A bicycle odometer (which measures distance traveled) d n c,:o;6rpymis attached near the wheel hub and is designed for 27-inch wheels.6ydmr: ;po,nc What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a porxhsx3u 3w*q 6int on the rim havxhsu 6xr *3qw3e radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value ho+:wb* b+ kl juv zn+,x)lvv/of the angular velocity $\omega$ Explain.

Can a small force evert2f +sa u gm*v5:-shrlc nubwr(8; e6q 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 do a sit-up with your hands behind your head than wh vzp.y)87vo/ui g3wo hen your arms are stretched 3h7 .wv8zpgivuoyo) / out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three a66kr wlm+z 8hot 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 to pedal, a small front sprocket or a large front sprockk8zlwm+o 6r6het? Why?

Mammals that depend on being able to run fast have slender lower d / oik*e-r4vc.ylkm) legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this dkoykc4dml/ )-vr*ie. istribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narroja ,-on9aczi1l h k7ikk+ p9m)w beam?

If the net force on a system is zero, is the net torque alsnfz31z*-j f9 ( iypnxb- uvi5no zero? If the net torque on a system is zero, is the net5y -u- nfnif3n*b9jzxz(1 piv force zero?

Two inclines have the same height bcsagz:as1j 1p2vq r9qaed*f b8o -:)l ut make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline wil av:f 1bzs:a1deqj8-cqg ) as2r* 9poll the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) dobo9j1r 1 7lirbwn an incline. One sphere has twice the radius and twice the mass b7l1obrji 91rof 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 haz2 b5a71bst5t vc ex 4y6au5slve the same radius and the same mass. They start from rest at the top of an inb6ts7e5 ay41 ub xvc5s25azlt cline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conservev,;*)j-gmg9vm rytx bpxt 5.j d. Yet most moving or rotatijx,r.;v p )-59byg mxjtvmg*t ng objects eventually slow down and stop. Explain.

If there were a great migration of people toward the Earthik 6hy qj+z(9( 4zyjhn’s equator, how would this affect the le y6( +khyjj (zqz94hinngth of the day?

Can the diver of Fig. 8–29o (5q8ojcoh+t u v.e,v do a somersault without having any initial rotation when .jeo vqc8uh 5v+too,(she leaves the board?

The moment of inertia of a rotating solid disk about an axis through its.n-4mg s*g3bg)eejcv5rg j9q center of massngrgjm 9 cg* .4veg)5 js-b3qe 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 5x ijum3rks -m.jycv-g.0n 5moutstretched hand. If you suddenly drop the masses, will your angulasj-n. gmmy5 r5jk.ixmcv-03ur velocity increase, decrease, or stay the same? Explain.

Two spheres look ident)e7 x,xx yj9;ticd twe4f/jw 5ical and have the same mass. However, one is hollow and the other is solid. Describe an experiment to determine 54wxe ex;,cyd wijxj7 9/tf)twhich is which.

The angular velocity of a wheel rotating on a horizontal axle 4tr.xez5 kr+. rc0yng points west. In what direction is the linear velocity of a point on the top of the wheel? If the angu ze.r.n k05ryc4t+grx lar 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 oszl9af t/3 ga,7tgvps bj +h:9f a large freely rotating turntable. What happens if you walkb39 :/,pg+a7hzsva tgftj 9ls toward the center?

A shortstop may leap into the air to catch a ball and throw it quicklhnq-ze-vfqs),nz6f 8v; o r, ly. As he throws the ball, the upper part of his body rotates. If you look quickly-qn6f qzs;r8zvl,h, f) e-von 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 momenu h.bc /9r;f6hw2ahdn tum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propellewb.a; 9hunhh2d r cf/6r can operate to keep the helicopter stable.

  Express the following angles inh)et1:sfk+ ku, hgoc9 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 coincidence. Calculate, using 1ohw-38ed gknaxb0 .r2w ur/hthe information inside the Front Cover, the angular diameters (ink/w 0dbg-nuwa 8.2e3 hrrh1xo radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km ug2* dv g.,zbko4*itpolu(xkgq13 7 xfrom Earth. The beam diverges at p(4loud 3* x*gx o.,kugz2 7iq1gtbvk an angle $\theta$ (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. pr+g8mx24 2fl9jjz gs.s w8ppWhen the motor is turned off during operation, the blades slow to rest in 3.0 s. What9r.24 8plxjs2gp8psz m+jgfw is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another chi-bgfd52v gj9d d-e1mild. If the ball makes 15.0 revolutions,b 9fd- 1ji-devmg25gd what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutions d z6if bgn94 i-ikru5-o v9* on(wugxw/*oe/jeo the whvkfb5z ei6*ngr9 w4 ou-io9 xgon-/ujwie/* (eels make?    $rev$

  (a) A grinding wheel 0.35 mly4t7zbreky wnyg f+4va,4e,) 62sa r in diameter rotates at 2500 rpm. Calculate its angulaeb yy)z r4y4t,fw,a6lek n+74v2 rsgar 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. docfd0u(k 6w,0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from thku0wd6c od f,(e 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 Ear3 f xvnvrr,1/h3lcv2hth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear spkg r17uxoj.g* eed of a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle u8xknnw6spbysbj4 udu6/)1; 7.0 cm from the axis of rotation is unkyj6 x)1 swupd/;4b bn8su6to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpm to 2875z c+has fq/sg0x*u i0 rpm in h7 *f+cqiagx5 u0/ ssz4.0 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 radiusmhu fa2..a g6kd u7(ug $R_1$ 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 themse0zmekio072p i4xmu- 7+nx yyilves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accm07zy7yn4 x2pi+k ux mio- 0ieelerated from no rotation to 1.0 revolution every minute during a 12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rest to 15,000 rpm in 220 s. Throu4tnb4n(m apypte.coz (, -li: gh how many revolutions did it tuptet an.no b4:( 4(-i, plyzmcrn in this time?    $rev$

  An automobile engine slows down from 4500 rpy0)2i 4vvrk g-w/ ae; iui/) cbpgcs)rm 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 stresses of flying*3*p ;j njutos highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions befos*jnt ujp3;o*re 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 rpr 3ka4l.gbeg 6sscrdc06a) , jwxz5 g;m to 360 rpm in 6.5 s. How f40srg kr5.sw, )l6d;a g zjabcxg63cear will a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850r k l(qeeakmh7xn7.8-ji4d1jt ev/min It turns 1500 revolutions before it comes to a stopk lqtej4e7(- .1 i8ankxh mdj7.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed uniformlyvi8,yqhfn4r zq(pai lfpgy3 l3 3 :7,w from 95km/h to 45km/h The tires have a diameter of 0r ,l3 qzy7q 4f lpv(:83iafp,gwyh ni3.80 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 unifoa 4sltqpa:2p9: dmjuz . +l,msrmly from 95km/h to 45km/h The tires have a diameter of 0.80uss9dj:t4pmlzpa. +m , 2:lqa 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 bik, g)w:mbla o1he puts all her weight on each pedal when climbing a hill. The h 1,mw)g :alobpedals 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 o8wbj:u,eru03e l :xsl6- iijxu8*l ljf 55 N on the end of a door 74 cm wide. What is the magnitude of the tor l0e:xirj ux8:-* , bj6ijel8s3luulw que 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 pmm ar i,p-5 t*ulo)yf-5zlv -axle of the wheel shown in Fig. 8–39. Assume taf,u zivo) -* -m5 5ptllmy-prhat 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 ae1(grkm ldy5n: 6bnh/ 6kbw5uttached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and directiowhk db5/:y(6ne6lm5 gun1rkbn of the net torque on this system.

  The bolts on the cylinder he/h +k; ywlz(u /ys6xurad of an engine require tightening to a torque of 38 /ws ;rzxhu/kuy(yl +6$m \cdot N$ 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 4zfek3f,o)kzn bc, n-radius 0.648 m when the axis of rotati3c z4nbf -,)okzknf,eon is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. The ri)oj/e7f b4bdq qyg:m*m and tire have a combined mass of 1.25 kg. The )b :j /d*4e7qoqyfb mgmass 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 rotatedwn2 bvft,zxx 1lr(:3 i in a horizontal circle of radius 1.b:2nw xt(r ,lix31zfv2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotaj . -kzl6sf9a) rkiy-qting at constant angular speed (Fig. 8–42). The friction force between her hands and the-s 6krflky-j.)aziq9 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 poiv zr s1iie*,b*oq1,8 ubjfrswrx1o ))nt objects shown in Fig. 8–43 aboub,sji srzr *we,o q1o )ux 8fvri*1b)1t
(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 oxm v.um*l-/ sdoy rfr fslh*)l.: 4-xnpygen 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 satell:9tupq q7 tsh4o2 ,tivite spinning at the correct 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 74,htto: 2tqipvs 9uq 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 radiukf0 ind5s, hbhn(rj. 6s of 8.50 cm and a mass of 0.580 kg. Cal5bhrdhnk(f0si .6 jn ,culate
(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 froz qh7ch3z nn0-m 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 merru(w*bue bec bg:pgh pb2;b/87y-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. 7webgug8/*b:(cu bpbh b ;p2eAssume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass 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 to ;ggkg bq0+,kz :x;hat 10,300 rpm is shut off and is eventually brought uniformly to rest b;k k t;0zxgh:gb,qg+o y 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. 8–45 accelerates awwn08ws6 xt e p569rp4(lz9wzw9l ppp 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 oa 1oi.mc.yi d3f the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45.1d.. aocim3 iy 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 thn,1qk)6tx es xin rod, as shown in Fig. 8–4k)sxx6 qe1 ,nt6.
(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 a+0tu rs 9:+sjhqu)pe3gcio +of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest within four full ty6m fcq:6htcy:xsj qk3 1qi,-urns (revolutions) cqiq:kqcsjhyt,6f3- yx:1 m6and 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 3 wl1 a5q,u5pob o5a 4m+mqucomoment 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 -vlzhc.k8 k/tpsd/d9b c87 yxtorque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls without slipping down a+zp8, 0itbaze53 9bqqfvch obgcv5 *0 lane at05fv9hqibv3 *o,czcb bpeat+gz 08q5 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respec oz)lv-b m5m,p1 dhqe0t to the Sun as the sum of two termsqmmp)1-b 0d, 5zeov hl,
(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 7c8; :o /iqagnfad v(f:.50 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 is a solid cylinder.g/ 8n;aovq :dia(fc:f    J

  A sphere of radius 20.0 cm and mass 1.80 kg sta1syyv3u u,j p2rts from rest and rolls without slippinjpuy 3uvy2 1,sg 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 v*t :f1t3hlju dh)zw b.ertically on its tip. It starts to fall and its lower end does not slip. What will be the speed of the uppezd3jh).:1thf b*wtu lr 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 y 26v u.xpnu-rt.ei 8zo khdw3g:/yu*end of a thin string in a circle of radius 1.16-g3d/ix u 8 p.*vue:.2wyykn urtz oh0 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kgecg8-y2 n/pu hscm 8i5-kg; uj uniform cylindrical grinding wheel of re-5 si8 2/cnh-g; jcup8u kgmyadius 18 cm when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at 0kw2f9/wz,cd i-vi ;x .ksqil his side, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotat qwvsz/l.,xw9dkik 02;fc ii -ion decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce hgw iu :fr8y 37g9- u.icgvpqn+m/cw1zer moment of inertia by a factor of about 3.5 when changing from the straight position tomwg.8pf9- nu+ii /:u z3 7c1gvqrygc w the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation ra,: xe n;ynbai)mtl3/s * hhy8lte from an initial rate of 1.0 rev every,ya*i;3n/ thhy xn8lb mls: e) 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 rotags.y1oy /8; pxu1me(boz xr;hting around a vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk ofr/8pe;xhoy1byo;1sxg. m u (z 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 spinning at 3.5ye39/3 ;wpbq*hr qm yv $rev/s$ 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 momentsvu05;+ nio 7)u8.o2smyztgknarw 8l um 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 ; 4ac b f gur8q66)v9/1afgnpigzs:opan identical disk rotating at angulc v6 qfn:/ asb;a689g1gur) oz4ip fpgar speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at o lu/lnr29; 2es utc.n2.4 $rev/s$ 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 l)bf:fndu roa em5;5:stands at the center of a rotating merry-go-round platform of radius 3.0 m and moment of 5r5melan:;fbfod:u ) inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freepe1.l 9 elbxt2ly with an angular velocity of 0.82.tbplle9 e1x $rad/s$ 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 kro1+i+nk 0 3xgwfr b7about half its mass in the process, and winding up with a radiusrn07 x 3+ogwrk 1bi+fk 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 wiv*supi 2s+llx(kd n27nds 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 z(f sxibhp(9d lp ()fj(s0f/ .k8zfxx$ 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 withou2)h6pn;dy9i. kelhnl t friction. The m)idley;n9p6hlk 2h . noment 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 stands at the edge of a 6.5-m diameter merry-go-k1)knj1 zxc i7i0 a2p7rbkv4 fround turntable that is mounted on frictionless beara4)cn72ki 1 fzr7pivkj0kx b1 ings 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 of8zq 4+1ui6pwul9 w bsc the rope lying on the top edge of the spool. A person grabs the end of the rop sluwp 19c+b zu6w48iqe 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 aa. yc -zvx5ogcc5: /lz);;jsr4 xroadlways faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axiy 5sdo;x:/orxrzjl 4 5g.za-c cc)av ;s) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates8/batb(e 5,vijucu13okw54 xgu 4ifz 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 t;7wa-z fmz ,fnhe shape of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at fzazm,w 7-;nfconstant 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 kg and yzxdrp6*u0 ,5ru et a 1y3l7pqdiameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Usee,r3d176qt5r y0uu zyl pax p* conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how iswga)7;g 0 kww1a k;wn:wzy+ qx the angular speed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 time 9x,r nyz/:dczs as great, 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 process, what would its rotation speed be? Assume that the star is a uniy9 dc:z ,r/znxform 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 +bn0 po4.ohth to use energy stored in a heavy rotating flywheel. Suppose such a car hahbo.ph n to40+s a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “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 illustrates6 *bna -sl-x/x7, hhdztf *4au fytwzgxr83 7g 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 rltzfw rhn,p8n6th (66olling 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 jm lpc(t9*upxq p8kst7 yo :9h- *xup5freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally ant*yjs upp9xp-c 9xhl *k :mu78(5pq otd 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 verticallyp3y0b i6v; zl)tm3d cq 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 height hbdc y0 3t;3 zl)mpivq6, where h

  A bicyclist travelinjfv+ka295o ucg 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 the norm9oj kcf2 5ua+val 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- o8-v;*1pa lbzp j9e,m;uakc ckg rock into a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, acc9kau ac;lpcpbj1em,v 8z-o; * elerating 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 cylinder k w 8+n;0t q)k6 amq5gf5vgtkyand her arms as thin rods, making reasonable estimates for the dimeg6) k ytk+5f 5vak 8qqtn0;gmwnsions. 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 whiy*x out8fk *cgq1 +s.kch consists of two cylindrical plates, of mass * t gksc81xk.*+qfuoy$M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the looped rough track oz-+1)h 0loa-o xyt lwaf Fig. 8–58. What is the minimum value of the vertical height h that the mo-xolyaa1+h tw z0)l -arble 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 dxtusyw 7f 7b,457yzl po not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reducqjm x1ma hgir76d5+fie+8lg 9 es its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. im1 +r7l6ixfajgd e q958hmg +(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