A bicycle odometer (which measures distancef// 9;dc.sypv4r9d dthy sa p9 traveled) is attached near the wheel hub and is designed f.4 p/dhfypy9t/sd9a 9v rcs;dor 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point on the rmr0di4 g,2szkim have radial and/or tangential acceleration? If the disk’s angulars4izg m,0d2k r velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the angular v 2 ngf5mbv8hv:-7giq u;)lal9ny yyc(elocity $\omega$ Explain.

Can a small force ever exert a greater torque than a larger force? Explain 0k09dt +koffgtz.x2b .

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 han: vdf e4utu-r+aj4se 5ds behind your head than when your arms are stretched out in front of you? A diagram may h:v uu4 rf4de-+ta5 sjeelp you to answer this.

A 21-speed bicycle has seven sprockets at the r tgc/n/ uoi,/t:g .c-afs9izxear 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 // n u:stigt-o ,zfc.igc/a 9xharder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower legs with*p1 bu3faa1+ca3u (gdi ) xbjq flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution ofq)1b+(acdagx3juf 1 *u3a pib mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, ntiy7u4 4wavhi /uov.5 arrow beam?

If the net force on a system is zero, is the net torque also zero?c 5t6/rdf nr3d If the net torque on a system is zero, is thercdfr5n63 d/ t net force zero?

Two inclines have the same height but make different angles with ,*z u+al mixupc vt58 *xw2l6qthe horizontal. The same steel ball is rolled down each incline. On which incline will tlumpu2a*zcl ,xq+*w 5iv8 x6the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down ;xzg 1ijh l+4man incline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has th14x;i +lg zhjme greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius- ei/ 3 d/i1bia5mqefe and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has i5b1dim- /eqaei 3ef/the greater rotational KE?

We claim that momentu+g sbm9k gq8+mm and angular momentum are conserved. Yet most moving or rotating objects eventk+8bqm9 gsg m+ually slow down and stop. Explain.

If there were a great migration of people toward the Earth’s equator, how wo,0(t mjfo ji/muld this affect the length of the d0mt/( ,o mijfjay?

Can the diver of Fig. 8–29 u-ysv,t ;:v- szf, cowdo a somersault without having any initial rotation wh;s-:wvoyt-zfsv, ,cu en she leaves the board?

The moment of inertia of a rotf:o 6 uyv(iab,ating solid disk about an axis through its center of mass is ,uyi :f(bvoa6 $\frac{1}{2}WR^2$ (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 rotating6(e 2na1ncf6rps nsv- stool holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will yous6n2f6( cpe-nnsva1 rr angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identicate,ytn v/za q)uvva:33;ih. ml and have the same mass. However, one is hollow and the other is solid. Describe an experimentv ;ieu t v.vanty33/, haq:mz) to determine which is which.

The angular velocity of a wheel rotating on a horizok7fy0a3otfd8znt +8 c ntal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the anguyo 7 f8tdntk08afcz3+lar acceleration points east, describe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on t *icdw y(d n).)qcvt /xth63ashe edge of a large freely rotating turntable. What happens if youyvt3n *()6dxti/d.c)q cwhs a walk toward the center?

A shortstop may leap into the air to cati*b;8z 4 sy,7wjp7a+tw hmef jch 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 direction (Figw*f+m78y;sh,jz4jti bp e 7a w. 8–36). Explain.

On the basis of the law of conservation of angular momentum, d aj,ugw;(3 ,yohlxx5e+kugq d466 fbyiscuss why a helicopter must have more than ou+ay b4h(wdxy3fxk q gg6,oe6 5;j, lune rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stable.

  Express the following angles in radians: (a)26 f,qf6hdkudfbx; 9e;goq/o 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 t8amve rv0 +xi6/qka 8jo+a-sghe information inside the Front Cover, the angular diameters (in radians) of the Sun aasevik /gjrv868-0 +qoaa x+mnd the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed1i )ehabj z3kgc ;6hi4 at the Moon, 380,000 km from Earth. The beam diverges at an angjk;b ze6gc1)ih3h4 iale $\theta$ (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.g(og*un26 +o4o ouuxw When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angula w4u2+( o* 6nuuooogxgr acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child. If thw1 7jb5kf+rsp e ball makes 15.0 revolutions, kr pwj7+fb5 s1what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutions do to/w vqd)g0lzdd e(-zwust/6 6he wheels makzvwo()/zge/-l uwd t0sd66 dq e?    $rev$

  (a) A grinding wheel 0.35 m i jghj 1t;,wobb6 *+5 jyju,*/vgh nqlyn diameter rotates at 2500 rpm. Calculate its angular velocij1ljhjgq/ ,ytbnyv ub+g **w56, jo;hty 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-renn)n7 o9j2 x 9bvyg5;kupvj* ound makes one complete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m frbex5 n27vj g jk9)p yv;nno9*uom the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocitete1/m6xm9d4o o,2gbpc/ *m4wfo mqr y of the Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a poigei( zqm8s4xx2jdj a8(c y;b (nt
(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 centrifd6oc- )ileb 3v9 .ozu0o)s exsuge rotate if a particle 7.0 cm from the axis of rotation is to experience ae))eob3.- sxsuzcv0oo d6 l9in acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about itws) 6hw,rv hfs;hc .qw/y4yz,m2 9e dns center from 130 rpm to 280 rpm in 4.0 s. Determine y nwqf/ ds h9he zhwwcv62r 4)y,,.s;m
(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 radiuz0 j rh(wfr-p9q;. r3txg /bvas $R_1$ 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, astroglfu2wje36 7extu b flm)hu+)x8 mw //nauts aboard the Apollo spacecraft put themselves 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 interval. The spacecraft cfg)uwemu u8+eflbxw36ht) j2x/l7 /man 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 fro.f9f,nb 9e hzvm rest to 15,000 rpm in 220 s. Through how many revolutions did it turn in this time?f9 .,fnb9 vezh    $rev$

  An automobile engine slows down from 4500 rpm zn6klu0zdze(9i m eu:* xn6t- 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 s xh1g/rus9xzv/*f.zkq1z5su; fj )sjets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete . ) r1jzs/ 9f;u*z x1uskvh/xfz5s sqgrevolutions 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 from 240 rpm to 360 rv(xm,6 ob yb0bpz k0d4pm in 6.5 s. How far will a point on the edge of the wheel have traveled in thidbbzymxok6 0( bp0,v 4s time?    m

  A cooling fan is turned off when fz0s1gee39oq it is running at 850rev/min It turns 1500 revolutions before it comes to9e 3zgfse 1qo0 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 mg6m y2202a + f,wjsicyht(vkmake 65 revolutions as the car reduces its speed uniformly from 95km/h ca6yg, w 2s mkyhm2t(v2i 0f+jto 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

  The tires of a car make 65 revolutions as the car reduces its speed uwi;q7 ; ba2cm85icnoze) gd 0ot* )gpnniformly from 95km/h to 45km/h The tires have a oaw z)td)2ni;8p goq5iegc;0n*7cbm 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 pet kt:*lbdv;3hdal when climbing a hill. The pedals rlb: d3kt*vth;otate in a circle of radius 17 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a door 74 cm wide. What is gpep0w )f j02cv2 dt+kthe magnitude of the torque ikpwt e02dgc 2j 0p)+vff 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 th2qr 3xq 4)jrip ymb+-ve axle of the wheel shown in Fig. 8–39. Assume that a frictiirv my3 qj p-r4qb+)2xon torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are sc8b*8l6w dn4c qv4 7:yo jt4tei(kfhattached to the ends of a massless rod which pivots as shown in Fig. 8–40. Iniswi 4j 6n(hdklbe t*qf8:cv4yo4t87ctially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine requz p r*vu5)3j x:lt(uzpire tightening to a torque of 38 ) trzu* (uvxp p3zj5:l$m \cdot N$ 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 inerti q)ky/; yjfc9ma of a 10.8-kg sphere of radius 0.648 m when the axis of rota jkmq9 f;/y)cytion is through its center.    $kg \cdot m^2$

  Calculate the moment of in*mc;q.bcd,jo v9) pi sertia of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of m,oids qp;.9)j vc*bcthe hub can be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball on the en55jp5b: d 8pqmn)rxk ed of a thin, light rod is rotated in a horizontal circle ofr e )xk8: pdmnj5b55qp 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 ::(tx2ht bge fat constant angular speed (Fig. 8–42). The frf (2x: tg:etbhiction 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 thefz b0fq;6 1hxb array of point objects shown in Fig. 8–43 a0bqz; hfxf1b6 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 w2g5 6b35uvoyv 1h7jdxu:uof8wcrl(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 spinnindrbs+:s0ykwewrt+cjqs 8b7v. ,8 u l/g at the correct rate, engineers fire four tangent8+0r usbycd:w bet7slq. rkvj s+8w,/ial 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 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform1;pmz nm8k6kr w 2lqn* cylinder with a radius of 8.50 cm and a mass of 0.580 kg. Calculate k p*q kmlzr n26mnw1;8
(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, acceleqqo 77cmkae(l66ihvsf e 40, 81laukg rating 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 small hand-driven merry-go-ro::gl* qsn njn9und 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 opplnj n:gns 9:*qosite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut offiiw( r+1iwn71sb j7uq dy5me+ and is eventually brought uniformly to rest by iqwb7dsi(17e+ny+1uwm5rija 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 acceler ,x;sw8oq tr6 6pgl:lyates a 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 tht60-d ;w*2cc kkgdr-y p-gqns rown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly dg2-c;0 dyqsncwkr -*p-k 6gtfrom 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 cana)yii 4n7a sp* be considered a long thin rod, as shown in Fig. 8–46*ai7np a)yis 4.
(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 o km2:t q b.tqb6j-n d)0mo*wiif 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 witheo/ju4ek 3.k hin four full turns (revolutionsj./3e eu okk4h) and releases it at a speed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has 4(t /rxvjkfo*iy3 *kl a moment 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 o+ yh1x5 iwbgbc :30hj9-syjdwf 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 k *1br31yp57apftdrv q g and radius 9.0 cm rolls without slipping down a lane aqp75vp a*r1 1t b3fydrt 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the g.civ/ rfu 5ij52ik0e Sun as the sum of two igif0u./ji2ekcr 55vterms,
(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 m. How mucgsvze/f/,-by v5tgp zpczl :y*u0h 44h net work is required to accelerate it from rest to a rotate* p/g,/: 4yp-bhz5fz0vt4 ycu vslgzion 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 k8.c vd82v2fx d)-d)obo vtfn from rest and rolls without slippd x8dt)v -fncv.fo )o8d2vb2king 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 balwry - 2i:xh/inanced vertically 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 ener x/wi-in r:h2ygy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the endxzspyo8zg3 9 9 ,9gmbf of a thin string in a circle of radius 1.10 m at an angular9g3omg8f xy , 9sz9pbz speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grindin*wrdcc) oxhgdl 0: i8r3m4l2ng wheel of radius 18 cm gdrc 20cw li4*h3:n )lmrodx8when rotating at 1500 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his side, ont e*xv)j(r:wl:) zdsw 6kbzo5 a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, thewto:r*s j)b :kzl x5z dwve6)( 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 mok;gjpmu flv;2n6c49 )pl gdt3ment 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; 4p9 kjfld;nv362mu gpc)glt 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 increasmbr9ey tx0,oo1b*1g ne her spin rotation rate from an initial rate of 1.0 rev every 2.0 s gbm*9 1bo0xn,ro1 ty eto 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 aroub 7dmt/ ec+. umy p6naz)hunf 5/+af9und 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, an n+/bh ue+t.fpfa5a6umm)uc/9d zy7 pproximately 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 mot4fz.4q:o6zl ratc f(mentum 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 of the7k yxvb/0q-sk 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 ism5 j(oya,0onx dropped onto an identical disk rotating at angular speedj(y50m ,xona o $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at8ynqk89ieb9n 4/ w dzg 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 stanm8fc o6fj ke1a7sj p18ds at the center of a rotating merry-go-round platform6fojjka8e sp181fc 7 m of radius 3.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating njqb.:5)blnoj 4m+x8fz h/ .cbx wr0u7fwi6ofreely with an angular velocity of 0.8h4 xj/i b c6lon8b:f+wmo.5b0 rqn).wu jzx7f $rad/s$ 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 eventucqju2:k- h7hx ally collapses into a white dwarf, losing about half its mass in the process, and winding up with a radius 1.0% of its7 xhk2h q:u-jc existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds wy2zqse 1qay-w,m:q- k9h 5oj4wxi0l 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 q9 kp4 j 6tvzjze2)8ts$ 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 withou0lt ggvqn2.i hs//p dvc3:m-h t friction. The moment of inertia of the person plus the plat s cthg:-l.vp/3 iv q0/h2mndgform 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 merryurn9 3qgj/ lz4gp9m -v-go-round turntableu n439- p/ggmzjv9r lq that is mounted on frictionless bearings and has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the3 42cr jm/iwxw ground with the end of the 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 the person without slipping. What length of rope unwinds fro3r jiwwc42mx /m the spool? How far does the spool’s center of mass move?

  The Moon orbits the Earth such that the same side alwa7hw2l))o1gr r c vcm9qys 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 casg)hv2) cw97rmq orc1l e, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate of 1 hiw f +i ukw4dyj0l2*ex.: c)tm/$s^2$ 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 i*wk i59ixi78xlef0 op)tcuja:xf 29of a uniform cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate the torqui5iitl29 xkj)af*x8 pf7ce90 w xuoi:e delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, eyw -x596wts6 l,fbcutach of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the enw,w9xscuby-tfl5 66 t d 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 angulaic.a:f4ry, kdr 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 masj(m eswek*,,bs 8.0 times as great, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational c(kb * ,m,eewjsollapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use energy stor al )ecq,ugfi k7uhug:tm*0t2 l7:v5xed 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 an7ltf0kig,qlmx:a uhg2*)u:e7c v t5 ud 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 illustrates7 ist -j9 ykhbc1s3(px an $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) ixskq6;mier.-r2.x ne2 fc 8snn.pj9o s rolling 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 6ou/1o 0ckfrj /fu),m /ochyur37d vg M and length L can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleration of tu fjy//7omg 1u/rh rod6)ku3 v0coc f,he 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 haq3ojrae lmje5/4 :fi0;pz j5a s radius R. It is standing vertically on the floor, and we want to exert a horizontal force F atj ;e0ma3f rj :4q/5ozaeipj 5l 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* ). 3qhgn6twfpb rg3i speed v=4.2m/s on a flat road is making a turn with a radius The forces acting on the cyclist and cycle art w3n6pgig3)rh *f.bq e 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-kktv j..1-1ecm 7ei amkg rock into a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a mek-7tiam.kcje.1 v 1 rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body am:0qg.*bjz4s2 1sgln0 otcd ss a solid cylinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calc0s2:lb1o z*tn.qsm dgc4g0js ulate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindrical pla)d9)sc mbp1zm.d 8 jtglm pq;;tes, of mabp m)dm;8ctq;m9jz.)d gspl 1ss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ 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,6pg 955p d*8qmn+8h /tpwi txuljkknk4ei. z 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 rt.6 i58wunk+g ej/d q mi 54pp 9klktph8xnz*,each 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 not aseah+ob.vlzelo 5rr/ -8(gn: vsume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car redm(,qc5+j4p +k9hvrxwcuh h7 b uces its speed uniformly from 90km/,54 p wjx+hqbc(hck9r hv m7u+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