A bicycle odometer (which measures distance traveled) isawh1)pkby4h*0 vhj eua5u8b 6vf)n9n attached near the wheel hub and ispe hu)jaany8 h4)0vu nwv15*bkh 6bf9 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 velocity. Does s :tt-frrc,4q rgj/ 8ca point on the rim have radial and/or tangential accelr,r-fttg cq c:4jr8 s/eration? 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 component of linear acceleration change?

Could a nonrigid body be described by a single valuzl (p xz/(dar9ih7 w: qfv93ole of the angular velocity $\omega$ Explain.

Can a small force eve d:x5(qqusrq g1gf3)6*j2 ytn2 szvf35picrnr exert a greater torque than a larger force? Explain.

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

Why is it more difficult to-loi( w9o6l 9jmwzy,mbe0e50 wfg4yu l* c-iw do a sit-up with your hands behind your head than when your arms are stretched out in front of you? A diagram may help you to answer tb009zuj4emmyfeciw y(5,6 *w-ll9 wo gw -olihis.

A 21-speed bicycle has seven sprocketso4k.uoye7 ij 7bcp ol1(9d0s o 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? Why? In which gear is it harder tu9ocpobdo7( ke4iy70j o.s l1o pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast hai6*lvw5mlq+t ve slender lower legs with flesh and muscle concentrated high, clmlilv 56+tq*wose to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow by i;2c uq n. xt9gp90n7+c8eot +dcypheam?

If the net force on a system is zero, is the net torque also zero? If the net tos6mkn,d0gd:in f.* s vrque on a system is zero,f*ismdg :, kns6n0dv . is the net force zero?

Two inclines have the same height but make different answujrot. +i/ z 8,.ctr1vgro1bv 7 4xjgles with the horizontal. The same steel ball is rolled down each incline. Ogio+ow17 u,t.jv 8v1tx/js bzr rc4 .rn which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from1 q6a ;/i uqyykkn7sv3 rest) down an incline. One sphere has twice the radius and twice thev6 k n7 iuakysy13/qq; 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 the same radius and the same mass. They start fromp,zqwm 9m j0 nzeyci 5j4;p:9o rest at the top of an incline. Which reaches the b;z 9jp,cmni0jey94wo :mp z5qottom 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 an+n+t t4 ydh 2sg)i3wxpgular momentum are conserved. Yet most moving or rotating objects eveni g4p3+yxnwd+st2t ) htually slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equator, how wou:s+/z 4w+dhn1sf beyild this affect/ bw:sif14 yzdn sh+e+ the length of the day?

Can the diver of Fig. 8–29 do a somersault without having any initial -a1:iu 3 f j bgjb;udau*4cze:rotation when au1:gu j;ud fjba-bei*z c3:4she leaves the board?

The moment of inertia of a rotating solid disk about an axis through its center wz8te k8 bx3-vof masszx8vwe b3- 8kt 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 yp37c6v,zm:coc q 1treach outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stav:6t1 czc,7qpr 3omcyy the same? Explain.

Two spheres look identical and h6 pkq; wx ycr50zm65 a.5yefumave the same mass. However, one is hollow and thezmaxpc5 ykwe5;r660 5m.y ufq other is solid. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horizontal axle points west. In2i njmd5/u mh7k2l /8.e zrxlyeou+ 5d 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 8lu +yru255ezmde7/k.lhj /mx d i2n oangular speed increasing or decreasing?

Suppose you are standk nvlg 1*8pdk5o /kac9ing on the edge of a large freely rotating turntable. What happens if you wadn ag klk95okp8*c/1vlk toward the center?

A shortstop may leap into the air to catch a ball a1c0p ip0*e uzpnd throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his puc0e1pi0z*p hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation ot 3hw6t3pf5txf angular momentum, discuss why a helicopter must have more than onet65f3 tw3xpth rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles vt z (; 56dtsf:agki90bui k.bin radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coinf3re4,u zl*pdg fc 6z6cidence. Calculate, using the information inside the Front Cover, the angular diameters (inu ,l6f3ze rg dfz6p*c4 radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is dire tdwxey6fdjce. *i3du/ + o-t(e7leg9cted at the Moon, 380,000 km from Earth. The beam diverges at an an*+ed dol(.6jiecet-ty uge xf7wd3/ 9gle $\theta$ (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 theyr7 7v/ hjax) dez wz*2z3(wyt motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades stdy 32(* ewz7xyzj 7a)zrw hv/low down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another chil g4bztrox 0-9mh( .urhd. If the ball makes 15.0 re0r 9mbt4 gzo.rxhhu-(volutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutions do t*dmvypf.1db,sh/t rgk 5qd 15v3, lm qhe wheels make?d/t1* 5km 1 l5.,svvdr m3hdqf,q bgyp    $rev$

  (a) A grinding wheel 0.35 m in di ;.z y )ooqrk1u3/5dixhrnw:j ameter rotates at 2500 rpm. Calculate its angular velocity15 xnwryooij).d;:3rh k/ qz u 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-ybeci. e :/m+sr 9lkck(b i i/8cql.s6round makes one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated/ i iisq.:bcer6 9y(bl.kk+/ lc mecs8 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 ohor0sva1uk yq:)7 h-bf0n j,wf the Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed ofhpzh6c0gho;9tfy ff 0cl-( 1h 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 7ku2r 0 o/wkwy0bga (.q.0 cm from the axis of rotation is to experience an acceler urqaw/02. ybg(0 wkkoation of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about it5q7bdgynt8zeal,5 -us center from 130 rpm to 28ty g ,nlbaquze57d-85 0 rpm in 4.0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radiusuu*b18hk rzc-gm j 7req-d0g- $R_1$ 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 themselvcpf a.2d g(8oles into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 18f2ldcpg.o a(.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 tx8ni636h m sqs. Through how ma8xmn36hs6tiq ny revolutions did it turn in this time?    $rev$

  An automobile engine slows down fro6fifb0 u md//cm 4500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresse*bvfg, yt0:zw9 s4l zws of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn t4 b:ztws, vlwf0g9y z*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 uniformly from 240 rpm to 360 rpm c, 4o, l- pcht(we/wlqin 6.5 s. How far will a point on the edge of the wheel have traveled in w(,le wctpqocl4, /h-this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 15 goflps5v+de-3h:hj 500 revolutions b3 hgo5 - :5hjfsp+vledefore 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 revolutionsz r+;+u bh1n5)as*whdhprz3,wf 3h/ivv u2j i as the car reduces its speed uniformly from 95km/h to 45km/h The tires h bnrshd)vw 3p; *uvzwiuh/+z3a52 ir+ ,jh1h fave 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 ky6z69bmmdxq 9 3zz9b as the car reduces its speed uniformly from 95km/hyz9x3 d9q69 b6kmzz bm 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

  A 55-kg person riding a bike puts all her weight on each pel6 y53ke:sjtr un8;kvdal when climbing a hill. The pedals rotate in a ci u 8kse6l3yj :trkn5;vrcle 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 2vf-jr-s ccq,s8 t;qi a door 74 cm wide. What is the magnitude of thef8tcsc j-,q;qi 2v -sr 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 axleglgu / 2 1s3ca*gigfg64 rkm6s of the wheel shown in Fig. 8–39. Assume that/gfgr *iul6gg3gscm4s6 21ak a friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are a4x dinyrh1.j9ttached 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 anjn41.hrxi9d yd direction of the net torque on this system.

  The bolts on the cylinder head of an engine (r0m;kek ,b:jlhr:bhks0 )ku require tightening to a torque of 38 k b:l h0kku,be: 0rr)mk;sj(h $m \cdot N$ 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 wh5 zg3.ojbv ayo)4 su(q 0l.ahgen the axis of rotatilh.zgaso( v0 4gqu 5)3by.oaj on is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wh6 csz+ngvw15ieag n8a6f1eh .eel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignored (whnizafgsv ege +n8 w11 6a56c.hy?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in a horizog 5nfsi,9lfc3-ov (ilntal c-cofvgi59,fis(ll 3n ircle of radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a pot+tpvvmr/y 28qy- 1dm9 ntpjf-uj )g2wter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and t+1/)jmy-nfpv9vum -2 tryq8 j2wdg pthe 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 obj k2th n-of3853cs9hb cldcbt1/ dh(wo: uh9sjects shown in Fig. 8–43 w3 hhh5 3298lth/9nc so subjccbfd-tko1 (:dabout
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen atoms whose total mau hgwv:dia 8c1. x3zjo7 wd)c:ss 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, engigy 9 e651weqip mxm5f*ph:i1lneers fhpe i5w19 f ix6m1ylpmeg :*5qire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the required steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder wit ec2q:9+n04lpic jy4x8 vdjro9q5bpez ;b k6kh a radius of 8.50 cm and a mass of 0.580 kg. C6p jl4pcne jy dkv9ircbz+90e:b4;qkq2x5 8oalculate
(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, acceleratix*y 1eyr (nd2qng 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 onk-1kp t*wj f4dh )/qdvlx3*l/ wgdkd6 a 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 disk of radius 2.5 m and haxvwf6*/1g*hld -k4tlq kj3)w pddk/d s 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 evep h pu,al+lw+9ntually brought unip+,wp9+ lluahformly to rest by a frictional 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 ynn1iyzvr:d -7wq0*y–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 balm) mu* e8nbbo vl/d8ncd*h3v+ l is thrown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceduv*cdoelb8+n *n /) h3bvm8mps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long th9t5* qm.ls s )arkp*d pi-pb.pin rod, as shown in Fig. 8–46.trpps)*-* pd.s5i9 aqk. p blm
(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 consiso7,rrk/fa 85np4 wiow ts 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 1l s )2-oxmfnq: cqvfaixd8o5.-ex (ijnr,-h from rest within four full turns (revolutions) and releases it at a speed of 28 s,-.qixxqn(5a-ff)- d xcm lo1iveh2rnj 8 o:$m/s$ 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 inezt --cgze7zh z/tp.yl4h) m8 rrtia 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 7qj2 ldq;u2g wof 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 6kktm578i9-y7c zjow aidj j7slipping down a lan7 mtij7awk8 jodc 6i7-5jy9zk e at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as tusyc-t ck 8-4y93 iesuhe sum of two t8y y-sktu -49sue3cicerms,
(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 radi5-: slzlbqp4re p zf23us of 7.50 m. How much net work isbplrepqf z3 2zls4 -5: required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls without v8khz ss5 -qaxihr1b 1a ,f49,18btisis+aki slipping down sravfkx - zkt1+ i4ai 9b b,a5818hihsqss1i,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 vertsx7 lv8jihp4 *tby 61vically on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: U *iyps8476txbvh v l1jse conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg bal;v ec)c vog1*gl rotating on the end of a thin string in a cirv1eg v;) ocgc*cle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of zpmjk4r633pomb 9n4za 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotati kp9j oznpmmr 336bz44ng 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, hanx5gh4gn+v x 5lds at his side, on a platform that is rotating at a rate of 1.3rev/s If he r4x+ hggnxv 5l5aises his arms to a 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 / )z wi0ku7i0gw9lb7hc8amopd3 j/xkshown in Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If s7kz/wkuxg d)ihbc8ja l9 70i/0ow 3mphe 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 rotiz 7 lj6/x0ilbation rate from an initial rate of 1.0 rev every 2.0 s to a final rate ofzi0/ x l7bli6j 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 its centej1 p ;z,c )a1dnoql6zyr 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 throzy1o,c)ad 1 6nq p;jzlws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinny i+ maiyx5d-4ing 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 momenny x3by- s(m, ;g-a5c for e,6t8rhkmktum 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 i9:+ xpt u1ralge-azs0.8jr lidentical disk rotating at angular saatu-.srg :1j e80rl+xiz9 pl peed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.)w. j 8e*dt;lmb9j bap4 $rev/s$ 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 rotatiy98v*v:c v susx/7y xfz3m/ lnng merry-go-round platform of radius 3.0 m and moment of inery/ 7xlz3vcmsn* 9xvs8: / yuvftia 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 merrys2 +y)meq+copcm8cxot 0/n5 f-go-round is rotating freely with an angular velocity of 0 5/p sm2ce qtxc0y+8o)foc +nm.8 $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventuau6t6q5 nrb 3z-ad,ny5uv:kbd lly collapses into a white dwarf, losing about half its mass in the process, and winding up with a ra65y-kqzdbbu6 d5 t 3anv:,nrudius 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 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 windst./4xf4 bldun3 pw t jhj18mf* in excess of 120 $km/h$ at the outer edge. Make a crude estimate of
(a) the energy,    $ \times10^{16 }$ J
(b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 $kg \cdot m^2$) of radius 100 km and height 4.0 km.    $ \times10^{20 }$ $kg \cdot m^2$

  An asteroid of mass ,)/o 4)3xv bdl bffspz$ 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 pbxbgud( r.q 4+ a5ml: hq; fq.w5u.hgrlatform, initially at rest, that can rotate freely without friction. The.fwgq b. q5qraul ;rmd4 (+5 ghbu:xh. 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 person sta;2gfs :5teuxp nds at the edge of a 6.5-m diameter merry-go-round turntable that is mounted on frictionless bearings and has a momentxp2 se5:gt; uf 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 ropenwor ar1cq0ak2rqyy .u 2o* 26 rolls on the ground with the end of the rope lying on the top edge of the spool. A person grabs the end of the rn*2rwrqo2u a6y. acq0ok 2r1yope 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 the same side always faces the Earth. De5 :ny id0f/0xkusc t6dtermine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a ps6dki0 uf/y:x0dt cn5 article orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates -fe9+ +xgkw77yi(k wmx qdi6z* mpyq/ from rest at a rate 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 ,5 ;ijnt-zo6sj wzid- the shape of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at jt, nod z;6-j5zwi-siconstant angular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cyl30jibjf:8a6 b*t 1ydd l m7gatindrical 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 :y j tad7*bf6 3dl0amg1jt8ibreleased from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of the rear wheel ;t7erka vgcs )p.c ;g3:/d3nv jwql)e($\omega_R$) 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, burzt,u(8ww bc9t with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gra9ww(b,8r tuzc vitational 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-m2,z;. xqwfns pollution automobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flz;m 2,n.sxwfq ywheel “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 vs5//ei;yallbx6 d zw rj-9e ujun-4-an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) j.gkke;id8-k.hbf 3(dw0x,zcz/i 5 fc7sp a x is 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 can pivot freely (i.e., we ikrqco54g k)y / rjba 01f;eih;npq;k2gnore 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 center of mass of the rp) qf4bqor2y;gihe5j cr 1/na 0k; ;kkod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing v/ h lyt1o4 ayyvz)x:b)p/f 72;fkne qjertically 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 henobvyaz/ fx 4;lej2ytq7)yh/1pf:)kight h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat ql1, b +q ugr;f6w)ald.,dgkoroad is making a turn with a radius The forces acting on the cyclist and cycle are qwdg,q fuol+61.,)labrg d;k the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.5 m and begins whir;4mg s1bwh;357owt ,,c7womexizl d dling the rock in a nearly horizontal circle above his head, accelerating it from rest to a r;too 1; bmw7exg3wc5 l,z d,4ishwd7mate 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 soli+ ftgm7s.zbw0skn 7t7 ut7u/lz5y4au d cylinder and her arms as thin rods, making reasonable estimates for the dimusa0ytz7 .7u7un+ fw kt7/g4 zmtsl b5ensions. 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 assembly which consists of two cyl:-lc,-x2x fh f57nmxg n ktj8x 1ds7oiindrical plates, of m1ki x:787dlhnf2x,goxt x5fn --c jsmass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 tr aqlc)4g.ta g:ack of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reachq) a 4cl.a:gtg 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 a9 5 nkjz/ (kfdlxcoc80ssume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revojd gkur 5;. *lw4bpvt,lutions as the car reduces its speed uniformly from 90km/h to 60km/h The tires have a 5*gw d.jk;utv4p rl,b 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