A bicycle odometer (which measures distance traveled) is anzw;29n g ,cr rl:n8vitsht- (ttached near the wheel hub and is designed for 27-inch wheels. What happens if you use it on a bicycle with 24- irrhvs(ztwl 9:nnt,2 -ngc8;inch wheels?

Suppose a disk rotates at constant angular velocity. Does a point onr ) 4jko/r1-/pyrh hi-d 6uvbw 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-ruyh- p/rdrh1ji4) wk6 vbo/ linear acceleration change?

Could a nonrigid body be described by a single valuei:sne5h i8 v0xr;n bnrl ,xs+- of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater torque t7 a,d qh r.3gu lzap)dfh8 6o6/ct:nmghan 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 beh fpuv.dfd*/u (ind your head than when your arms are stretched out in front of dfvf/dp*u. ( uyou? A diagram may help you to answer this.

A 21-speed bicycle has seven s1 8uswk;gh6 (lw akk6aprockets 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 geaak1guk(6skl8;w6w ha r is it harder to pedal, a small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to r06 1i/jiv rx bkj3* ssvy/hte+un fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distributio k06 /3v1 vjbrj/st*siexi+yhn of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, narrow beag*fckn b m l0)9stl6n(9j1v b-i ku9iam?

If the net force on a system is zero, is the net torque arj+7u; tdrwh fy; 14ugr/uky,lso zero? If the net torque on a system is zero, is the net forc/ 4jurk; fy 1wgu;h,dru7rty+e zero?

Two inclines have the sixxgm 5x5f ,;mak1ww :ame height but make different angles with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed of the w 5 xw;xam:k, xmfg1i5ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rollikgpnb 2m:zu +o1 yzt/)ng (from rest) down an incline. One spheg):z 1u2tm/nz+bpkoy re 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 has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same radius and the same mass. They start from znrbe+q0n ,cj(8a*h v ero, ,qrest at the top of an incline. Which reaches the bottom first? Which has the greater speed at the bottom? Whic0( oqb8, e ,,j*z qn+hrcrnvaeh has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum e(9 8 , amvblzw8.o; 80madiwhxn(tnm are conserved. Yet most moving or rotating objects eventually slow down an wmz8nt m;08xh8adn,vlowe(9.aib(md stop. Explain.

If there were a great migration of peop0(-e;z h *j j4wf+jg lpv0hyemle toward the Earth’s equator, how would this affect the lengew4h-jmg zjy*+j( phv le;0 0fth of the day?

Can the diver of Fig. 8–29 do a somersault without ,k3tev70yw7vik 2y lkhaving any initial rotation when shivy0l 7e k7w3tky,kv2e leaves the board?

The moment of inertia of a rota45femmdses fa 6 4mj:7ting solid disk about an axis through its center of mass f4 :s7m4 sfj65eam mdeis $\frac{1}{2}WR^2$ (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 kr;gu xhd7sp(/rze/1cyl.o g44k; aui ze0 p8stool holding a 2-kg mass in each outstretched(cs xpiokp z4 u4a;0ergedl;.7ur z y8//kg1h hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical 8sy w 7b-5vifdq,rqi 6and have the same mass. However, one is hollow and the other is solid. Describe an experiment to determine which is which. y8bwv r7 d-qi,if56sq

The angular velocity of a wheel rotating onaf 2j -mg a3u1q9wiuk j3,bj5d a horizontal axle points west. In what direction is thejqmj1aw -b ,3gf iu2dj5ua k39 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 or decreasing?

Suppose you are standing on the edge/pw9cw*xt eg 5 of a large freely rotating turntable. What happens if you wal /wx 5wcg*p9etk toward the center?

A shortstop may leap into the air to catch a ball h.oznc i3m3leej24 2n 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 om2.zen ln3 ieo2j4ch 3pposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discuss why s er*f tju64mt 88uns(1r*bjza helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter stabl **tteu8 s u48 jrjfmn(s1rzb6e.

  Express the following; txh32yh/cbpy/i2 ee, foqb:, zgnm 1 angles 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 m* /xpyc cmdu coe,+k94s,7g hof an amazing coincidence. Calculate, using the information inside the Front Cover, the ayx/7p suh e,d9mc cocg *,mk4+ngular 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.45dse ;uwp o,ajpo3 w; The beam diverges at pspe , ow5;ad;uj4wo3 an 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 6500 rpm.r( rp m8a1ebp dgrc1q8(c9 l 4p9;ykev When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is ;cp(kmp8gv189er 1cyr( rd4 aep b9 qlthe angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a6pztj5-jqi 9 u level floor 3.5 m to another child. If the ball makes 15.0 revolutijjutz65- qi9 pons, what is its diameter?    m

  A bicycle with tires 683jn q88-kyb3tsq wrp+ cm in diameter travels 8.0 km. How many revolutions do the ww8prt bj+3sykq qn 8-3heels make?    $rev$

  (a) A grinding wheel 0.35 m in diu2 cr2e*2grecameter rotates at 2500 rpm. Calculate its angular velocic eug2re2r*c2 ty 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 comp0tc7 )3oopdb:c 2na c:j45xs s e*aonglete revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m frc2jnooap ax5 )0sedoc 3s cg7*:n:t b4om 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 k8fmwp tb 48yi vet*nf4 lr1;8the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a poln .ry98r3 s, wwc:lzcint
(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 axe so z4d6 )(98wc / xprx/buvpp0m, bmecf*h4nis of rotation is to experience an accel6m(dz )v4fnuph4m* x r8c/ocbp/0 9ps xew be,eration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpm to 2nwgak)ck( uq;iqsg 4799zi/q4n ;zh +nmuj g380 rpm in4a+kqg/ ms9(g3 nkh c9;j7nu 4uii)n;qzwqzg 4.0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radius f m ,6 ze5op7ilzrqbyl9bc zg4699ge4$R_1$ 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, astronaf9zbdc7co8. 2hth*l quts aboard the Apollo spacecraft put themselves into 9q 8c.2 7cth*d fohzlba 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 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 fq:5d 6lxdmo(xuj2vl/ rom rest to 15,000 rpm in 220 s. Through how man :5u l(oqvxjml2d d/6xy revolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calcula.sb-f es 8 mgv7t:/nlute
(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 fl( gw2 kdc0r*w;p2c wblying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final speeb 2g*dc(;lkwpr0 2w cwd.
(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 9g/.m -rrrt morpm in 6.5 s. How far will a point on the edgor/grm-mr9.t e of the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turn 6 )1 tomv5ihaad/e:ffs 1500 revolutions before it comes to oamh 1fv6/tad:e 5f) ia stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions as the car reduces i /nm 5q,ft7upmn0hax8p0m; z9ggvad*ts speed uniformly from 95km/h to 45k 0nt9p ;m p7/nd 80zf,aqg xm5mva*ughm/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 red 0 /k4*u)xg+ u6,knd.rh.6fzpwilgq aufnh g+uces its speed uniformly from 95km/h to 45km/h The tir.iz.+pd rl4 hnhgf xkwfn +6g/k,a qu0guu6)* es 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 eab fnlgl .y8a*(ch pedal when climbing a hill. The pedals rotate in a circle of l y(lng8b. *faradius 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 / akz)r yurkt,te1+ j5im3a ,uthe end of a door 74 cm wide. What is the magnitude of the torque u3,) i/et 5zmrjtkra1 auy+k,if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fig. 8–39.6tmfsuc4/5r:2dra l *oninp4 h 2r ,qt Assume that a friction torque of 0ao: dtfur, 5i 2 nnchrq44/s ptr*62ml.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are att pwi9p:pej +2bardf3 p 7l95fqached to the ends of a massless rod which pivots as shown in Fig. 8–40. Initially the rod is held in the horizontal position2 a9+brjd3pqpep:ff5 7 pw 9il and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an engine require tightening to a torque of 3d.vx6 1db2 ebk8 kdxb 2.1b edv6$m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a 10.8-kg sphere of radius 0.648 m when the *yskwy;94bfi+d o 1vb hmq6h 7axis *4 qmhb;yo d1v7wb+f9y 6hi ksof rotation is through its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm in diameter. T.3zdy9l:.of puocl: mhe rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignored (why?).pu3cm :dzl.y 9o:ofl.    $kg \cdot m^2$

  A small 650-gram ball on the *q vfr3d ttr/6 3u*yplend of a thin, light rod is rotated in a horizontal ci rpf3yd3/qu* tt*vrl6rcle 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 speeicl )pb9t*mo:c( c)l bd (Fig. 8–42)ob* cl)b (tc9p:) lcmi. 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 objg us9r cvd5fx52ff28 fects shown in Fig. 8–43 abofsxdr v252 f8fuc95fgut
(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 consistsphb(8ou z :b(i 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 corrtnp,4i iu: py3ect rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radin up3:4,itpiyus 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 k*nq7u0mp9*ca)*alerrt - i xw w41ciuniform cylinder with a radius of 8.50 cm and a mass of 0.580 kg. Cacq*a w p*wu4an)i7 tr9 ckre1i-* xlm0lculate
(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, f x,0zoxt:;5p 1 tfpxwaccelerating 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-gob5da8 5)lh4c bqj( w ryz5v.ru-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 ra (zj)qrbb5h54lw5du 8 yv c.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 rotatinbppt6du ;z /*or8gt,qg at 10,300 rpm is shut off and is eventually brought uniformly to u;p p 6ttb/oqr, 8zd*grest 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 b 7z;s.3x wuvv q8joun5all 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 actioney8n0xmt cr x1i.14,jil r ,ku2ir vp. of the forearm, which rotates about the elbow joi.r,2xuipkl ,ice1y0tr .vj4imrx1n 8 nt under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin kaaso.+to,uv f+0: nqu8iy 7i rod, as shown in Fig. 8–4aka ui+7:0,uvyo o8.+itnqfs6.
(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 consistsewio pqnm,5g xk0u1 ),ue+2bj 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 accelerate9cw8; gair 8vas the hammer from rest within four full turns (revolutions) and releases it at a spe 8agc ;w9i8rvaed of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a moment of inertia ohn1duy-; b :bjc pwqw,s /b2)lf $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develomody +je1 zok0y: 6fm:ps a torque of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 zb1y:u4 izdo t- ff2j+cm rolls without slipping down a lanz 1u4-jtyf2do:+b f ize at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with respect to the Sun as th6i-iig .rbde-w4;mef / 1twyge sum of two terw6b r-w1igd ;tiy4gei. mef/- ms,
(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 much net w4 nupj7 9hzgxg;0i:pwork is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume itjgp4pwn09 iu ;7:zhgx is a solid cylinder.    J

  A sphere of radius 20.0 cm and mas bsizilx:x163f o0r0b s 1.80 kg starts from rest and rolls without slip:3lxbr0i10f xoz6bs i ping 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 and its ,q ijju21ps ,rlower end does not slip. What will be the speed of the upper end of the pole just before it hits the ground? [Hiru,1 jq ip2s,jnt: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 3dxp5h y1tpolix 8t520.210-kg ball rotating on the end of a thin string in a circle o 123dh o855 xpytlitxpf radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum o6j sp/624jwfo lchk:of a 2.8-kg uniform cylindrical grinding wheel of radius 18 cm when rojoo 6c:jp4 w/khl26 fstating 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 rotatg4-28v m vdr3jh)l6ssucjrw f *zt4.ging at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fig. 8–48, the sm)t4g s*6 -2gj8z4ldvh.3cvjwr fru 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 rpq+ b *k:h.vahb/ hmb2rwjq-/n x t,wa):egb5educe her moment of inertia by a factor of about 3.5 whena,:/ kb xq5pn).hrwh/:w 2beb+q gbajtmh-*v changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate xfk(-tg+ kp-h do7rb mb;ohv 063d5qy from an initial rate of 1.0 rev everymrqd+5 xd 36; -khfb(0ghb k7 tpoyov- 2.0 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 around a vertical axis through its center at a freq ob; y-rf8zxn)uency 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 fo8 rz)f;b x-ynlat 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 figulvabj.zu)oc1 g (lc83re 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 mr/ t 1e+zn +ufwxty:z.f2f b57ejbqp9omentum 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 uvnl2 nu/bx 1q**od:n,b an7c moment of inertia I is dropped onto an identical diuvnl, b2nndbo* an:7u1cx /q*sk rotating at angular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4 ye5g9j3u.iaq $rev/s$ 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 merrb z1;c;j.j(psgo acc+ s6w7i0*skj hqy-go-round platform of radius 3.01 k.zoa *;jcb0s+pihqjscc6 s7wjg; ( 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-rouns9tk5 iec k tdh04r1ny 9mvt0:d is rotating freely with an angular velocity of 0.t5yk 9e0 td sivtmh:9cr 14k0n8 $rad/s$ 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 intot, v svl l21bcfiko1,1 a white 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 Sun’s new rotat1vfckosvl1l21b ,,i tion 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 exc w r7 nxar,it:unr- - x+dreec2ke0.;qess 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 *r6zry; nuvo8 l4mo:i $ 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 freem q1x-gki)4hbdxh 0*cly without friction. The moment of inertia of the p*ghbdiq-) 1km4x chx 0erson 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 )*1ynjkzzae-+s m se8 diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia of z)j*- syek18a+ezmns 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 x q,ww*a cryt77t5vt2with 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 ro7r2v *tx5wq 7tta, wyclls 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 alwaysj1s x++vjg ja8 faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its ow+ 8jva jx+sgj1n axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest at a rate w01lkv6vut .a b+q t . y1v2ovg3b,hgkof 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 of2ux - zi(b/n9he/smu0fs3pzm radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calculate ezu ms-h/zu0(ip2bf/m s3n 9x the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylinm7ntk q r; b5b3kmtkf07 db9-ddrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long t7dbkb5b nqmf3;9rm -dk07kt 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 wheefa3gj m.s0b7kznsn*ddu8. t( l ($\omega_R$) 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, wrvkznmb.xa) g1w 5q3: qz7m, )lh/ol were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 km, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at alm,wq1l5l bwnh7z)x/ zk3rm : )qo.gavl 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 storu;teu8(2 h 7 t*sjzuuved in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter u82zvujt7e( * u; uhts1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates an bk 7+6n oxhl2r$H_2O$ 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 oncf.e6 vk*bzo1 a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., we ignore frictsgz ;cj e0a 7huaw.1h2uwu y20ion) 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 the tip of the rod. Assume that the force of gravity acts at the center of mass of the rj 71w0 cegs.0 2auah;u2zuywhod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radius R. It is standisyjx9 i7i(1ggng vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests ygii17s x(gj9 (Fig. 8–55). The step has height h, where h

  A bicyclist traveling 0jtiq7++fv3 vz (or m2fyg7vbwith speed v=4.2m/s on a flat road is making a turn with av7tr v+7zq mg b2f+fo0jvy(i3 radius The forces acting on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of lengthfzm)uixec1g 6 w;5o 3d 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 thieumw)z3 5fdgc 1 ox6i;s feat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her ari7ix p)noo8; dms as thin rods, making re i7)ix8d npo;oasonable 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.   

  You are designing a clutch assembly which consists of tw*l+z t,s 4gedv 5bmytt(/ pwk)o cylindrical plates, of mas*s(v t5 e mk+4pwl)t/ ,tgzdbys $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and r.c9 2dthrt plohb y0 sy18/g2;hszkv :adius r rolls along the looped rough track of Fig. 8–58. What is the minimum value of the vertical height h that pc 0 ty1ys2gdtb9;/sk: hz.h2o8vrlhthe 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 no7vk8f z7x ns5pt assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as thee0oj +syrr:9o 7y -l3kf9 aapi car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 9 esala9oi0fop j:kry r7 +y-3m. (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