A bicycle odometer (which 3fl0hn ) 3ptr*ki5+ee5coiydmeasures distance traveled) is attached near the wheel hub and is designed for 27 he3t5i5)n *+iepo3y0 lcrdk f-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocittu52 nc3mzaxg im1 60oy. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular tnu620zi3coa mg xm15velocity 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 bodyr e8)1rgi p bmjykm).0 be described by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater q. a)qs3sfwpqyvxui 6g4vzr9o -f(a-j r82 .xtorque 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 twp2kz/)egt7 a han when your arms are stretched zkgpe/a7w )t2 out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rea)pqhw.g8 *mtvfwz.xm, 4 io,ir wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a lavo4z qmfhw ,,*tp.imw x)ig8.rge front sprocket? Why?

Mammals that depend on being able to run fast have slender lower legs with f ,.2xiiwk7ragyf 2gy8 lesh aw8fyx,2gy 7gk .ir 2iand muscle concentrated 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 a3,o8mpq5b u jj long, narrow beam?

If the net force on a system is zero, is the net torque also zero? If the net t* epo70 ppd0u zfx1pb0orque on a system is zero, is the net forcpdpb0 0eu1opp7 * 0zfxe zero?

Two inclines have the same height but make dch9 iu1d45jj w:zh iswc)h5 o0ifferent angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the ball at the bot 10wwoi5) 9 jh4d uch:zijc5shtom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an incline. One /ch 1p8i+ftq) e13hf lq vrci+sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which hait/l+q p8ifqvhr ce)31h cf+1s the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. They stae6)v6fqsg 4 +queqb-grt from rest at the top of an incline. Which reaches the bottom first? Which has the greater spe fbgu6e) 4+qqsq- vg6eed 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 conserved. Y a9qemuem t9ka3g;4 w3/ ctwzhwkzm5 g1p a*46et most moving or rotating ob* kpq1ta g4m5wgm4aa wzu9ewm69 ;3c th3zk/ejects eventually slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equator, howof7+ttgtqcbz 3v;;/w would this affect the length of v ;7wbgz/t+ 3q;coftt the day?

Can the diver of Fig. 8–29 do a somersault without having any init:mspw( *wof a 30l hypzen4+(bial rotation when she leaves the bpl*zn(mf4 +o(y: w a0p3bhwseoard?

The moment of inertia of a rotating solid disk about an akr dqme1m .a5;xis through its center of mass er ;.1k5mdam qis $\frac{1}{2}WR^2$ (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 maup7)h/s,r wxbtzy18 vf:.l ffss in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? 1,l p 87ywvtx bfr.fs:)hzf /uExplain.

Two spheres look identical anv-ym6gt-; pq rd have the same mass. However, one is hollow and the other is solid. Describe an experiment to determine whicv6ygqt-r;p-m h is which.

The angular velocity of a wheel rotating on a horizontal axle poin6e8kjt ozp:;m ts west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing orkoepzt :6m;j8 decreasing?

Suppose you are standing on the5 *pb .-midx0 clwbm9k edge of a large freely rotating turntable. What happens if you walk t mm9lic *- p0bd5kwx.boward the center?

A shortstop may leap into the air to c2;+bu (p;fv 8 ee,vqjd9 r3(llxnaqyy atch a ball and throw it quickly. 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 dire( ; 2ql,vqn3dpeer98yj ;fuvay (+lbxction (Fig. 8–36). Explain.

On the basis of the law of conservatancqni+dqh .9 8k si88ion of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the hs9hdc q+i 8annk8qi8. elicopter stable.

  Express the following ang8)t n1 qv*jiriles in radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coinc x; 5zr(,ibcnd m+pvi*idence. Calculate, using the information inside the Front Cover, the angular diameterz;mcri5pi*d(+ vnb, xs (in 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 from Earth. The beam diverges cdlqg6 1vrtni : ;qm- qrt(q(0at an ( qir;t:m6ngd-rq1 ltv0cq (q 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 b03 3/d5 gne xhn82c b0jbtznm yrv3h46500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angular acceleration asbnv28xmzjgnbec354nyb33 d/0 h h 0rt the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If the balo +we k( (ezbtwi2ebg/;ki52pl makes 15.0 revolutions, what is its dg+i2o5 z e(ke/2; b(wbikt wepiameter?    m

  A bicycle with tires 68 cm in diametyl dgy:6wtn7w. pl57yer travels 8.0 km. How many revolutions do thew p 7lynlg d57ytwy:.6 wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 25o7 17jhvvcjhsl)u* et.;ci y3 00 rpm. Calculate its angularh)c otv. e*l; scuj1ij773yhv velocity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.0 s (Fig. 8–3h--zec7p/xm o4 amlu1x ,tas: 8). (a) What- -lsachpum4: x1tm/,zx7 oe a is the linear speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) : r-yavt k83sxin its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear sp skv iu0):d;3 b+ye+)dsawcmk 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 7.0 cm from the a.z sojz0ig n pj+s*:r:xis of rotatir:j*gzsn0o:j+ i z.s pon is to experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpm to 28 )6o,+ ajiajhvb:9esv 0 rpm in 46oab ,+v he:9jja vis).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 3b yq7,jm(obr $R_1$ 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 themsel v yfj2-kizoh-u12rd7rq6ty 7ves into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a 12-min time 77 t rhj-k1zi2 qy6ru2d v-yofinterval. 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 2y a :;q.ibl96wgswep. Through how many : yb6e2 qsap.9 ;lwgwirevolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm ind; tz9r v/ fp.anv+a8oeu /ud: 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 highspeq l h;vpvm+**xssg4k;ed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete rmpx;g q klv;hs*v4*s+evolutions 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 agq: tw5vlvo77ccelerates uniformly from 240 rpm to 360 rpm in 6.5 s. How farvq7:5 7gtowv l will a point on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned oev lvn5ojb; 9uz98xv( ff when it is running at 850rev/min It turns 1500 revolutionxl5 zb(en9 vv8v9;uojs before 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 revolut gm 3;/mfngb8)8hptjler(wsh/ h.kiw5 ca9w1ions as the car reduces its speed uniformly from 95km/h to 45km/h The tires have a diameter of hw );3il9h5. ts chm8w/rmbp8gf1n e g(/ajkw 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces its speed univsgi. s2,uwkn s3 ..kec. bbw18z6kjsformly from 95km/h to 45km/h The ks.s sw3j..2cu n k iv8gwb, k1esb6z.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 when climbing a ei41b7od ryy ( +e*vnhhill. The pedals rotate in a circle of radius 11iy e7vy4b+d e(h *orn7 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 qd w)/-b ysr8fdoor 74 cm wide. What is the magnitude of the torque if the force is bdr/y-qsw) f8exerted
(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 shown in Fig. 8–39. Assume 6b5m*-gny dxithat a frigbd*nm6y 5-x iction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are annlr7 ekx 77 e0ez/a (zump0m9ttached 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 magnitunmmzxlr7 (09k /77pe 0unzee ade and direction of the net torque on this system.

  The bolts on the cylinder head of an e 0:h :r upy-n;wckbxz,ngine require tightening to a torque of 38p k bwxrz-:yc;un,h:0 $m \cdot N$ 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 radq5.t*nu i9* yh2l3sm.2d /hga pobgww ius 0.648 m when the axis of rotation is through ito*. liup*2 . w qyn/bdg95mwg3hst2 has center.    $kg \cdot m^2$

  Calculate the moment of ine2 + f35u-uk fho,uqwasrtia of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of ho2s u+u3w,kff5 -qau 1.25 kg. The mass of the hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on ovsab seu a/4*67ob6s the end of a thin, light rod is rotated in a horizontal circle of radius 1.2 m. Calculata oa6 sebs/4vb 6su7o*e
(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 rotadfv h;sg h-4-fi.fj (cting at constant angular speed (Fig. 8–42). The friction force between her hands and the clay is 1.5 N tj v( 4- cffs-hih.;gfdotal.
(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 arr ho/wj(5t i:(;tdeg pqedtd(.ay of point objects shown in Fig. 8–43 ajw(dqgph 5 ted(t :(;e/i.dot bout
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen atoms whose total masszu c 0y*dmfj)1 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, uniformu0/e*gcpc a4p cylindrical satellite 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 4.0 m, what is the required*4/p ceaug0cp 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 g ylq/blb67c5pi6ie4( pc/y smass of 0.580 kg. Calculablcs 5p7c/bqyle(g ip/66y4ite
(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, acc /4j1hqfpgnglp/s :)v9do+wnelerating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially zia o*37s(id 5i tvep(on a small hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 1izv7 *i3aditoe (s5(p 5 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 required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at2 c 502ct wjcmk 6sh0gyaq6kp+ 10,300 rpm is shut off and is eventually brought unim+ ytwk5k2q6scg60hpa j2c 0c formly to rest by a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8–45 accelerates a 3.6-kg ball a9dtq on4 3 bi +xjjtn3; vf:x4m+eyl;x4tuv4ut 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 th7xmc(a+ 7/lt1-i dcemfdrt m5 rown solely by the action of the forearm, which rotates about the elbow joint under the action of the + mr e-57t da71dc (ci/mmftxltriceps 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 bla erlh8d*n k ;o,o2ioyt69qkt 4de can be considered a long thin rod, as shown in Fig. 8–46k9 q 6t* ;honok8ritdelo2,y4.
(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 masnx/e*qv qow40m vc/i.ses, $m_1$ 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 re48 9bn no,8uwekj1)o iadwp, vst within four full turns (revolutions) and releases it at a speed of 2k apnu,) je9wn8d48 iob,1 wvo8 $m/s$ 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 tmyz36y5mqyz8 l )*ibof 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 t+ oc )ak1qcb)wzds1. vorque 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 radiu1 a0+p -mlr:hdyxdjqp5- l7sws 9.0 cm rolls without slipping down a lane l0p l + r7as-m:d x5y1pqjhw-dat 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as the qesr/xzq*)h) rjkh c7jx)y4y i/a,, lsum of two term7 q /,se x,/jjryl)hkcia )*4)qrzxhys,
(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 1 yj5 *tg4u6 vgo+1qedn yc*nn9640 kg and a radius of 7.50 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 c1 n 4 yun 65g**jdo9vneqty+ga solid cylinder.    J

  A sphere of radius 20.0 cm 6ia hs-k,a0n*h2ihswp p/j.fand mass 1.80 kg starts from rest and rolls without slipping down a 30.h- s 6pa,2hp.k ahns0jfii*/w0 $^{\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 verticalv:c g 0u m6pd9br dwpu)l--qygw.+8bss+ 8lgtly on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of-8brug ppg wlmu906)8db+.qd w:ct s g-vyls+ 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 rtkam ct8m:x;4y + 1iwthe end of a thin string in a circle of radius 1.10 m at an anttk4 armc;x:1iwm8 y+gular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylind8 xvak)d):n+in1qdo nbtf+)jrical grinding wheel of radius 18 cm when rotating at 1500vj+:t)ox ) )kbqn i8fddn+a1 n 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 rotating at a rlx)42rc g8zo (1lzpjf ate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the speed of rotatir1x2pofzl)4zlj g c8 (on 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 het48 y;ditixzf ljfbs4;-k pue4 *(j2xr moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s whejt4;s(f ik fxx8p2u tzi ;yl4*4dejb-n 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 rotatioe wi;lck/ 7c+yn rate from an initial rate of 1.0 rev every 2.0 s to a final re7l;i+k /wcycate 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 t)) iq5 3tov9w,irnrxa vertical axis through its center at a frequency of 1.5rev/s The wheel can be considered a ti o5qxtr,r )9)vin3 wuniform 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 figurt i r*s2bfs,,ua5u y6xe 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 xr- :mgzhr7-d momentum of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is dropped onto an identic(yklqq wx/04 bal disk rotating at angular speed 4(kx q0y/lbwq $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4 ao y6y4ypyu +8$rev/s$ 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 rot j v7lzs 9oh;b73s1rxlating merry-go-round platform of radius 3.0 m and moment of inertia 7 7lo1r ;s sbjhz93vxl920 $kg \cdot m^2$ 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 rotatwz8w9c )pl -py4uhe4pdrhk ;2ing freely with an angular velocity of 0.82p)p h4ep;z 8l h- yrkcw4w9ud $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwarf, losing about half its ; hz noeov3e.nl a7m.q9uv,e kppo)-,mass in the process, and winding up with a radius 1.0% of its existing radius. Assuming theooa- z, nve3h,;un.e)pv k9p q omle7. 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 winds in excessw8ms4m ebm vek) .jtmnh2 )/fw8er4r6 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 qoth,5 uwx,v -$ 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 withou2hy zjvdjn6p+p:-q * it friction. The moment of inertia hiqpy*v6jjz-p2+ :n d 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-round t;7 bfy-ux3d l0 -( tmyu-imauiurntable that is moun;-b -u imalfutx3 07u- (myydited 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 of the rope lying on ther2/sxbj 4g i4p top edge of the spool. A person grabs the end of the rope and wap jbxi44r 2/sglks a distance L, holding onto it, Fig. 8–50. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side always faces the Eahr-2+tyt.+q yt r ,sxjw/6lrnrth. 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 orbit+t,nq-x r/ +t ry 2sjh6r.lwtying the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate rflnme5owq.m 5 oj+3.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 cyl dfa5 )knm9 85+mss-1p luteexinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque deliver9u pn)mf8 l1admsx ee5st-k5+ed by the motor.    $m \cdot N$

  (a) A yo-yo is made of two s a b1nfn:2z-cdt5m(gcolid cylindrical 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 snnac-zc fd 1 (tm:2b5gpeed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular speed of the rear je.n3r8de gq,0k9tx c;p+j jn 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 great, a 0jywi9r7jy7 w 7s6iog5ml4jwere rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of rad w775lowjig7a9mysjij0 46y r ius 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 ud6;mw,dvnm* q se energy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.5qnwv6, ;d*m dm0 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 an v3j 5j d)u avf0bek93j(psja;$H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal v9e b1b7qp1idsurface 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 ignore ferbxjvla 64xfvke3c6u 4*fu4 wr i79 5riction) 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, deee64cjfrwuxf 5r6kb4x a7 i3vvul*94termine (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 standinga e.p,c*.ncle 9;qvun vertically on the floor, and we want to exert a horizon ec,vqepa9lcu n ..;*ntal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/splcxlyb 3kc ue/1j0 ,0 on a flat road is making a turn with a radius The forces acting on the cycliejc k 0bl lxyp3/1,u0cst 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/g sakn2cone2lh 871g5 *rff16a kgo o 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s./og5 o1ahcs knog6g*fe721k2 nrfal 8 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 / tom(1msoqwx q5ulnq1:, n/vand 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, and with arms held tightly against her body.1/u (q5xwt /: qvl nm,soonm1q   

  You are designing a clutch assembly which consists o)qu*mwens5g w7-9 ie ef two cylindrical plates, of -g em)q5e*wiun e 97swmass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 of Fig. 8lx2, 5suz(gx3eyi vd:–58. (i5y3xlg:dsu 2zx v ,eWhat is the minimum value 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 do nokif q 7x jftj+)yswv2 .es3cn-)-nc)w t assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the cey (gi,8 nj 2hcg gpr8d+rot-,ar reduces its speed uniformly from 90km/ggydtreih-n rc2,8op (g+ j8,h to 60km/h The tires 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