A bicycle odometer (which measures distance traveled) is attachx(72v xth eqhh o*)d,5 x1ud;zuh a5omwnnt6 9ed near the wheel hubnh;v qodnwx zx,thx* ah e5 1uh7dtu965 2(mo) and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular vl +izjqwn7 ,y;elocity. 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 whwq j7;, lzniy+ich cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be descr iwj it/23+yk n 2l,f y/aeotoqw;-9ngibed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? gqcla m , euzy/5ri,4o:czzu/-)o7hfExplain.

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 t7e fps(v k/d.dhan when your arms are stretched out(fkdsde7 .v/p in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprocket o-u ib ej gxqv/-(z,6;7zc1e bm;unnss at the rear wheel and three at the pedal cranks. In which gear is it harder to pedalzg vmeu;6/c 1-oxq in 7bb,;nzsu-(ej, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender low)jqtj )wo r,m,gt3 r8eer legs with flesh and muscle ct oj8,)rje,3wrt gq )moncentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkerst9ztpkb /t1ler2( w6 o 2+xgxr (Fig. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the net to7ki x*zki ktnvph)x y4940, ubrque also zero? If the net torqkz 0k4iknt ,9 vh)y7 *u4xxibpue on a system is zero, is the net force zero?

Two inclines have the same height but make different angles with the hori x5(3p xqma r9ehh9bvkhg57 (izontal. The sa59hhem5( a3prk7b9(xvghqi x me steel ball 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 rest) down an inclik: 9:sh o0pkm(rn*fn sne. One sphere has twice the radius and twice the mass of the othhs*ns:r :09(fkpnm oker. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the s +j wqvf.r0r3op wxu;/ame 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/r ;f+3j.r wwxop0vuq at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum a knc+ud)+t4 ylre conserved. Yet most moving or rotating objects eventually slol tuy4ck+nd+) w down and stop. Explain.

If there were a great migration of people toward the Earthgzht*j.sfy x h5 ;z79y’s equator, how would this affect the length of the dtj. *7fgxz9ys; 5hzhyay?

Can the diver of Fig. 8–29 do a somersault without having an:7 :rlmdvq y3 +itgv2vy initial rotation when she leaves the bri v:3d2tqlvm+g 7v:y oard?

The moment of inertia of a rotatin3vvc1doou y9 j3xdydfa,: 4n)g solid disk about an axis through its center of n)3 acod 1vx ddv:oy9y 3fu,j4mass is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2-kg mass in each outstre enugq0-p+h i ke: kd5r.x1qm:tched hand. If you suddenly drop the mkd+0 1i:.5hqre ge m -kpqnx:uasses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is )gq;gqpwh7ij.z2 -r4bgnvc 4 hollow and the other is solid. Describe an experiment to determinnq2-ibggq pvh)w4cjr ;z 7g 4.e which is which.

The angular velocity of a wheel 96ybfixjaec d- :n) o.rotating on a horizontal 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 poi ia- y n:d.jef9bx)o6cnt at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge z2 jd)ww,l 4t9usa7dx 7jjagm+s3u(v of a large freely rotating turntable. What happens if 3wugt( j svda7 jslu) 9+,x 7dam2zw4jyou walk toward the center?

A shortstop may leap into the air to catch a ball and throw it quickhgt1u)e m8t p /fndprq+i91x3,y.qm xly. As he throws the ball, the upp.pmyd8 q+/t9iqfep),g3 t1xmxuhn r1er part 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 angular momentum, at l/ysdc2. 3znl*0sgdiscuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propells*t nll2a3d. ysc0 /gzer can operate to keep the helicopter stable.

  Express the following angles in radians: (a) 30 v:ygt 03yr; cu$^{\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. Calub y,vopn -edi44vq)r1zbm78 culate, using the information inside the Front Cover, the angular diameters (in radians) of vvb4b4n,7pz1d)e-mr 8yio uq 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 bea1vgbb-w,q 9i d3,smmlm diverges at an ai31 9,sqmv w-, lbmbdgngle $\theta$ (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 t-,0 k xnfe-dbi (pox/ahe motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the bl xdp-e0bnfi a- k/,ox(ades slow down?    $rad/s^2$

  A child rolls a ball on a level flo nwa uw*z6cjt2msud5yv2k( :(or 3.5 m to another child. If the ball makes 15.0 revolutions, what is its di5w *kusd( (vauwn2 zj:2ycm t6ameter?    m

  A bicycle with tires 68 cm in dia*.0uc k (y(v k)of5e pw0phskvmeter travels 8.0 km. How many revolutions do the wheels makekk cvk0.yp*fs05pu(ow( h) ev?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate its39j-kod xxr :g angul 3jrx9gx- dk:oar velocity 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 4o) 8yrr hft3l,o sd(0d.0 s (Fig. 8–38). (a) What is the linear speed of a child sear03or t8s)o d(d ,hyflted 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orbit around * o+zvo p c)wih5n)r+mthe Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear spet mb njm w8 c4;/dufs;f7/l2eced 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 aku.7 ccsl2ph , particle 7.0 cm from the axis of rotation is to experience an acceleup2s,k . lch7cration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its centervq f:j545kxxnp+ :,mnoumtil3* 9uef from 130 rpm to 280 rpm i fi,jm:+e p qfnxov5n4 lxt95m3:k u*un 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 radiu87i8k 8wnif mzs $R_1$ 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 tlxja6ory5(wv-ij *z ;vy uy22hemselves into a slow rotation to distribute the Sun’s energy ya lwr 2*z jx2(j5u-ivvoy6;y 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 i s7eb4q1njh ( t4p:7,mxwrjhqn 220 s. Through how many revolutions did iet:7m x4(p qjq hrb,jn71s4hwt turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 wed wyaq5 8p4j+,txz; 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 stresse2 cx1yrbcrk7w7lfp,2 s of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn tcfw7b lc7px2yk 21r r,hrough 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 unifo u+nnazr7p+vh+x i/u/l*4z hzrmly from 240 rpm to 360 rpm in 6.5 s. How far will a point on iv7hz / z lnh+*zu+x ur/ap+4nthe edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/minl*m/zdlt2v v92wr itbd3e2l* It turns 1500 revolutions before i2l 2*t* / wetvdrv9 z3lbli2dmt 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 revolutions as the car reduces:sotald:iozd(n; 9z 2kv/n* bhk9ng, its speed uniformly from 95km/ o, *;9tnab9kz2/no (:sdid :lnvhkgz 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 reduces its sp 9;cs4: qnvkmoeed uniformly from 95km/h to 45km/h The tires have a diameter of c4v :mo9sk;nq 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 bi)3 , j:ldzt v-zxd5cvfke puts all her weight on each pedal when climbing a hill. The pedals )3ztc5z,dj:vlx v -df 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 7 3pwzzk +)s ;ss1vpny 1 b4dary12y0vl4 cm wide. What is the magnitude of the torque if the force is expn1zws+sb3y y4;1kv0 ) zl2aryvps1 derted
(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 ab r rz,wq5*moj )sbkodf3; .2iiout the axle of the wheel shown in Fig. 8–39. Assume that a friction torque of 0.4r zo5 sq )jf3bm;,d .wo2iik*r $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the e:r,jhp0pwk*b nds of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal0,kh jp:rwb p* position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head xv)g. ypjh17lr30 oi bm1m.n;tsx xt:of an engine require tightening to a torque of 38s3 r)lxb.ojmm0th;1v1p 7 : .igxtynx $m \cdot N$ 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 th5j/ sb( i 2ylper.xzm1y61z hhe axis of rotation is thrsz.5 ji6e ymh1p2lx r(/1bzy hough its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in 0ue (f-s-ta1v,f njhwdiameter. The rim and tire have a combined ma e10 , u-h(jfsa-ntwfvss of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotatedsz4 uk7;wb*cr in a horizontal circle of radius 1.2 m. Calcwzs 7 ;*kb4urculate
(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 bowlyeii,:ryp6muj :g3;fjrsw;k,- u sq9 on a potter’s wheel rotating at constant angular speed (Fig. 8–42). j6 ;gkreq i;w3sirf ,:s9 y:puujm- y,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 objects shownlhie ,y f5ko9tqi,2jn50+si f p(o q:m in Fig. 8–43 ao2me0tfhi5p o+9kfi:q5j,yl,i s( qnbout
(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 consise5s6q b 8pxide1) g(ktts of 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 correcasa6/n5nz0.v kmpn +( :ap amgt2s z2gt rate, engineers 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 ia0an25(amstzzgpn2 /vm sg :a+.6 nkpn 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uni km554co rug5sform cylinder with a radius of 8.50 cm and a mass of 0.580 kg. r c4kogm555 usCalculate
(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, accela.7mqua5 5g fzb /b o3iet6pw.erating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven merry-go-round and i/smuk 7* ysmvf;1 lqyk )7bdl/s able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform diskds yylk1um*7q l bf;sk m)/7/v 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 eve9vodog8d12m7qntr tq(m qsj yc 6(o4 1ntually brought uniformly to rest by a fricty2 q8(qdm cq74ovdmr 1o1g(t o9 jts6nional 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 accelerates a 3.6-kg bayh3pvg*g/va.e t 8hsat(w)b* ll 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 thrlp0 cua7f31soown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball ispol 37 c10aufs 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 consid )etf cg0pr l;89hes( 7zd0usi1/fvoo ered a long thin rod, as shown in Fig. 8–46i0l 0pdszc vfo1(e)/u9 7t r;hsgfe o8.
(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 ofnp+p.x64dd (ew;lo i x 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 fr30fsidxt1c7 ts,fq )o om rest within four full turns (revoluti)d st1t7c ,xfo 0qifs3ons) 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 momnt5t8ry9sfp0lih/m7 ent 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 torqul+ q(4 gui o5onvv 3a:fat,t-ke of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mase(5 pmglt x:2g s9dpb:q.dt.qs 7.3 kg and radius 9.0 cm rolls without slipping down a lane at 39ld(..p q5 qxbep2:dts:tgmg .3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as theq:4zv 2g9unix 1 bap0x sum of two ax2qgnuixp01vb z 49 :terms,
(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 7.5 ,2414ym1 lkfb2exte +hdk)+c wnusphxh 7lb *0 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it ,e tb+lfs2y7bp)x2hmwdx*1u lh nkkc4h 4+e1 is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from xro1f o (mbx5ucy314wrest and rolls without slipping5c 1f13xy umw4orox(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 vertically on its tti6*h3bjq;bkq: hz7x ip. It starts to fall and its lower 6h h3*:b 7;jqx zitkbqend does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end of a kf7x66ay+ ) o5lzurf bthin string in a circle of radius 1.10 m at an angular speed of 1)rbalo uz6 ky 67+f5fx0.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of cymug5 cd7s1jp o v0*7a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotating at 7uo jcv 5csmd170yp*g 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 :a 5gouhxug ..that is rotating at a rate of 1.3rev/s If he raises his arms to g.hugxo 5.:a ua horizontal 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 moment of inertion9;9ac0j a ,xfzw m6ta by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations;t ca09jmxw,zoa 69 nf 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 rate of sfyevq)17n*qy ) m b*boyi:2o1.0 rev every 2.0 s7 *2obqb y q e:syf)ym1noiv*) to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axis through u8 fyrt -pu5oj0g 3s(hits 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 thgo-tu sp03r8yfhuj5 (e rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinn uezwg f5cy02t8z4yyi4 rh2 -ving 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 momel1 b +n( :dic)8 owq)+b,kq-prns3lyxn9lc okntum 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 of7cgex;g*r j0j42 htqo moment of inertia I is dropped onto an identical disk rotajjc;*x240 tqho7ger g ting at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.sn :-gq09q4 2ykubiub4 $rev/s$ 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 athd(o p 0tgy: 6/*pz3u/pka 8piqecb-c the center of a rotating merry-go-round platform of radius 3.0( yptp u:aib8/6-peg0c/ 3zdp ckq h*o m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merc uot)n(fxg76 ry-go-round is rotating freely with an angular velocity of 0.xo ntf(gc u6)78 $rad/s$ 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 colld9 +anig6ov+uapses into 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 li9g+n+do vu6aost 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 exctyy zgdox,xzc8e;fa q104**)h:pr i hess 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 cb: r5go:x4xpeng*xyhb 88e+(ck/ zq$ 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, initi8-80 f;trz j gxgejuu0ally at rest, that can rotate freely without friction. The moment of inertia of the person plus the pla0 xure-fz gu8j8 j0t;gtform 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 persos9ujdw7b 9;f-s8vm r,ahy 3z yn stands at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictiowb;8 m dj3f s9 ys-hr,za7uyv9nless 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 th wza96uvc :4mve rope lying on the top edge of the spool. A person grabs the end of the rope and wal4zamuw9:v cv6 ks 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 the same side always faces the Earbos:b 4q. p1rgth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter bpqb4s.r1 og:case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of a/v 6 .krni:cv1 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 uuu.x j u 5smgm jt3wv51f: r:bbk;g/.tniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular5.xu rkmu:w; ftgb3bg./jt sj: um5v1 acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylin3hk wjt97(ef v:bq ,jhdrical 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-yo when it reaches the end of its 1.0-m-long string, if it is releasedj h(3q,jf:kwhv9 b7te from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed9g5cy/b nb6op of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times as great, were ropm 2ric1e7(3 ty* leestating 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 13 *r7lms c( eeeyt2ipprocess, 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 to use energy stor;*zo;a*:h j hbji,,n sx ysl(aed 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 shouldix;;(h ab* ,*,l onjy zsj:sah 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 tzsgo1s2 n;1 -jlp0wjan $H_2O$ 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) (zft/ii .4ge ypr+ )imis rolling 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 capu (b p3 xfc15kj5al;tnq) qa*n pivot 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,a;k b1*t3pnf)l p u(qj5qc5ax 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 sra73y *lgsof3+9 /px ahbe4x 5xxdy0utanding 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*07oas/9e xly +3xaur hfd5bg4yx x p3–55). The step has height h, where h

  A bicyclist travelin-9a, jsjyk 3fqg with speed v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist and cycle are t3j9yfsj, -aq khe 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 rockf8lq9rd6i1 ks into a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, acceleras f89rd1k6ilqting 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 4sztt43 g*y.h)cbbkx8sn cf aah26e. as a solid cylinder and her arms as thin rods, making reasonable estimates for the dimensionse*xk42 4fsgs z3. ath ayc b6h)t8.nbc. 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 assembl-z l n2pti dg47) zp2nyw(6r)6c2ciu;rpk tht y which consists of two cylindrical plates, ofp 6-t7 2r;lkhzwizdu 2)tccn(4 inrpy2t) 6gp mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the looped rough ,u;4l hu-ixoo 6a)oy,rxk0nj track of Fig. 8–58. What is the minimum value of the vertical height h that th oh,ro-uyxa,o6 04nj;ikul) xe 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 assum3kyx1ddn jk/0,y8ty( k;n tbz e $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its x0 (wm citucb 1h)ztj*2k0h9/riyp ;r speed uniformly from 90km/h to 60km/h The tire2/x ( c1miu;wyp rktjb0tirz* 9h0h) cs have a diameter of 0.90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s