A bicycle odometer (which measures distance traveled) is attached near th f;jpk4m -f;hhe wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wfhhfk 4pm-; j;heels?

Suppose a disk rotates at constant angular velocity8o ya;6fjtbr6 . 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 linear accele tob;a 68yj6frration change?

Could a nonrigid body be described by a single valueamrgxr + )n4v r55ljy3 of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque tkyi/ u1u0yw2iod f3h5han 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 hs fbah4;n)-coead than when your arms are -) ob4hsafc;nstretched out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprocketwj 0r;f /9cjx:gq vbb)s at the 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 harder to pedal, a small front sprocket o;bcgrjwqx )/0:fb jv9 r a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower lv2cvu-xjz(bnty) a/yw6g+ b 6 egs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynami 6jxw )y6v b-(cy2agb+tvunz/cs, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow beosm8 p5vnj7 0zam?

If the net force on a system is5kv7y a xp0bw9 zero, is the net torque also zero? If the net torque on a system is zero, is the net 7vp9wk0xa 5 byforce zero?

Two inclines have the same lwwxllv /fa j2 6k(1/vheight but make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline kj1f(lw62a/vxwv ll / will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest)4cix)uk4 ,hl,x /f2l. bvpw;fpufca,v v c6o0 down an incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the inclio6k c0bipxu2,fv 4l )wfa,4hv clcux/,fp;.v ne 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 radickq3j,1qb qt2rqwzfn +j 5 4u/us and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the g4r qc t/jz2q,3bn+ 5 ukqqwfj1reater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are zp)rhpm 53 .j5pixx)e conserved. Yet most moving or rotating objects eventually slow down and stop. Ex5)mpj 5pxe). xzihr3p plain.

If there were a great migration of people toward the Earth’s equator, how wpyv -7.5jx+g hgr4 tme1 r)g9uxoqdm4ould this atg1xd -4 h+em rp) xvo7yu4.m5 9rqjggffect the length of the day?

Can the diver of Fig. 8–29 do a somersault without having an 8bp5r.os4;c h+uz owyth6y2qyh* zi 7y initial rotation whent; +hzryyu46zsb 75oyi oh *h8 qw2cp. she leaves the board?

The moment of inertia of a rotating solid disk about an axza6j: z8nz -nkis through its center of maanj -zz z6:8nkss 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 hos7)1czxx xb/rv,t )ki lding a 2-kg mass in each outstretchxc1xbt rx7)si kvz /),ed hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, otd0v +m7h plj; l4vwf7ne is hollow and the other is solid. Describe an experiment to dvd lmv l;wpj0f7+7 t4hetermine which is which.

The angular velocity of a wheel rotating on a horizontal axle p :-5hhyu ;c0:axdj ;l zn-ody htx,p5mpkda.-oints west. In what direction is the li uah:p,y- pjo0xny:5lh dx-5t czm- ak;h;. ddnear 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 or decreasing?

Suppose you are standing on the edge ofhbcfvss uj+rpb 0p*c/ 19z c/6m/;red a large freely rotating turntable. What happens if you walk touz /cj61c9s/emrh rbf+*d p0s vb/ c;pward the center?

A shortstop may leap into 3 u+bmvco-,dqthe air to catch a ball and throw it quickly. As he throws the bu,mvco+bq 3 -dall, 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. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discuss wvi3k66w - i9ltpswdef)4 t2yshy a helicopter wfe3sttp v 9id-yw 6kl42i6s )must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles inlmuq hvmsz s( 6/7y)+qv l9;ib radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincidence. Calculate, using the rp d9or.:yd*)drd8c c information inside the Front Cover, the ad*ypd 8 :drdo)c.c9 rrngular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the Moon, 380,000 km from Earth. The 6np-jog-vps 6zy; gc3beam diverges at an angley3z p6cs- pj-;on gv6g $\theta$ (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. Wzerji5.rfq;m71zz f 4 hen 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 4j5zf1;7rz rim zeqf.down?    $rad/s^2$

  A child rolls a ball on a dio7 ,7-eu zwaufo98 hlevel floor 3.5 m to another child. If the ball makes 15.0-ouu h7iz,w8f eo79d a revolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutions w 8y*537xm ocdhj*swo do the w mx ws*3*o hjd5owc87yheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculatef7 dbmzl44c6z*.qp ng its angular vlbq.m 4dgpf47 c 6znz*elocity 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 makesevb , mka /aa8v0s81yx one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from 0a8, 8ks1ba yvaxv/em 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 around they( olkp8- k1h03v2nshf i3nkw Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed yroy5)li 6m9f 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 centrifug4 -fjq(x( q/dvv3h.tcfs6zyf e rotate if a particle 7.0 cm from the axis of rotation is to ex- v3fdf zt.4qc6/(qfy vjh(sxperience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its 2)s2:4gz gous3h grqtcenter from 130 rpm to 280 rpm in 4.0 s. Dz3 s2hrg:o ) g2sug4tqetermine
(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 radiusf/mr/;zj r-nj d7 aeiyx;8,kr $R_1$ 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 aboardz ep:f6fm.)mt the Apollo spacecraft put themselves into a slow rotation to distribute the Sun’s energy evenly. At the start of thei fmmz ).e:p6tfr 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 uniformly from rest to 15,0 s(ezlbes km17e9aro 8 t;-fz(00 rpm in 220 s. Through how many revolutions did i bzeret;(a(7so9e1flsk- 8zm t turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1269u-lqijz*jp.d - k:fud ht8f 00 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, 5.jxuzs5w7j noynp4 highspeed jets in a whirling “human centrifuge,” which taken xn5spo47 , ujw.5yjzs 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 acceleratc6j;qwhe : 2 za+hwf8b/77ll t nys5ejes uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a jj2hlst5qa7yw;wl : he e+b f86/ 7cnzpoint on the edge of the wheel have traveled in this time?    m

  A cooling fan is turned off when sl l.7q8paf 1 m)cv no tzdoht5,k8 1fy1mvx,0it is running at 850rev/min It turns 1500 revolutions before it comes to a lozk8ax,o80 mf5 .)q11tmn1sp 7c f lht, vydvstop.
(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 a c0p i,k-adlfp4rx* o)s the car reduces its speed uniformly from 95km/h to 45km/h The tires have a dipk l)0x- fp ,ircod4a*ameter 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 reducl8r70/ fle+rqlaznv /) d(tgpes its speed uniformly from 95km/h to 45km/h The tires have a diame7dl8p lznf+g vl)(/r0r/qteater 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 ridinero -w3r;akla( dt 3.og a bike puts all her weight on each pedal when climbing a hill.td3l.w -k r(oa;a3eo r The pedals rotate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 7iz-9ohpo: qo,q l1eri g0 7,ks4 cm wide. What is the magnioo91- qipi q,7eg0zr ,h:olkstude of the 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 thty )5ca*d9w 91 4 t-r idjqxs2cyf.xe-epuo/ne axle of the wheel shown in Fig. 8–39. Assume that a frictionn*de )j-e94 yct-tis 9wpqr / fy2a xc1uo5.xd torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends ofjl1r i41 icds( a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude l( idcij1s r14and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tight03eiaut97 ve djg1;qkening to a torque of 38i0k eeja 3u1;vtgd79q $m \cdot N$ 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 spher.3wxvii* e rajp:i82b e of radius 0.648 m when the axis of rotation isj*vx 3p iew82 i.a:ibr through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 j3yo+mlkm 2 u2,vh0rd cm in diameter. The rim and tire have a combinelm3 o2rujhk,2m+0y dvd mass 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 thu bm1w c*jsk76in, light rod is rotated in a horizontal circlm6c1 bk7suw*je of radius 1.2 m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating at constant angular speeeym/ nuu+m9 u.d e/+. 9 myunumu(Fig. 8–42). The friction force 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 point objects shown bkpm2h:m; j: din Fig. 8–43 d2 m:hpjb:m;k about
(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 ocaprguv l 23fj90c-p/x8n9 g 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 the correct :vyx3l e-ypn/kg*r68dl* b lp 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 rocket if the satellite is to reach 32 rp vlkxlyp68p* :e3glb/-dy*rnm 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 ala 21fyg3wo -nskxr 25nd a mass of 0.580 kg. Calcul2k o5xyw3-a2 rl 1sfngate
(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 rvgkixp,+ usn,wt ,-ap: z5(tbat, accelerating 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 on a smaw871y q1t gvxull 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 of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting fric7yqgv u t811wxtional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off a2bcdtsgxhnvn7 a /hcxp- 4 ,)1nd is eventually brought uniformly to rest, cxsca) gtb -n7 /1h2nh4xpdv 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 1sgz0 oblv25 ,/eyk4h9 jvna oball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball is thrown solely by the action o-cg6zw,:1 nda43vx 6b-cvur:ejqov xf the forearm, which rotates about the elbow joint under the action of the xcn :qre1zja6o w 6xu b-dv:-v vg,c43triceps 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 considered7ywo:a.;6ejc: x r3ylhe be q6in5m3se :)we k a long thin rod, as shown in Fig. 8–46.eoe3l6 eb:q 6yc5 ja;3:wyri)ek.7xs:nm hew
(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 consffu 0n-qt di 6cl/wdv8;l 8l8oists 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 accelerartz y2*gom )rl pc290ltes the hammer from rest within four full turns (revolutions) and releases it l9z0rlt p2 mr y)go2*cat 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 moqw1x 1z4m uricbn 5ol/+ g0s7gment of inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque ofymt.sr 2nu niu. -e6a8 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls without slip q: f2,nadof(jping down a lane at 3.(:qffo 2d, nja3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with np r87:jpkpgot5 *f0c respect to the Sun as the sum of two terms, :g t 7*5cpfpkpjor80 n
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius of 7.50 mi vfhco- :oo4trjq1naw79dvf c.20-h . How much net work is required to accelerjhdova4w-7n 0:tf2co9 .o -vfic hq1 rate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls9 n3o4ck a+ypc without slipping ocy+acp9k4 3n 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 starts to fall 8, 7jdl7wvenky 1zz3*npe on 3and its lower end does not slip. What w jlz ywd3*,8o1pnkeev7n37znill be the speed of the upper end of the pole just before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of xwb gu:mi -0d)bj0i;x a 0.210-kg ball rotating on the end of a thin string in a circle of radius 1.10 m at an angular xiwi-u m0j :b0 d)xgb;speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular 7 akbr,; q2sjxfdl*5 9t 0v2f7txyighmomentum of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rotatxi,as7xftq y*2 bjtg072;k 5fvhr l9ding 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 side, on a platform that is rotating at a rate of cp atxxjo+/ d34p, b*y1.3rev/s If he raises his arms to a horizontal position*+dxbt ,/4o jyacp3px, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce her moment of inertia p4kr-.. xis z19sj3f y1v humxby 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 when in tu4x zx91 yssi3km r-f.jv1hp.he 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 rate of 1 eey+3j td9);j0bim1hvz6,i:0y ( z znznftgm.0 rev every 2nnei6 t)(ti v fzz;bhyj 1e9gdz,jzmm:+003y.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating aroundoc()o3mfej p5 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 disk of radius 8.0 cm, oc)eom3jo (f 5pnto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure fmw 0ird-a-e+ 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 momf+ w,t i(dyhf2entum 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 diskly/-ooo8 fqnk/ to4sllg8 0k- of moment of inertia I is dropped onto an identical disk rotating 8qg/n- oo o4/ykt0 slkllf8o-at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns .xt ecrt2 )bs.at 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-round platfo :coi e3udc(x7rm of radius 3.cux7c oi3:de( 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 rotatinbff-( d-ibs2jg freely with an angular velocity of 0.8 -fbi 2sb-f dj($rad/s$ 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 wh jz7v8,dfygmek:wq4q ls,1 x 4ite dwarf, 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 Su zxj m8 ,l kdsw,vq4g1fy:e7q4n’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 exce*vb+ -q1tek urss 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 wcz-fyyf8 -cve *1o7fq(des 0 c,x/rn$ 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, initiwqs9biqxs-)vc1i (t ; ally at rest, that can rotate freely without friction. The moment of inertia ob(is 9cwti-q)vq1 s ;xf 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 diamb2x) qm,e7xvr eter merry-go-round turntable that is mounted on frictionless bearings and has a moment7 ,mrxvebx q)2 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 ly v0*9)t2e++qivonlf0s imxjq ing 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 the person without slipping. What length of rope unwinds from the spool? How far does the spool’s c 9j0m+sn f*v 0 iqoiveq2+lx)tenter of mass move?

  The Moon orbits the Earth such that tydcds+sel .rn1p 31p*he same side always faces the Earth. Determine the ratio of the Moon’s spin angular.3+d1s1dnpc ys*er l p 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 mgw d,w jw 0aczl)t7:+from rest at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of radiu;ijod erj( rc6i- ak;0s 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the t0-;ciao jkj6e(r;i rd orque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 7izb,7t-nqqf9y u mf u-mjb8* 0.050 kg and diameterqnb8qmjui7b zy* -mu7-9 ft,f 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 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 wheel lf)5n vos(g) bw-bz2c($\omega_R$) 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 mas68c))g qfx6u-eg sv zoi:k 9svs 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational cole6x:gckvuv -ssf8 6 giqo)z9)lapse 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.a3f-gciwu m3qlremxd 3 /*hfv g.k)5miv t64 low-pollution automobile is for it to use 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 1.50 m and mass 240/akc5r)ufii 3ml.3d6 fm- qe *x3 .vgh4wmgvt 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 h,3 qnnl9 ), kjime( 2ti:cb0:zlhtpa $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on a horizontal surface at p 7v+*fcomlg6b zlt-v,yku +7mgaja44t 98vq 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 pivoq+pm m/qg.a3omfd7s +t 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. qoqm37 m/dps.g+fa m+At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standing vertically on the floo/nza 9o:+d7 szom; e du+kc4ylr, and we want to exert a horizontal forck9;yc usa4modd +zne + zol7:/e 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 speebu 1j-p-3g wpod v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cycliwb3o-p u1pjg- st 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 lengthc4:al.xba79f /qd wnj)i dm9q 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. What is the torque required to achieve this feat, and where does the torque come fr./dc :dbn)9iaqwxl 9qa4jf7 mom?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her armod9:ztqzlcnj 1v 988cs as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular sq:l9cc9od8 nzt8vj1 zpeeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two 3+wskojww .5-p e xzo1cylindrical plates, of s -w3 pex1zowwk.+5 ojmass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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 radiijm/8frw: )bk8/pa5v d5 lqfrfgx/y - us r rolls along the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach thep bvfyq5f5 axdgf//8 k)wm/ rri:j-l8 highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but+4f d;k0dlblm do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uniformlpj q1rh1) s7t*tbd xt8y from 90km/h to 60km/h The tires have a diameter of 0.*rt pts)7q1h18bdj t x90 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