A bicycle odometer (which me q+xheg, 7bfnqv-7g;oasures distance traveled) is attached near the wheel hub and is designed for 27-inch whe ;+77ofhvqxbegg q,n-els. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Doesw /5-mmjq80q j-qnns b a point on the rim have radial and/or tang8n0/msqbjmnqw j-- q5ential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the angul bg+ +8tbk)luai9 4kpkar velocity $\omega$ Explain.

Can a small force ever exqbbpj2h8hrs d .s 395or)ue9d ert 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 do a sit-up with your q4r e gb*k+v/hhands behind your head than when your arms are stretched out in front of you? A diagram may help you to ansqvgk+*/ h r4bewer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three at the 83cduq-v,4e4ldf v a1: typxipedal cranks. In which gear is it harder to pedal, a x,v3v tp1dec4f:u-8yl4aiq d 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 runs+t4-po: mt./bhkcgy b:w y6p fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynh-s / mb+t kpcy w6:ygbot4p.:amics, explain why this distribution of mass is advantageous.

Why do tightrope walkers +x0fv7qxx) 1 p0cwke1x0gzge (Fig. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero? If the net 8 qrco ck/1st wje 7379ewneo3torque on ac97nq1eecrs3eo/ w78k3 j owt system is zero, is the net force zero?

Two inclines have the same height but vdvmz rn:.,w33fc mx5/ wcs9zmake different angles with the horizontal. The same steel ball is rolled down each incline. On which imf.m9 r c,nz: 3/zwdw vv3s5cxncline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an incline. *b)9w9,exdhip jgo1kkk 0 w9aOne sphere has twice the radius and twice the mass of the other. Which reaches ti)hxd a,b0kj9w1 kw9 o9*pk eghe 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 g xwvap+)c* (ythe 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 kiyv+a (cgpw)* xnetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum andyjbm; +ukl b36ce::-iim7 yf z angular momentum are conserved. Yet most moving or royyk:ce;:+uzfb7l-6m b i i 3mjtating objects eventually slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equajdl62* ae,bjyr 8zwf4 tor, how would this affect wf 2y8 ajj6* ,4dlbrzethe length of the day?

Can the diver of Fig. 8–29 do a somersault without having any initial rows8v,yeqp :f: tation when she leaves the bo eq,8fpswyv:: ard?

The moment of inertia of a rotating solid diskp6 t6:v ne 9g( ulln+;aiiugb) about an axis through its center of mapii(v u6lgb:l)na9n +6et gu;ss 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 earfv3p3k1 l7ul kq16* /t gg8amntstn:ch outstretched hand. If you suddenll8*6s 7lr:nmtukn1p3k1 t q gtg/3f avy drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, ong*7n sa;h cdn* cw4o9fywpo1 (e is hollow and the other is solid. Describe an experiment to determine which aocnsh1f( n ;c7 p*o4dgw*y9wis which.

The angular velocity of a wheel rotatin.g3y2 8bud3swvl x2 bog on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angsvx 2 gl833ywboud2b.ular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large freely rotating ttea*ghk z2g 9p5r( a7turntable. What happens if you walk t7et2* r ah(p9gkag5z toward the center?

A shortstop may leap into the air to catch a upu)hp;k.)p-4x 1g/z *pfaco xqb6abtd3u4qball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you b;q fo3qx 41pa)a)d/u x6 ztp*.gpu p-4chbuklook 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 comvf)3g(,kbe vnservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more,v(3)mvgb efk ways the second propeller can operate to keep the helicopter stable.

  Express the following angle o6:ro 5c ax0hnnl;;pus in radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazw.h a g0m:q .eaqp5p /lhm7o7fing coincidence. Calculate, using the information inside the Front Cover, the h l mw.eqq57ap0gfp a hom:./7angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 3 rr0 nr+ s1s+3g09o2jptcm pni80,000 km from Earth. The beam diverges at an 91s ntpn ms3jc+20r+ri0ro pgangle $\theta$ (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 rp0clg)z cb(gc ;/ eqk*nm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration algk0/c(bg; )e cc*zqns the blades slow down?    $rad/s^2$

  A child rolls a ball on a level ,s sl24(pg9j)z ah:et 5jr rybfloor 3.5 m to another child. If the ball makes 15.0 revolutions, what is its diam: zj5(l2y4psjr abt,sh )9e greter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 i5ndx8d)kz cxgr fn4on .;r.1 km. How many revolutions do the wh 8rdkof g c1nxxd)nz ;5.i4nr.eels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calcfz +6p )oh;kmuulate its angula o6+u)z;kfhmpr 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 4.0 secqq 3r brs03( (Fig. 8–38). (a) What is 3s3q(e0cbq rr the linear speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its or eur: wv-)-tglbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a poin j870n9t dwl0:e sxaltt
(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 o5kr9vq zk9 3va centrifuge rotate if a particle 7.0 cm from the axis of rotation is to experience an acceleration of 100,000 5 kq9zvork 3v9$g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its ceutmq-p 0 9 vqv2al2if)nter from 130 rpm to 280 rpm in 4.0 s. Det2 iqmvlutpfv0 2a-9q )ermine
(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 radiusx 1p9bxy(d*h0e 9svy c $R_1$ 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 themselvesy 9xnw3.5oyc n into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determincynn .9 53wxoye
(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 r9 1ltc0 a r/vdb+blz(qpm in 220 s. Through how many revolutions did it turn1trzl9qcdb +v/ (0lab in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1tr 8 4y +dgkj;do+*qqm ;f0saf200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses *i..i9ww3x c/ i h rzcv0a nna4,k/pmoof flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min tr3 k.ixw aciph*zn9 ov.wc40 i/n/a,mo turn through 20 complete revolutions before reaching its final speed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240m+ykr ;hnblc) mn stc+-; g5t2 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel hak ;yns5ttcblchr+2mg+ ) nm;-ve traveled in this time?    m

  A cooling fan is turned off when it is running at 85ytt6qce-g i2b e7z7b k9. tuf30rev/min It turns 1500 revolutions before it comet - yti2u eefbtg.6q79kc3zb 7s 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 revoe slu, s2zqn61lutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires 1n zu q26s,eslhave a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed uniform vqi4-n2o gjo3ly from 95km/h to 45km/h The tires have a diameovg2-j on 4iq3ter 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/eph),a 32dy r l;;v ds8 uciqq+4iwyt puts all her weight on each pedal when climbing a hill. The pedals rotate in a circle of ra/ +;sria,huew)c; qdtpl24di8qy3 yvdius 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 em)c25v93we wb)wqsy ond of a door 74 cm wide. What is the magnitude of the torque if the)c b9 w3w2q)vwm5o yse 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 of the wheel shownftc;v ob (p3n) in Fig. 8–39. Assume that a friction torque)no vptb (f3;c of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m,.jh,e:rqpod1g2bkxd*q4s ( n are attached to the ends of a massless rod which pivots as shownr4k qp2.g1s,(d*jd oqex :bhn in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening to a torque of 38yrg8iyyk ( hvl9,)cnq d3am:c p( :me0 :aymi3(gm h0ynlp) 8: ekcqdc r(9,yv $m \cdot N$ 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 d6pbis0lr i7s6 9*)wwdt/dgo.ts f: q radius 0.648 m when the axis of rotation is thrr60sb t6wq/wol*.ddsfi:97)tgsipd ough its center.    $kg \cdot m^2$

  Calculate the moment of inertia of av:kvl.x)pdr+ y6k87vtjxjg9 bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignorx j6yx dl+)8v.ktk p7r vgvj:9ed (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rob cdq24ph-1z)o,/t +q atimqnd is rotated in a horizontal circle of radiu d/+q ),t qzq2po hicn-4a1tbms 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 bow f /gpn(s-p/pel on a potter’s wheel rotating at constant angular speed (Fig. 8–42). The friction force betw-se/g/ pn(p pfeen 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 shown in Fig. cnr3 s0kjr f-68–43 abou jkrf0c6-r3s nt
(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 oxyz3.z w)hwhd(v gen 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, eynmtabost y 9z2d/.(0 ngineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 nz ot.yays mb0t (2d9/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 8.d (-zk7tkfm,m 7cs, no50 cm and a mass of 0.580 kgm-t7doz,k 7(kfcs , nm. Calculate
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bat, accelerating it from rest t3/ uiqvo 0/ d)rpi 3ptiae(9dao 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 smalgdyt ml()6uf0l 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 dis0 6fdu)(tlgmyk 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 rotatiot-a+6qy d)gee )ula)xln/ *v ng at 10,300 rpm is shut off and is eventually brought uniformly to rest by a frictional toly/d))el*ge + xon)utaq-a6 vrque 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. 8–4gyb: y 8cw6v. szo*gv85 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 solely by the action of the foh1f. er 2okekmk8hwlm6 *7 ph9rearm, which rotates about the elbow joint under the action6 21om. kk7hpfk hm erwh8*el9 of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, as shown in *a q,;0ubf: n2 ixgjctFig. 8–46*utf2a;bnjcx q 0,gi: .
(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 ofdo o*idjq 9n:r553tmte/ qf w, 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 within four full turns (revd6w(n x er1xe,/yvfrw;u(u 4q oluti,rw4xy/xr uudq;16v( wee (f nons) 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 moment of in- aqf t9ub.(tsertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque of fbmtj5er l 4p+7 / m5pc-fx8ls280 $m \cdot N$ 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 slipping down .djx+r ne78ba a lane at 3.3 7n e8+ar.jxb d$m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the sum of xf;ap1rwl6xpdg* k05 two termxd 0 kl5*1 px6fpagr;ws,
(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.x aw,r5( ydj2 ig4v7df50 m. How much net work is required to accelerate it from rest to a rotati4 jiya5 (72v rf,xgdwdon 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 rln.awe,r25,k t* jph aest and rolls without slipping down2ktrw al,hna ep j5,*. 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 o 9e*w wmgj-rj4ua.3-f(ma qmon its tip. It starts to fall and its lower end does not slip. What will be the speed o g waw(a43fmjr .o-j -*mqm9uef 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 t ,bec7kfb8f ju/giv,q* w/dd)v -2xfq hin string8 jb*7d ibfudvv2f/qg,-kfe cxwq) /, in a circle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular mnp9nvk21y/h*7 cen/h 8sobuf omentum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotating at n sbncf*v/2198 /hn7eou ypkh1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, on a platform that (is k.shnpku fy8gn2 q /*q-b6is rotating at a rate of 1.3rev/s If he 2 snnh*y(g bik-suf6p.q 8q/kraises 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 shown in Fig. 8–29) cu t- tv z z(cr ,.q3mlqbth++nci06l5can reduce her moment of inertia by a factor of about 3.5 when changing from the straighthtcc 5mzu-(tinb q+l.t 3,0 qr +vlz6c 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 heyg-2c otvm z*:r spin rotation rate from an initial rate of 1.0 rev evem-g*c t vo2zy:ry 2.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating9kw)*; juec7f p ag/d v5lejz, 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 pzwjklg /vfa5c,;eeu9 * d7)j m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skaterzew8yn7m)lsg g2e;u9 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 momentsql9r . /y+miohc+w7( x do w7+lgt9lxum 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 momn3uwion0ec e0sm /.(qent of inertia I is dropped onto an identical disk rotating at 3s.ineqn00mc /(uoweangular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at : t4il x1g x(dub4thcigs;3f;2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center of(n7fog e n-.u:f;rs7 xnj jp8y a rotating merry-go-round platform of radius 3.0 m ann- y . uf7(p8njjo7 ;:fgresxnd moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-gw71mnn k,pg6 ao-round is rotating freely with an angular velocity of 0.8 wn7a1gp6k n,m$rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, losing about half i *rs2gjjsd;ud;l.4j)3 afo.,g i vv jqts 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 to the nearest integr3oj*g;,q2 liva4s )jd fug;d.sjvj. er)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in excess of 120(gmcqf 8vj3zc-w/ ptfb/loh.0a h6 (e $km/h$ 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 8p1tt ;8l ufhab+q7 jp$ 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, that can rotate freely e(rzfwpn 4h6+iwymom;2a7 .swithout friction. The moment f z .46ion;2whm( ep7+ayrws mof 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 stands at the edge of cko:++cqaw*g z+t hz*q0t fb+a 6.5-m diameter merry-go-round turntable that is mounted on frictih+fqz0+:o k+wtc*ztq b*ag+ conless bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of the rope lying on z1x+ jeu-9t6b zn6jg cxofcb7y( o7v5 the top edge of the spool. A person grabs the end jevxj (xo-fb7 z6o56gu79+n tz1b ycc of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such /5lr burqs 8k9r2vro)that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentu9l)uv5qr 2or8kbr r /sm. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest atwnieug)m2v:2 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 the shape of a uniform cylinder of radius 0.20 m acqrwrefi/ ygmvjh/l0ov 325n 0*uires a rotational rate of from rest ovg /eyf05or3i02n/hvlmrw*vj er 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 0.050 k1ka ,03 33 wuengro4oimh.an mg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservatik rum0ae1.moig4n3a 3hn3 o w,on 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 angular speed of taya.6c. m k6e 8uo5n ixzhfkay306/ jwhe 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 : 0i owwow+.ly;hqudnxwd8 *8great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the proce*wo+h ouy:q ;8wilwx. n8d0 wdss, 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 r4(ixo y1omt9 energy stored in a heavy rotating flywheel. Suppose such a car has a total mass oy4m iot(x1or9 f 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustratewdnh8 0w z29aos an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal surface at speed +qpo/ocx)j)m 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 ignore frictmny-. wre4a( wion) about a hinge attached to a wal4r wa.n(em-yw l, 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 rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on the floor, g 8lea- i;3m 6h gw1z5j+umwue7k5ndpk)eo2 hand 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 heightp)e wj nhh i5m2g ;-k 1wl7 k8ed3uzg65muao+e h, where h

  A bicyclist traveling with speed v=4.2m/srux *leamr*e+6o2t7u wv)s .c on a flat road is making a turn with a radius The forces acting on the cy6 7u* .oam)u+s cxe*2rlvwe trclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of qzs,/ysvu6 aci-vl, -length 1.5 m and begins whirling the rock in a nearly horizontal cir la,,6sqsz y-vu -v/cicle 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 as a solid l( ysu12 )l/p nzlzu-hp,2qmxcylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio ox2mzp-us hn(lul) zp ,y /2ql1f 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 consia*m:kr.basylryhh75yhmz:j i 9 h(+w4 g. b4osts of two cylindrical plates, of mass4l: h.grhz r h:.syjyb km +o4i9aay*(7whbm5 $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 amayd j9ga1o5.yal5d + long the looped rough 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 highest pointydom59g+y jd5 aaa1l. of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not assu ;gsuxyc96n +i6m.mcj me $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed unifoepvcenmr v,,.8o(6k ibp8lj9 rmly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each tire? lr ,ve .kpncv9 (mi,8o6p b8je$\approx$    $rad/s^2$(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