and diameter 0.075 m, joined by a (concent( ;. im.qtr +izzi2if ofjej:,ric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angula + +i:clz4gm; dc8wlcpps*3 6nmgqhv8r 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 b4-qlcvt.h8)6 rbe h9//qha m+9a aonyvcq1ttimes as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitation4 am/9n 6qa 1. hyth/qlt- vabevo8)rbcq9+chal collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it tm7 hkylmsb b5f8s 1()to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinuk5h8( )7mfl1 bysmb tsp.”
(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+51n m-6n+i j,(3guktgebawekf c+vm $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal surface at speed v=3.3 in4 xu *syws2*$m/s$ 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 anngv.wt 7e.:y .jlqhdf)z)1i j d length L can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held j:dwzy 7g1ftl). i).jvqhen. 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 center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on t9s. 5tal k m+s7kvg:(n7h9y,jixv jgphe 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. 8–5psg x5k l s:ja.tv9+9 kyn7im,h(vj 7g5). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a ti lsk: j48x*ryu6x. 5vlp /afturn with a radius The forces acting on the cyclist xa :i*rl4flxtk58u6p y/svj. 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 6naex 4udc1 8v ,taro-into a sling of length 1.5 m and begins whirling theao vd -6 eat4rcn1u8x, rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body k0nxlcf7tdt 9:;.gp3wv l+wp as a solid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds fo0:f dxt.gp39+7vl;n wtklpwc r a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consil bp)c.:3zceq4(l e cvsts of two cylindrical plates, of mass:pq( ebl) vlccz3c e4. $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 roua sy k0 pbg-l9a;lv1zyi/ e.b6,av9qe gh track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the highesy ayl9, /v .aqi16bkezepvs9a;g-bl0 t point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but d. qc,n9mh 0 j xw p45)aqkl,,pgw.ajudo not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions s8jw7 2kw(mr o--juccas the car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of ea cwrc--sokw82 j(um 7jch 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