A bicycle odometer (which measures distance traveled) is attac l2 yl6f +gqef*it8g5ehed near the wheel hub and is designed for 27-inch gffe68 e*i l tg5+l2qywheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates z7m mfyy0sd v5pdp+8.at constant angular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of lidm 8mf 50syd7v.p+yz pnear acceleration change?

Could a nonrigid body be desc6fm-x ejx: p:yribed by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exertv6zstca gd8;9muz0rfa3 n k;6 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 5/ 8j9 rpdx:( uzua72fuwwilfto do a sit-up with your hands behind your head than when your arms are stretched flpuj52rf9 wi( 7a8:xwudu / zout in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the*91iwz zt iz-xeri:no p9fm)8 rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it hard 19z-ti:r)oxzmnei9iz *f8 pwer to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being ablhf;fz izf u qb(8ts h 6-);vd84h5bupze to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this dist svtf8ph6 ;fbhb) uf;d8q-u zz5z4(hiribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a longp*rvacop bo74l4 )k d3, narrow beam?

If the net force on a system *i l1bth *0omlis zero, is the net torque also zero? If the net torque on a system htio1bl * l*0mis zero, is the net force zero?

Two inclines have the same height4yo1ogu0n; c f but make different angles with the horizontal. The same steel ball is rolled down eaco4; 1gn uyocf0h incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an ini(rbx-l .xv m0v6:fswcline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of twr6 i 0l.mfbsv:x(vx-he incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same 2w ae,p8rkv. ngto5( nmass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which p.v ank,g58(t2wreon has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are cons, 7 eu+, dy,7yqu-ey cxzdiuu4erved. Yet most moving or rotating objects eventually slow down and stop.7e uziyu4,ye-c ,dxy+ u7 qud, Explain.

If there were a great migration of people toward the E,bvt b2g 0s1 2mrg(k hlgsi/;larth’s equator, how would this affect the length of the dah bt21(ss, igg/ 20vgrlml kb;y?

Can the diver of Fig. 8–29 do a somersault without hav xakh+;bnns .f:ha).a ing any initial rotation whe af.:n)ahna.; bhsxk +n she leaves the board?

The moment of inertia of a rotating solidlkpfl*zbnbz ) o)69g.0ty9pe )7z vyz disk about an axis through its center * zt)ok ebplyp70y9)f v.)9lzzn6bzg of mass 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 mae0ga(aizyg 9t0+ysr3 h)x u)ess in each outstretched hand. If yogeayg30+h0 aur )teyx z( 9s)iu suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. Hfopfg za182vq .6j zsy6zz; 1vowever, one is hollow and the other is solid. Describe an experiment to determine which is whizj;8aq z6y12fozv. s 1gfzpv 6ch.

The angular velocity o0wfv44 . dbpxrf a wheel rotating on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If t4 vbp.wd4rxf0he 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 on the edge of a large freely rotatingslqb:,9pm . di turntable. What happens if:,sdm.9b qi lp you walk toward the center?

A shortstop may leap into the air sl u a.ys-;gt5b)d x;ato catch a ball and throw it quickly. As he throws the ball, the upper part of his bodytbuxdgya s- ;)5l.; as rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discuss ve640mw ,vu j*.vfny+ s dg0gdwhy a helicopter must have moredv. v+* j6gs, dnfmw0 e0uygv4 than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in r rx::y.m ,g;n9mfgi kpadians: (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. Calculqh c+/mvma ;8iate, using the information inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on q+v aimc/hm8; Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from e*acuj u.w56mp l-zv 2Earth. The beam diverges at an an.2m *up jeazlwcu5v-6gle $\theta$ (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 65.u9m0jum/)d ( z mwmmv00 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow dowmj/ z mudm.m0vwu)(m9n?    $rad/s^2$

  A child rolls a ball on a level floor + 8q:a goe-aev3.5 m to another child. If the ball makes 15.0 re-qe8ag:oav + evolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutionsyx12 l4:fs vfs v 5cci+r9qv/b do the + vc /ffvsq4bv9sx521cr:i yl wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate itso8zjg 7nrk t.yx),g)zf+ ptw j2c(cte,nh.z7 angular velocity . egf. hzxtk,7wrcyc))g2z8njp jznott +(,7 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 revoliuau m87xry,(;ttur 5ution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from theruytxmu5u,r8 i7a ( t; center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) in its orbit around ebrwaab1.x)* the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a point b. nl4js /x7z fh yo8 khy9k.jvypb+//
(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 from tnc8:jhphoqmn 2 lo nq568u+9yhe axis of rotation is to experience an accelp92 n56o:qc 8 q nnuy8jhhlmo+eration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpm to hzq ehxc7 u*79280 x chh*e77u qz9rpm in 4.0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radius )qgm 6h1i0lcbb 2oz*jv3j o:v $R_1$ 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 sd*)2* 2e *8fdjy9gta isua;n7 bzcx caboard the Apollo spacecraft put themselves into a slow rotajs x*7csea*niuz 9;a8*)t fyd 2c bd2gtion 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. 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 unifo(ayzdfzp6 .6yuvr p xqqf5 (0sv.n1 w(rmly from rest to 15,000 rpm in 220 s. Through how m1p0dw. yv a6nfqz.qx5p6(sz v y(fu( rany revolutions did it turn in this time?    $rev$

  An automobile engineld )xnnz6cz5) slows down from 4500 rpm to 1200 rpm in 2.5 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested for the stresses of flying highspeed prlcyu6f)l84y;:m t hjets in a whirling “hummc:pfu8 ll)y6 r; yht4an centrifuge,” which takes 1.0 min to 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 frhr5i3 89rbszw om 240 rpm to 360 rpm in 6.5 s. How far will a point on brhz983wis5r the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turnsk9 oy qjxs*l/2t9) gszl ygn/a7,f,ul 1500 revolutions before it cou/z, l ft xl ln7qy,s*/gyko2sga9j)9mes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed uniforl : vsf:c,txz+/evyu 7mly from 95km/h to 45km/h The tires have a diameter of 0.8uclszyv /:t :v+e7f, x0 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 revoluti*moimb qwq,c; :otaz v ,3,:fr0zblu0ons as the car reduces its speed uniformly fromoz;rw0lt,: m0a bbu i,fq v3om:q*z,c 95km/h to 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each pedal wdnndoxf ah0hx02 12e7hen climbing a hill. The pedals rotate in a circle fdo xdxhn1a272n 0h0e 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 5lfo00 kvjq .m+5 N on the end of a door 74 cm wide. What is the magnitude of the l0fk.0q vmoj+torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shjql90d c foy/7own in Fig. 8–39. Assume that a friction torque of 0.4 j7cy/0o q9dfl $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a5kwl3 ll;op gg en)oh--t p9j( massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and thenwj-h;n kgl-po gl5( )to el39p released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an eng 7wc+y+rdaj r3ine require tightening to a torque of 38 +jw+ar yd3c7 r$m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of radius 0.648rxn 0jj;-d4yg0x. uxk m when the axis of rotation is through its r y nxju0d-g.04x xjk;center.    $kg \cdot m^2$

  Calculate the moment of inertx 9r6b6nt7iu4 x,lkiy: (xd ybia of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass i(uirt lxk6x4xyd :7b,9n 6yb of 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in a horizo 9ir*xsq)nzb, ntal circle of radius 1rnsq 9xb*)z i,.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 potter’s wheel oj6 f0pgem3wzys5j:0/ x5c bo rotating at constant angular speed (Fig. 8–42). The friction force5fz/0gxmy w60j:sbo oj3c 5p e between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of inertia of the array of poin0 4bk3s udrxq4s o9k+lt objects shown in Fig. 8–43 abou4rd3x+qk 0l ok9ubs4st
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen atoms whose tota0axxsgq/ 7sl)u,se t20gvf t (l 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 thig)vrqz.) nu ;e 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 iivzg n));q .urs 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 8.50 cm and a mass og hv-9/4 h5+y)g crgeim4slajf 0.580 kg. Calculate h4csle5ggygm-a+hrji 9v)/ 4
(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 xxfxm h )7xu2+ l4hx4ofrom rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driven merry-go-round y,nc aa+x el vm:,e8g (9gpp6w+at3lpc7u*amand 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 of 25 kg) sit opposite each other on the edge. Calculate the torque requicel p8a,mlt npgua6 ewpg+y ,a:mca( v7*+9 x3red to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rota 2 f tlaqu8;yg2t3 cgdkx5qmn(:h/d*ating at 10,300 rpm is shut off and is eventually brought uniformly to rest by a fri (q :f;thlkqxgd3*/aatud252y n 8gcmctional 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 r j3;e9obq5+:tge.c ledmcs5a 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 dfpsnp6;xs - s+j6:rbsolely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The :+xbdr6 sn ;sfj-pp6s 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 x0uf.+z7;y bq9 2z;eitz yipkh2 9dn jrod, as shown in Fig. 8–46. .ybufzdtq20+yz2i k7nhijze9 p x9;;
(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 coy6tn3;uuex wz,tyl, n my6p o l607o9bnsists 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 turnhxp d,v6( sdk7ijx+(y s (revolutions) and releases it atk dvj,dxs7x6+p(yh ( i 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 momen1j.c f f/or 1 nkz:0tx/mbgoe/t 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 devesf 0/84zzbojqlops 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 wit8a4hpgj dft.ej08rk1mt 16q9msy a)j -alde4hout slipping down a lane846ghj d)m.ejta 1aart pd 91 4j ysklmf-0qe8 at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Suu t*sbo3i 1v(cn as the sum of two terms, sbticv(o3u1*
(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 zj5 m5lbz(,mnradius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolutizmn5,j ml(b5zon per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and ma* tyap46o, fj/dtu jq)ss 1.80 kg starts from rest and rolls without slipping down u 4qdt*/fjo,p )ty6 jaa 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 vertind23qxz -vl-k47hjt vqrb ; f(gzv,g6cally on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of-3(fg zq-6 k;qz4l2vrgjvdnb xv7,ht energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotatibw bbo60m /3mbk/wyi. ng on the end of a thin string in a circle of radius 1.10 m at an angula 3mbym/w w0bo/b6.b kir speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momez j,f75g* qyqtntum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotat*7 ,5gq zftjyqing 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 his sid3/autcflja 3ah jns51ie6:2ne, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal posiu sjf/26tie3 aa1nc:hj 3a nl5tion, 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 on2;c9cq nwukfgr- 9-z 940zzpw l7a gqosad 15qe shown in Fig. 8–29) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck positqnzask5wo - 7ad1 c92lw0rzpg4-u9gc z; q9 qfion. 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 an initial rau:tw d(f m4gg4te of 1.0 rev every 2.0 s to a fuwmt44: fg dg(inal rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating aroun7kr+ ijiyrn9ve : whxvvb6,+ )d 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 of clay, approximately shaped as a flat d6xiw ivy,n +r+)vk:7e bjvhr9isk 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 momenty) w-,v f,yuxqum of a figure 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 momentum 3p ++kv3;z*gxdd m(vlmmdpm 7of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$

(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is dropped onto an idend5, vhp,tez4 sy:x-ln tical dxyp-:5thv e,4 s,dz lnisk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at ij myi;ky9 qx6re2m u5e5s*y02.4 $rev/s$ 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 k x0 9 s sahww0mlzvu 9h0f30im7w+cp25waq7cmg stands at the center of a rotating merry-go-round platform of rad w70w 59zm2cmmq3a0chh9psaw+w x00s ulf7 ivius 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 rotatin/yqsn2 srn* 7pds+/pi3nh -yrg freely with an angular velocity of 0sdy3sr7 +yn-*/pn nspr 2ih/ q.8 $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, q df*lwxr91z -losing about half its mass in the process, and winding up with a radius 1.0% ofw fd1z-xq*9rl 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 involvu6p f( gb0*sz*v ph)uoe winds 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 t:2;36d 4tpub,nx x41fhnz0s f gsi/ygfweb, $ 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 rotle xj7rls 45d3ate freely without friction dxr54jlsl7e 3. The moment of inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m +n 2oz0ol/cdmdiameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia ofoz 0nm2o c+d/l 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 lq ym65 -ij4wf lc8m8f1 t*yhdthe rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind l4 81it *wm 6fl5yhm8 djy-qfcthe 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 thq snhb/+n.x:o3+llzde same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiti3q ldhlxbs :n++zon/ .ng the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at k 8f y(dabptu-+ xaz9, j3skw9a 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 acq5d/ps:fyany3bz,su n s5lu7:yr c4 ,muires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delbn 75n/d5uucs a4 r,yl,:f3pm z:ysysivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cw 4dskbl/wi.4 ylindrical 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 wid4.s4bw k/l 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 anyi(x6eyoran) w z)gp5 *njgwj6s 8:k; gular 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 times as great1 s-i7vw zp3 2v9,9ksw rkumlt, 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, lo-wlrk ust z73wvpi kms929v,1sing 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 foreb9e1z.k i7 xo+x3 laikv1)ul k ff*9o it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 ki) l9 bk73 okxvek ze*fxo1.9if +a1ulg, 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 illustrates anjfr*5z7 k adx/jez m7. $H_2O$ 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 rl6:j4tae . igoolling on a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., we ignoreebv ,-vukr. ;++lo -gg qoaju3 friction) about-oe ugl+ubv -,a + r;.ogkv3jq a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing verticallyd1dm09+ y6ldafzmh.cb fra w 7d+mj3 id0hg20 on the floor, and we want to exert a horizontal force F at its axle so that it will cligdzj1h.yb6wdam+ 7 dimrf20hc m l0 afd+09d3mb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/sc6 lu:(f8: .pzjuez cg on a flat road is making a turn with a radius The forces acting on the cyclist and cycle are the normalul:8c (z uzef.gjp 6c: 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;sm)v gm r-c4mrvy4 s) m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate s rcm)sryvv;44g-)mm 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 andn 9)j3 pvjxdf6 her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, andjj63n )vfxdp9 with arms held tightly against her body.   

  You are designing a clutch assembly which consisjux i + 7-a.xgpzi)x-uts of two cylindrical plates, of m7x xx)izuagi-up-. j +ass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 r(()sibp k;+ztlw1+ogc(t ldl adius r rolls along the looped rough track of Fig. 8–58. What is the minimum valswlcd)+( (t1z+lbg ;lko (iptue 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 don x */udfjwyo10 q60qg not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed* 4koo.a3xxyx uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What.yoxo 4xx3a *k 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