A bicycle odometer (which measures distance travezrq ;ml,e8 2htled) is attached near the wheel hub and is designed for 27-inch wheels. What happe;,e 8lqtm rzh2ns if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotatkfjyw75 cdf 8+es at constant angular velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which yjw8ckd+5 7ffcases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single ; ++hqnqzn+(1f spunavalue of the angular velocity $\omega$ Explain.

Can a small force ever ex hl1crn.r4h4u ert a greater torque than a larger force? Explain.

If a force $\vec{F}$ acts on an object such that its lever arm is zero, does it have any effect on the object’s motion? Explain.

Why is it more diffic)hmais9bp4)l 7 zqvy .ult to do a sit-up with your hands behind your head than when your arms are stretched out in front of you? A diag7i.sy a)l)vpz9 4bmhq ram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and thjf9p s:spz gxs6w/(w)dp:1 eadyg 5q-ree 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 psa/ se j1gw6g )9dfy:pqx z5w(s:pd -small front sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower legs with x(axm5uz im/ 9rwn 2cf5ny4jev y3-m :flesh and muscle concentrated high, close to the body (Fig. 8–34).cz(n 4x5-m3:9u/ vnfj2yix eywa5m rm On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35),elu. 7cpyeci*p n8gg -z,8jbz6h z 9n carry a long, narrow beam?

If the net force on a system is zero, is the net torque also zer-ss (ih x1w5czo? If the net torque on a system is zero, h(sz51x- ws icis the net force zero?

Two inclines have the same t2-wpnnwou av.(.w z*height but make different angles with the horizontal. The same steel ball is rolled down-w * w.(zo vup.awn2tn each incline. On which incline will the speed of the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rolling (from rest) down an inpv00szypx gjq7s i-k(9 xc331 a8 emhscline. One sphere has twice the radius and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greate-z 93j(ys0cmsx pv 1ah xsk8i3qp7g0er speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder h4e +mc5m70w4nst5ctb /h ifprq0zl/b m3k5djave the same radius 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 cb5st l5ref imqw/0dm4 4 7 0jbhcz +3/npkt5mthe bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular momentum are conserved. Yet most moving q uf+.uwc0 2hvor rotating o+qvc h0.u2 fuwbjects eventually slow down and stop. Explain.

If there were a great migration of people toward the Earth’se(tb5b ym g6y/qaftn3 (z 89uz-izc-l equator, how would this affect the tcyzi a56 fnb-ztbze8l(y9gq/ (um 3 -length of the day?

Can the diver of Fig. 8–29 do a somersault without having a rdcj, ek me2k4difto a8 0s6qrtr524*ny initial rotation when she le* tm0rejf 8t22rs4, d5q6d4 arek kociaves the board?

The moment of inertia of a rotating solid disk about an axis through 7d q/8pz/ q 8vfrz0mht9c0 fbrits center of mass i zht8rq c8rd/f9m f700/pvqzb s $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2-kg mass in eac 7h8zhus3fp4v h outstretched hand. If you suddenly drop sh8vp 7h 4u3zfthe masses, will your angular velocity increase, decrease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, one is hollow6ed ,c nm3o-5ncodp;e and the other is solid. Describe an experiment to determine which is whicm6o-ndc d3,o; enep5ch.

The angular velocity 3lm.hls dyr2;xp p6e1of a wheel rotating on a horizontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular accelera;3 sl6rd lx.2mphy1p etion 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 o0zy znnmz972x ;uj; w;hplj;uf a large freely rotating turntable. What happens if you 0 z7u ; njz2upz lxh;wj;9nm;ywalk toward the center?

A shortstop may leap into the air to cs8w 9d xwbg6z)nt f x(xj1o54hatch a ball and throw it quickly. As he throws the ball, the upperbsn8tf4gjz5dow h) 6xwx(19x 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 com:3y1jvv of,kweq,p 4nservation of angular momentum, discuss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operof jwkev 41p:q m3,vy,ate to keep the helicopter stable.

  Express the following anv z./1mtts9 wogles in radians: (a) 30 $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coinzve 6e0wkzzs 3-tt 6g:cidence. Calculate, using the information inside the Front Cover, the angular d zkt- g66ze wts0:3vzeiameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at the M i7;/ldmyg a;noon, 380,000 km from Earth. The beam diverges at adnmg y;;i7/la n 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 blenderlx ydd/8yrx 9f8*-qkqc9batrm9 z 82e rotate at a rate of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. What is the angacy*rfq8ybdtk q/ xlm98 d98 r ezx-92ular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a lev-n9gm.(ycfe vuc7z +z el floor 3.5 m to another child. If the ball makes 15.0 reg7z. ycf-c nue9( +vmzvolutions, what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many revolutions+dr 7w97b 8 f4vzdf0;bhktfc+ z9r ezx do the 7x;rv9++rz9 4zd 8heffftkw7z0dbb cwheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate its a0fddx6g(a.n6vayox h0rj 3n - ngular xjho (n n.adr-vy6fa3dx 0 6g0velocity in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makes one complete revolution in 4.0 s (Fig. 8–38). *kck .vbn 0ih)tz7/pi(a) What is the linear speed of a child seated 1.2 m from the centeb0n pcivti) 7k /*hkz.r?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angular velocity of the Earth (a) kxk7 tgx 0e84hin its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a pointw+ .p o am1f0q)ao 5i,ewmxa;q
(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 pueuf2nzbq,p h7v 17izgp6)q 3article 7.0 cm from the axis of rotation is toi62g q1, )3bp nf7qzpuhuvez7 experience an acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniffujkzrax ;q s p7z81p+51y q5mormly about its center from 130 rpm to 28 5zkqjr;q1y 5psazupm x8f71+0 rpm in 4.0 s. Determine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radius h-o1kh 4q8:*e;0 ob0ol ccvv*saccqn $R_1$ 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, astronautsg.nv,anyu eqe z1. e)5 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 can be thzv.5u.ge1n e), n aeyqought 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 fi. daetb7.1f krom rest to 15,000 rpm in 220 s. Through how ma .te.7iabfd1 kny revolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 12yr yj s ha:u.d )cy1badzwq.h5( 9v*;q00 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 ofb *k;8yiscs vgd*0:.t aqbe1m flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 cy*; v8b*mskqc.a:s id b 10gteomplete 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 diamet+wkw dxq 0b)lqd yim)02/+l 1 z0vlctder accelerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheed1xl++b lqwd k d0tmw v2)zl)yc0qi/0l have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min Ix v5wj+9gu6he7g s1u, pc,nmst turns 1500 revolutions before it comes to a s m9g,6h1ep gun wcsjsuv5x+ 7,top.
(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 revolb/vt82niuau g g 0cztkkq p8u,z1p+;(f*o+mhutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires(*1;uvg8f,0o/8 tu azh2+ ptq+ku bkig pnz mc 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 revolutbn rcps;a)4m2 ions as the car reduces its speed uniformly from 95km/h cb sa )npr;42mto 45km/h The tires have a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight on each pedal whekjk y16-d;po vn climbing a hill. The pedals rotate in a circl -okk6vyj;1p de 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 th 49ab*tl)o)o-wwxy*i c23 wdz.ovrul e magnitude of the torque if the fo) 3 *u 9x odiorb.o-ywt4w)v2zwall c*rce 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 tki53u(x z0ycw he axle of the wheel shown in Fig. 8–39. Assume that a friction torqcxki u35(0z wyue 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 of a massless rod whichv4* /jzbk g0,afmt+urip5m8ygiai6 9 pivots as shtyag5ikbf /0p,m4um r8v9zj 6a i +g*iown in Fig. 8–40. Initially 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 require tightening to a togvrz )*ad)ft19u+ y +ws9agf zrque of 38 fgzf+9adutvw +y) gas* 9rz 1)$m \cdot N$ 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 inerdsk k,2 m7fx*xtia of a 10.8-kg sphere of radius 0.648 m when the axis of rotation ikk7fx *2xm ds,s through its center.    $kg \cdot m^2$

  Calculate the moment of iner llta7dp5xrs.n+1 z5 qhm(b7 +hnf(g ltia of a bicycle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignored 5nhnlm1. qa5 +glp (x zhs7(frdtb7l+ (why?).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotated in 9;u popbol, b1;c)lgqa horizontal circle of radius 1.2 mq;p1,; lpoog cb ub)l9. 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 rotatingl-ule lo3-;83d6a v-edx jc5y6: fzapv+m rzo at constant angular speed (Fig. 8z+a6-5o pfz: 6m3dlax -d3 l;r eou8ly evv-cj–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 objec2u pse05pd-:o2m e )ieq2tguxts shown in Fig. 8–43 e2g52t 2e xspuqu)p:dm0o-eiabout
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen atoms whose tod14(ea-5 dh1fz ixohxtal 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 ca78l3yttgjt9 n3dskgs-5gmfls 4 5a,orrect rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has k tm59ggls7 4 j-tasy 5fg8l 3,td3asna 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 unifor 2 v:-ibihalq0m cylinder with a radius of 8.50 cm and a mass of 0.580 kg. Calculaibq :i-lha v20te
(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, a9 mjdnhp*w4i yt-0 *:guei sy9ez .dd+ccelerating 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 p )chpjk ou 7k x u/w+/f +f7isr;gyhq+1)r)opsmall 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 gppi+u/)ry+s;o fpo7u1w/ kkqr x))fj h h7+c of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,30g .rz2x8har(ajqd) f 70p eg;i0 rpm is shut off and is eventually brought uniformly to rest by a frictionarfdag80 xe a2j hip.r7 g)z;q(l 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-.) 9s. pvxjbszh n ijo9:8u2xmkg 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-kgr lpop e76y359zroy6t ball is thrown 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 unifor or5z9tly7p36p 6roeymly 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 rod, as shown in Fil73(gwd 2sgh7gsuhqp *hg62u g. 8–46.qh7sl w dgh3 g*(2g7pugu26 sh
(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 consisjn, .2bxwam14nj31c mj6 voszfh3i,l ts of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer from rest within four full turns (revovqp 4b0h5xsw s :z6ir(lutions) and releases it at a speedz :50 sh(rxbi6wsp4qv 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 mome ljhw f/rddluc30*- *dnt 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 torq f6(oxm3yw.i . pbnpx)po6atf; m vb5)ue of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mass 7.3 kg and radius 9.0 cm rolls without sv ;fj r7mmy:gate: q6m86,xx qlipping down a lane af t76mvg r:mj8x;aqq y6 e:x,mt 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth weo,x*r j30ua *y4zht 1uoaq rexd h).glf-b,5ith respect to the Sun as the sum of ty,z* a4 )g*.eh0ux-bj 1oeqld5 rar,xth f3uowo terms,
(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 agb2enggkx,5 y/lia 20nd a radius of 7.50 m. How much net work is required to accelerate it from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cyl,5i2gag gy/2enbxk l0inder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rolls withou9/u k am9epo)kt slipping ) a9ku/9omekp 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 il.yom( nu;q w)ts lowel.) ow (um;qynr 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 energy.]    $m/s$

  What is the angular momenzjy0bcls; l*93a2w/k mlf1p ntum of a 0.210-kg ball rotating on the end of a thin string in a circle of*3 pz bmys lnkaw1;j9llc 2f0/ radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylin 9,2rut)ry phzdrical grinding wheel of radius 18 cm when rotating at 1500 rpm? h 2uypz, )9rrt   $kg \cdot m^2$
(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 ,uar-n4u dgr2rate of 1.3rev/s If he raises his arms to a horizon,rdur4 g -a2untal position, 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 b( h4,sn,ph.asflde9 of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s when in the tuck position, what is her asfdn,9 a (.b ehsp,hl4ngular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increi: t)0it 2dlfhh 4zx0- ppv+y3 8yvywiase her spin rotation rate from an initial rate of 1.0 rev evert+ f0xyv048i2pt )i 3 dyzhply w-hi:vy 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 centeta pkm cp,69 8wjnupu ),-fv4fr 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 8c 4,afpmf)jp tvpu9kwu6,- na flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spo ny.ysvjd *8k4rbe 29inning 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 ms+w;l 0 bna5jux-kv)yomentum of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is dropped 73f- uildxuutsh0 -cgwaw h7) 994wn gonto an identical disk rotating at 4)i3sw7ul9 cuu-xtg09dn gh - w a7fhwangular speed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns atopb ,ee4 *e : bbj :+lsdva5v12n6cgyh 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-rou hlckzd.jf(* 1nd platform of radius 3.0 m and moment of incf 1hld*zk.(j ertia 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 freely with alk :riyoy c:y7k) p7s5n angular velocity of 0.8y iso:yyl5p7)7 c:kkr $rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventually collapses into a white dwan9wrin*m 1; tjq;hn wn0+f mt,rf, losing about half its mass in the process, and winding up with a radius 1.0% of its exinnm9 ,m;r*th;w0t1qin w fnj+sting 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 in excess of 120r+gmpck w9ccm vt-9lgjm a+ms69ss ae8(1e84 $km/h$ 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 64 z(y, t6 -1e drebkhr9d hzv57rndwt$ 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 witk8gdd ky0;r-e : 3dxcn v5efvshg4:p2hout friction. The moment of inertia of the person dyrf v :hvxskk3 dn-8 g:e420c5;pedg 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.rz 2cyd 9sf dme:xh hqv:6(73p5-m diameter merry-go-round turntable that is mounted on frictionless bearings and haeh6: 3v (72rmc xddyfqz: 9shps 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 lbk;q j; j/ 8:xhyyoz*on the 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 from the spool? How far does the spool’s center of mayjq ; olj:y8xk*z; /hbss move?

  The Moon orbits the Earth such that the same side always faces the Earth. Detool kpp5q0m+))5ws s,hpg0mn ermine the ratio of the Moon’s spin a)+n g0sms0, poom5 ql 5hw)pkpngular 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 from rest at a rate of 1v,q+v jdx9h j:5(mhap 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 uniformnn 17 0+dio,p1tcakps cylinder of radius 0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angular acceleration. Calcuk+n o1np a7, dc1p0itslate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 kg andr8rbw2v as6 p5 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 ev28rwrb5ap s6 nd 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 b lo-4j j24asxx o., ycdc3aq6wheel ($\omega_R$) 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 size8:95xdmd7gg dql c vz9 of our Sun, but with mass 8.0 times as great, 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 sphev q m9ddz8gcl7g:5d9x re at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution rtd 2ll: 5;fumij7e7fautomobile is for it to use energy stored in a heavy rotating flywheel. Suppose such a car has a total ulji2fme:f75 tr;d7 l mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates ;th8rrlmb2:1-nff o49m0 keonr x x-o 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)oww9 +8grfvv) is 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 M and length L can pivot freely (i.e., we ignore fricti 3ay,vuxyf 79don) 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 y9vx3a 7,uyf dforce 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 th e(6 +s2ig6nsocc3wsme floor, and we want to exert a horizontal force F at its axle so that it will climb a step against which it rests (Fig. 8–55). The step has(+ cs26nswo 6g smi3ec height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a c*l:z2/ r6sje-oi rv; y)(3knsx dy ulturn with a radius The forces sy;ven/3(izjl :r losuykx )* c6- 2dracting 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 okjizn zo 5iab-/0i 67bf length 1.5 m and begins whirling the rock in a n0 oz5/6ia-i jzkb7ib 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 from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder and her arms as th-to45fdz0b i rin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched f-oidr 54btz0arms, and with arms held tightly against her body.   

  You are designing a clutchvgy m 8y*k+ms0 assembly which consists of two cylindrical plates, of mas+sy*mv8ky0mg s $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls al/w5y q/.guopqong the looped rough track of Fig. 8–58. What is the minimum value of the q p5/g/.wouqy vertical height h that the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, bl ,/qfxs m7 f4mx*:ybgut do not assume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car re/:* fxhao4 sn3dcqm 1up4lk3 mduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular acceleration of each tir43o hm fpq: 314dlma*c/ uksnxe? $\approx$    $rad/s^2$(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