A bicycle odometer (whifspe( 6l;f/hjn)(*bj gpnl s (ch measures distance traveled) is attached near the wheel hub and is designed for 27-inch wheels. What happens ijfnel( /b g lsf;n(j* )p6sph(f you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angul 23ah,fcezt.fnb,p o5ar velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases unifoh3 o2cbpz,.n,f5a fetrmly, 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 descd; yu;eqg/; zzribed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torqu (zx0l*o2hjl5fix nzu ,zawayl n)s8. :pp73we 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 sqvk 0*:hasu/a sit-up with your hands behind your head than when your arms are stretckhv :0usq*/as hed out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel0prdv;x7 0j5 uaqve:y and three at the pedal cranks. In which gear is it0 r;ypd0 jxe7v:5q uva 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 sprocket? Why?

Mammals that depend on being able to run fast have slender low1uemxj6,ge1 u er legs with flesh and muscle conc e,um116exgu jentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow beam?wskhep (5iumo h4u 0n2;8zg b9

If the net force on a system is zero, is the net torque also zero? If th v3.p6i)dsp fqe net torque on a system is zero, is the nes)fv 63ppi.d qt force zero?

Two inclines have the same thkp8m6blw8uy) l6jivmg x*;j tx5 36height but make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the bl hw66 6t* m8k 3yx8m u)gixjptv;5bjlall at the bottom be greater? Explain.

Two solid spheres simultan( 8tusc1 ayy+f ,) ai4 vjgac85 het3em+qrc5qeously start rolling (from rest) down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bo(3acq88a4) ycm5ju h,yct15 + rgtsiqefve+ attom 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 havq-ym )p,- ygote the same radius and the same mass. They start from rest at the top of an incline. Which reaches the by -)pg,-ytoqm ottom 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 angula ktfyve4k(v-ky9vp zef1g 3*pg773ve r momentum are conserved. Yet most moving or rotating objects eventually slow down and stop.f*v3e3e zvp1( vefk gytv k4-gpk7y79 Explain.

If there were a great migration of pejy2tz,e 9 wf.m.byc .rv0jh/vople toward the Earth’s equator, how would this affect the length of the 0f wrbvy.hmteyj,./j.9 2vczday?

Can the diver of Fig. 8–29 do a somersault witw:; fh rlvzsw3x0i1g zz;)8lghout having any initial rotation whenz swhgzvz;l: 3)80w;g1ir lfx she leaves the board?

The moment of inertia of a ro. o- n0 8c gfd2v5rpu -60fbilcyaircys7tw(:tating solid disk about an axis through its center of mas airgcv (lo05wf8i u 7.ryp:2f ty-c ds-bnc06s 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 ,yzi 1.7qmzwffab(fg- ( w oc(rotating stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velgy( b ,z7. iwqfz(1m(fao-fwc ocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, on. .5inv/q 8brrca8wmn e is hollow and the other is solid. Describe an experiment to determnc w8aqvrn 85m.r.ib/ine which is which.

The angular velocityuzj+9h7 u/s8clsuo 0y:iinz2 p 9g,et of a wheel rotating on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wh27 9z l:p+cniui/osy9 sh0uj,teuz 8 geel? If the angular 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 z(;ggnw2uaqv; j ht-5on the edge of a large freely rotating turntable. What happens if you walk towah(tu; jz vqwng-a;g5 2rd the center?

A shortstop may leap into the air to catch a ball and throw it /xo ow pck7p :-9zlug :n3co+bquickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–w7c39o/u+k l:-op cn:obpx zg36). Explain.

On the basis of the law of conz /fxq. r9).mekyjywnco * immf-8p86servation of angular momentum, discuss why a helicopter must have more than one rotonw *ip/-6fc ym8jef.z8)xy. qmm k9ror (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in ,lz 41baph n9kradians: (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,zxt ae 3l54f4q5 dc4zr using the information inside the Front Covlq454r fezcdzx 4 5ta3er, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directedt0opggki e:6h )i5ss / at the Moon, 380,000 km from Earth. The beam diverges at an an:gg/hioep5 tk 6)s0si gle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rate of 6500 rpm. When the motorq22u-mxztq pp;o 2j51o p xu42byjdi5/;n ll s is turned off during oper buo y2 puloi z;2p-s5 x p2qjt;4nl1/xdj2m5qation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If ua( a,q rq)ate6-1exz the ball makes ,qeq aeau-xz( )6r1t a15.0 revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How manyyk3 x xdf9.xed1)tsij.) w mu0 revolutions do the wheels make fdm) dxs9kj uie.x. 0yt3w1x)?    $rev$

  (a) A grinding wheel 0.35 m in diamete 5jzos2fewp -8l 9-dqlr rotates at 2500 rpm. Calculate its angular velol-dj5 92zslpf e- q8wocity 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 revoluti8abhqo8i1f ,x on in 4.0 s (Fig. 8–38). (a) What is the linear1f 8hb,xq8aio 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 orbit yipqkqk*t90+p 0 ulw5 around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a poin*xv o;v xskr(p wunodh )d.625t
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.0 cm -ojpnrrl /e86from the axis of rotation is to6rpn-/o8 j ler experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center 5st - 2zupm k7gm2jh;gfrom 130 rpm to 280 rpm in 4.0 s. Deter -umz km;72g 52gsjtphmine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radius p1b 1j2xcyh3m$R_1$ 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 a0/r/sov big7 themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation toogv/ rb/i7 a0s 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. Thgn q3+ko) /ndirough how many revolutions did it turn in this tinikdn )qg +3/ome?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calckjm- rq 1ybroq ;m9/9wulate
(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 hgf3rb5fq2 bnkum9,o1 f;5yg*s .q k peighspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching iyq9.mo3uk5bqs 1kfr*5nge pg b,f2f; ts 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 t0 isdv)o (8pxeo6bbt* o 360 rpm in 6.5 s. How far will a point on the edge of tv eoxi)6sb bp(dto08 *he wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 1;0fkfh+f hlkgfu z0 9)500 revolutions ffl00h hf)f; 9g+kukzbefore 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 revolutions as the car reducwnhq(d *7(vgies its speed uniformly from 95km/h to 45km/h The tires have a diamevd n(qwh7gi *(ter 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 spe3cmhca 2vo cx/9:t2 wzlk9 l;e3(f yfved uniformly from 95km/h to 45kmth z kmlyxco2 2fc3a:9lwvevf/9 c;( 3/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 pedal wixk n g+ 1l;mn0b/:dmchen climbing a hill. The pedals r/g0bklm1d: +ixm cn;notate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cm wz0ys o pi4-9qjn wde40u (g2lcide. What is the magnitude of the torque if the4 y0wdz- os 9egpi2c0nq4 u(jl 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 ) e6t,*kdz5 cojrlsa, shown in Fig. 8–39. Assume that a cr ,)*z5ostle6,dj akfriction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, p :rw-g-+w2kovuv9njgp t+e-are attached to the ends of a massless rod which pivots as shorujto+:9g- 2e k +wvpwn-v-pgwn 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 cylinde8xzs +tc4pu 1ns* )nzjr head of an engine require tightening to a torque of 38 c +j8s*xnzp14snuz )t$m \cdot N$ 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 r ,:vrb7br;qdwbl0tp3-mc4l t adius 0.648 m when the axis of rotation is through its centlqcr dbp,704tm bv:- w3l ;rtber.    $kg \cdot m^2$

  Calculate the moment of inertia ofk (bzp:49:0 h c6arfany8 mmmf a 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 igna( hyfmm b9ampzfc8 :r0:6k4n ored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rr bt3kp6ys -6iod is rotated in a horizontal circle of radius 16k iy6s3rbt- p.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 potterd1tgy- 2p kaxj)am1x6’s wheel rotating at constant angular speed (Fig. 8–42). The friction force between her hands and the clay is 1.-)ax2 g1dtx1a kjpm 6y5 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 pfzfk .j2a -g 6rgj7)aroint objects shown in Fig. 8–43 abra2ka6r)gf- .j fj7zgout
(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 cmtle ,6aqhs;ue6w9:* rp6p qaux;n *donsists of two oxygen atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at theu; af;q y+b,a;w1h jhx 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 4.0 m, what is the required steady force of each q;;hby w+a x,f1hj;aurocket 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 radiud5ey/ ww6jdj(sr(/iw8 y9qt ewx *dknv. n.2ss of 8.50 cm and a mass of 0.(n/5i /wdyjkn2dw69yss(erv. wxwq * 8 tj.ed 580 kg. 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 re*b h3f2 vgw9bnst 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 )*c5 tfpej+y uhjag)n6 (zfo( on 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 has a mass of 760 kg, and two children (each with a mass oc (g yznfj)t*he) f+(o u65pjaf 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 eventually bro 60nnqhcnx 0.rught uniformly to rest by a frnnrh600n .xqc ictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a 3.6-kg ,zyb.c1b(cq 6omw ;1x fuc)ygball 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 solel2.l m 53hplqp95hdxf ey by the action of the forearm, which rotates about the elbow joint under the act2mq5h.pl df h5le9x3 pion 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 considea r3:w dj lc( 0)b(y e+xxjbv89c;yydtred a long thin rod, as shown in Fig. 8–46 j8cl3+y0( j):yrctwx d(9yead xb;vb.
(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 of two m4m l/ m,g6 hip9g.l8m yy)agtoasses, $m_1$ 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 turl64xywwgkg9hx4y :.gm gk:+qhr p g0u 84 9jltns (revolu8wr x9gp09 6+hgykg.jq y4lku :mtg:4x w4hgltions) 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 inertia2*efylzgwjq 0j 237 umq9r ;zn 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 develop o 8 r9stpv,qd-ko*5d:zqdf:x s a torque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls withouw -ajae 4ao +o06erbt5t slipping down a lanaejbto- r5w6o aa4+ 0ee at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Eartb23ii,z4x0xax8puq(r vn8 w nh with respect to the Sun as the sum of two t 8nq,ix38 ( zubv 0wp4iaxr2nxerms,
(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 radiu 75+kecpn( kod;iwep 1s of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution o1(ide+kpk;p5 e7 ncw per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts 5trj)v5v q/gg from rest and rolls without slv /g)t jg5vr5qipping down a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

  Two masses, $m_1$ = 18 kg and $m_2$ = 26.5 kg are connected by a rope that hangs over a pulley (as in Fig. 8–47). The pulley is a uniform cylinder of radius 0.260 m and mass 7.50 kg. Initially, is on the ground and $m_2$ rests 3.00 m above the ground. If the system is now released, use conservation of energy to determine the speed of $m_2$ just before it strikes the ground. Assume the pulley is frictionless.    $m/s$

  A 2.30-m-long pole is balanced vertically on its tip. It star(lnuum/c6 8c*sf8rvm3z2e pm 7g m1cdts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservatiolm* u7mu(dcrc3mvzm 88c2s /1g6npef n of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg e/*p-/vkbd/, l* ve y*c v,q euyc)kchball rotating on the end of a thin strin /hvde*cbk p/l ce*/e *) c,vyvuyq-k,g 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 moq6 g 8gxvu0u3fmentum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotating at 1506ufgq 30xu8 gv0 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 e ov,5dh; tfu9nt(9ahjr1xb 5rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases to 0.8 5 ;5aft bxhe1n,j9(do9vturh $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce her moment of m3xy 0 bzp5cy-qxdn/d w( 7d7ginertia by a factor of about 3.5 when changing from the straight position to the tuck positixw7-m7 d0gy 5p3(yq/ dxzd bcnon. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from no/ci- *vg ,t5kk69l61u7rz imbx0 mtxy4yqjan initial rate of 1.0 rev every 2.0 s to a final rmj6b-z vk9/ g,56l0 nkyriym1 tox qcux7it4* ate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a vertical axisle,ky dxpm7+i3 o1r9 r through its center at a frequency of 1o,y 37 im+dexlkr1 p9r.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a fibguy03z,h /z63cibw i24o yyy gure skater spinning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular momtd 93pi3e emql*gm7 :.uv8eov entum 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 ont k j. epna96apaacu-u2ie w3)+o an identical disk rotating at angular )uke.ej- aaupiaawc 2 +6 p3n9speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at e h0p4x)uxrok 3n(hox :rvt5) 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 a rotating merry-go-um7n77xaq9b p round platform77pax9uq7 bmn of radius 3.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an angular velocity of 00o teqrj4tp ;1.0;q 4tepjro 1t8 $rad/s$ 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,+54w r k2qn;;cgpr qk pxsmrmi3ao, +, losing about half its mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(r, 42makoxw n,rgq+5pc mp+srik3 ;;qround to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in excess;k ah;3md.kjp4r tah ) 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 q,g:oyl- e 7bw$ 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, initiallympitqo-,q11:jo ur vm3tsb. bza0+z, cw.n5j at rest, that can rotate freely without friction. The moment ofqzjuz cpm+, 0:.,-nw1tv.bq3 5mts1b oa o irj 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 w v z+ aq,wl:a+4f1edtstands at the edge of a 6.5-m diameter merry-go-round turntable d+we4zl+: qw1,tvaaf that is mounted on frictionless 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 o z nn;*j/nfae88;w pyu85l rfxf the rope lying on the top edge of the spool. A personfz; 8we;5j8ny8 p /lxuraf nn* grabs the end 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 E3 l*w-roz j.p,8:achm ngzz,g w8;natarth such that the same side always faces the Earth. Determine the r8wcpzl .,a:*z,8 r htzg m-onw;gaj3natio 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 particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate o5p 7-psusnd5w7 r7oso x7mho.f 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 0.2w0gnp d9yr1z uapo 0-u8 i9;2t.ioqru0 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by w.8rii dp0 1ro0tnpqu9u oa9;ugz y-2the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cyo)vaoz6 cny+vi/eq 10ert 25h lindrical 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 r iq256vt vec0 oea )n1y/ro+zheleased from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angu3,ozms j 42j5y;wmhq+ 9vk;r qx8ofnglar 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 f)fq-cu q0)6(6ytsyn+ av dxdour Sun, but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitad6f60q uatfc-qv y)x (+sn)ydtional collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use en u90ev bpf-m6nx zu-iz:+ csb5ergy stored in a heavy+b vbc6- zfimx-0s5e n9:zu up rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “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 illustral a6bx 2cqj -uh/ipqs +eh2;1ates 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 rollingvc8) mt g-+qj4m,hf cmkx;p: e 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 lenl) *qgm1w1g qwxv1yn6gth L can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration o gw1xgn *)qql6vy mw11f 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 st1hnns6e2d g gbgc 8jb7neu 7+:tk.g 2canding vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55)t e76chgg2 n : ekg.nn bs712ju+gb8cd. The step has height h, where h

  A bicyclist traveling with speed v=4.2m/cpkan/v j;+ 6k,rq7m;etws 2n. r m*yas on a flat road is making a turn with a am;qsa. ,v en+7yrkp;r/ jkn*6 t2wcm radius The forces acting on the cyclist 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 length 1.5 m and begins whi dh.d9c+ecd1h42 ehuo rling the 9 uoe ch+12 dc4hdhed.rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as amwntn-;fwjq9y:uj3;mq vp5 5 solid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio ofywmm;q5 ;nju-9qw fj3t5 pv n: the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which3p:5nreo1c +f g v/wwu consists of two cylindrical plates, of mass1 /w5:g eurco3fwpvn+ $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 th d6/d bnjd9zcbmk xq-5,wi5 y,e looped rough track of Fig. 8–58. What is the minimum vyj-ikw5c5d6dn,mxd z/b,q 9 balue of the vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but done ckp:+ c*k+u not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as w/du(ggl.bj-4r8 o/ msdwp(b the car reduces its speed uniformly from 90km/h to 60km/h The tiresb (m(/8p wbwug/o.lj-r 4ddsg have a diameter of 0.90 m. (a) What was the angular acceleration of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s