A bicycle odometer (which measurevk2 i9cnli)s aad *q .*mw0.nz-e;mecs distance traveled) is attached near the wheel hub acc*me*k.i)nased.;za m w lniv02q- 9nd is designed for 27-inch wheels. What happens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angular velocity. z v,5 es19-yq0aav-y 8 fspmziDoes a point on the rim have ra,z iseq1aap0v yy- 5mfz8- 9vsdial and/or tangential acceleration? If the disk’s angular velocity increases uniformly, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be describat/;lm qxi3) +sbr9wued by a single value of the angular velocity $\omega$ Explain.

Can a small force ever exert a greater+r 3t.c7ujb e+ 4mvnao 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 difficult to do a sit-up with your hands behiqr,p/j70 jpuctoh . 7ind your head than when your arms are stretched7po jict.hqp7 ruj/0, out in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprocketsc5kw9 sku9r h* ou8dube3bc ;* at the rear wheel and three at the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small fr3owh 8c9e*ks9 kr; bb uucu5*dont sprocket or a large front sprocket? Why?

Mammals that depend on being able to run fast have slender lower legs wo+0ulmtd w*ls 62c:r with flesh w c2*wrlmd+lo0s:t6 uand muscle concentrated high, close to the body (Fig. 8–34). On the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers (Fig. 8–35) carry a long, nah -i8trt2ay j4 xxvm-6rrow beam?

If the net force on a system is zero, is qdaq3ps;ynho *ygo-8 n ,ha2. the net torque also zero? If the net torque on a system is zero, is the net force zero? y8s.nao h3,;haq-gnp2*d q oy

Two inclines have the same height but make different angles with the horizontal.zz;ve 1 5ssmq)u 5,ihz The same steel ball is rolled down each incline. On which incline will the speed of the ball at tmz ishq)ez,z u5;5s 1vhe bottom be greater? Explain.

Two solid spheres simultaneously start rolling (fmsr(qb;02 0futeqgb5 tj;tn)g4u z3l rom rest) down an incline. One sphere has twice the radius and twice the mass of the othez3)btf4 ; jm 0;qr ut5nq20( sgbulgter. 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 thg scle(2 04/6iqnnw 9xxaec7ge same radius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Wigw2cnx/9xles 7( 6n0aec4gqhich has the greater speed at the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and c537v5j w1bnw5kf jb o* kkc*sangular momentum are conserved. Yet most moving or rotating objects eventually slow down and sto 5wk3*f7j*ckonb 1 ks5v5 bwcjp. Explain.

If there were a great migration of people toward the Earth’s equaw)8. k:2cjkj cqeq rltpsz8*)tor, how would this affect the lj c .wl8pcket2qskq8* )zj r:)ength of the day?

Can the diver of Fig. 8–29 do a somersault witho;oe+kek nhn,7h :*bbq ut having any initial rotation when she ,ekh*kb h;n 7n e:bqo+leaves the board?

The moment of inertia of a rotating solid disk about an a )k 9hc iiq8.86q+ gztmu /sdn:/sbwjjxis through its center of kiqz98jn/6t)i8 q.jg cu m+b:/w sdshmass is $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool x ec1il5ma5b1a jeetcb-9(q2j*ys;y holding a 2-kg mass in each outstretched hand. If you suddenly drop the masses, will your angular velocity increase, deccjm95jbx15(; a ty2- yqiaels*eb1c erease, or stay the same? Explain.

Two spheres look identical and have the same mass. However, on (5b-uo2jqqt mnh-ier0 r/*mxe is hollow and the other is solid. Describe an experiment to determine whico /2u* 5e xnr(bqtmm0h-jqri-h is which.

The angular velocity of a wheel rotating on a horizontal axle points weszxb6mo6 b887rkcma up421i jbt. In what direction is the linear velocity of a poi7k6bz 1mj 42amxo 86rbc8ipb unt 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 of (au j.7v2qbxya large freely rotating turntable. What happensuyajvq.2bx 7( if you walk toward the center?

A shortstop may leap into therap a jpc4ml18fxg3k7 7 x.k7y air to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you 3a7k.7p1x7ya8ckrpj g4mx lf look quickly you will notice that his hips and legs rotate in the opposite direction (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discussrg94 5, b k,ggudlie2gr)-crg why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopt-),g 2u kr459g,rliggdcb greer stable.

  Express the followingt;q )ri h(wv(g 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 of an amaziny, rndt/ j*995 kihfvgg coincidence. Calculate, using the informatiodf9j,95k/ ngt*vi ryh n inside the Front Cover, the angular 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 frk; x1sgm/+tga / d5vlb9ozpw ,om Earth. The beam diverges at an s1gtlm9 ;,vk/da w/ x ozbp5g+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. We8g,rhgp y4rcq:gcir.vq mx; k16 x ;-hen the motor is turned off q1h4r: ;kcv gqeigxym,6-;r.px crg 8during operation, the blades slow to rest in 3.0 s. What is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a level floor 3.5 m to another child.te78d vpvg-z673g fvzl0 z8wo If the ball makes 15.0 revolutions, what is08vz8l7wtvg zpz6-fe 7vo 3dg its diameter?    m

  A bicycle with tires 68 cm in diamy8i ,29x7etb7:ewxv3yk b yraowr r4)eter travels 8.0 km. How many revolutions do the wheels maktywb x4)o7r7kx yr2r:a9 ,y8be 3ve wie?    $rev$

  (a) A grinding wheel 0.35 m in diameter r8o ax7jwpgxd km-.y,bj:+m02t hb6 k fotates at 2500 rpm. Calculate its angular j8bw-7 jdy2 ah+bf kp0mot .mx,x6 :kgvelocity 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 completeapkm(ydno.i5zx* )y . :) yrnl revolution in 4.0 s (Fig. 8–38). (a) What is the linear speed of a child seated 1.2 m from the center? yoim)dp ayz(x. *nlnr 5:).yk   $m/s$
(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 the xcru81:4q iaiyy 2u0g Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed ofd d0+daxv w aik 7r8 glie4d5)d8nq5t) a point
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.0 cm from /rt5 i pp;jk zktmu /1z,v khgq,yu11*the axis of rotation is to experience ytr ijzu k;,mkzq hgtp/*1 k1v/ 15,puan acceleration of 100,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about its center from 130 rpm to r+ik )6quw0lydfx b.5280 rp06kb5.+ifl udr)q xywm 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 radiu-t05 7;fjntsxf )h biks $R_1$ 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, astronauts7r ,6shz pt4ke ldh4n0 aboard the Apollo spacecraft put themselves into a slow4l6e zhd4t rk0 s,h7pn 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 fr6of v09x u/ws fjp60ozom rest to 15,000 rpm in 220 s. Through how many revolutions did it ts9vp0jz/6wo o 06 fufxurn in this time?    $rev$

  An automobile engine slows down f;13lw*aomza k rom 4500 rpm 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 hightflo:m8.g ./7n ly uhxspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutiong.u.l/m lo8 7 yt:fnxhs 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 uniform d34 j:-gryqffly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel hav d-3fy4 jqgrf:e traveled in this time?    m

  A cooling fan is turned off when it is ruot2u0epm8a , znning at 850rev/min It turns 1500 revolutions before io8za2 pem0u ,tt comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revolutions hy,66/t:w nr1mv4cgxs) jj .xlin/lvas the car reduces its speed uniformly from 95km/h toc )x4vj,s:y/i.x1nj mthn6gwr l v/6l 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 makgi 8k+mqhg5ja)i jv *u77o7rae 65 revolutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires 7a 7kmgjvuq8h+j*) rii5o 7gahave 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 u eyy-lq-d)w/ on each pedal when climbing a hill. The pedals r/dql-w ye)y- uotate 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 ofq 0:n(g 0hbvct 55 N on the end of a door 74 cm wide. What is the magnitude of the torque if tvn bgqch0(0: 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.m zd jp2*ob. 6,r3flwpae 1c.b Assume that a friction to.cwra. op2 *,bjl 6bdf3mzpe1rque 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 whi* 7rr.flbt(in. tt/xdch pivots as shown in Fig. 8–40. Initially7 t..nt /(rrdtxi* flb 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 requirob4 p,(ahg ws(e tightening to a torque of 38bha w(so,g 4(p $m \cdot N$ 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 tu7wfr hw7/kq )he axis of rotation is thrk 7 7quww)hr/fough its center.    $kg \cdot m^2$

  Calculate the moment of inertia of a bicycle wheel 66.7 cm txe.78e;k qfm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub can be ignored (why?. 7kt8em;exqf ).    $kg \cdot m^2$

  A small 650-gram ball on the end of a thin, light rod is rotsqqp7pt+ 8jn 5(pv cx 2w-ovd3ated in a horizontal circle of radius 1.2o7-cq(p 8 dn5xvqv+pwjpt s23 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 ovuc7ndt53q0;iu/ td 1 jye0 abb)qi4wheel rotating at constant angular speed (Fig. 8–42). u ay ;vqnub51e)0itdqbjd c/ 7034oti 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 zgnvb 8)d*1as of the array of point objects shown in Fig. 8–43 z8a* snd) b1gvabout
(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 oxygenc.v1. els g,fra2qqe , atoms whose total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinning at the correct rate,pma 7yjzz(leh6j0( 8yi r,ntdp i;n 9v5s.a6x engineers fire four tjt 57px nzm;hyeiyd6pl(nr6(vjz 9iasa., 08 angential 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 uniform cylinder withejt *6n4 ri;4 jmhg9 *lkvvzc: a radius of 8.50 cm and a mass of 0.580 kg. Calculate4lmzni*er4k v 6 9tgj:;v*cjh
(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, accelera1p9e1n6bu/ :6w5wu n iw/uhol ce1mziting 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 tangent ey6y):j dtz*aially on a small hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with)t yz:6yj* aed 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 10,300 rpm is shut off and is eventualr((:gmutj7 rqly brought uniformly to rest by a frictional torque of 1.(7u tjr(g:rmq 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 axt n uyjmiwm+h50 +8c)ccelerates 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 thrown solely by the action mjkw 4+p/hn(5 9t5kxp0cpzy1 yscqe7of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerm9+zt knew j4yycps1kh( c/0q7 p55xpated 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 mi-;)3b2 zazz7bh(ky akcik4iqvy(: rod, as shown in Fig. 8–4 i( iz;h bya:c(akvkzk b)q-2y 7m4z3i6.
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine consists of two massrnc6 mh9r nav:4i)icz/wh 2e1es, $m_1$ 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 turn q7;kqbdwx3 3 9whta2os (revolutions) and releases it at a sp 3tqw7x kw;a 9hbd3o2qeed 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 rd6k,6g nc79d xha kd2inertia 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 rr*jvcw bf8l))i dh/,of 280 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mas92,1 f+2zr m*uxm fy3ym zdijss 7.3 kg and radius 9.0 cm rolls without slipping down a lamzid s9y3fy+ z,2x2r uj1fmm* ne at 3.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy of the Earth with r; q 7zoh q,otx1foa4 2n2az+:ijnl1bdespect to the Sun as the sum of two t2n+,xioq;nzdzqfo 1hao 21l 47jtb a :erms,
(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 a-uj;9 tl u8sm:qhd:8qle )iws nd 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 is8qwldl es m; hq:i)t-juu 8s9: a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg starts from rest and rollsd ikw0-1:ix7 :zxscrh without slippr:d1k- iiws7 hx: zcx0ing 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 irb 6guy86q k7ps balanced 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 conservatu r7py8g6q6k bion of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotatiw4t6c3 eg- 8 yqey9bay3bdorym+0 a2kng on the end of a thin string in a circle of radius 1.10 motr3g -2460dywyece 8 kqa+ bmb3 9yay at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uto7gv( kpa+uh: ,h7wi niform cylindrical grinding wheel of radius 18oih7k h,:7ag t( u+pwv cm when 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 hxbxiu;o+ l fpo0ao9i h 61)nz5is side, on a platform that is rotating at a rate of 1.3rev/s If he raises his arms to a horizontal position, Fixoo5pu69fn ib)xo0;l1z+hiag. 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(yuxn(az zdm uz1)-zad/c f57i +; fes moment of inertia by a factor of about; z(ysa-u/ fni f)1mcdza du(5+xze7z 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 angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an initial rate of 4o( q. jvxt3dluy6tp(1.0 rev everyj.lop(t(d ytvx u4q36 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 centnnvunvdf2(w5 lv8/l* er 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, approxiwv5dl/ nv8v*nl nu (2fmately shaped as a flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spinnin npo0ys5v,tt (g 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 the Eart )-7voxdqc-p p8e.zbxh
(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 momennhu o 50-xx2 l*nr.j2clwatn 1t of inertia I is dropped onto an identical disk rotating at angular spo *nat2n5h1 nl -c w.rj2lxu0xeed $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turns at 2.4dzj cju ,o,1v( ;f+nmj $rev/s$ 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 rotl d5(llp jni.q4al*sgz n8rq- h zw;l3dfk6 1-ating merry-go-round platform of radius 3.0 m and moment of rl qq83d4l-zg1jzn;5k6pns(llwi da.h l *-f 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 freely with an angul o; c2zfuc sx/.m8bm;aq ea0(war velocity of 0z(wsfc2oqx b; e8am.;c mau/ 0.8 $rad/s$ 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 dwarfl.g3ugrfms ug4w5-h5 , 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 rotation rate be?(round to the nearest integ3u w h5gg.45r s-lumfger)$\approx$    $rad/s$ (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 involve3zmz,f(sv5+b 5b*wsd 9hewj3-gu ryl winds 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 ar*qq+hssn/x2 +2 d tf$ 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 roge/zg(k;n :xwtate freely without friction. The moment of/n wg:ke(;z gx inertia of the person plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diameter merry-go-round turg-o a8gqm5 n sqx+a)/vqa3y:gntable that is mounted on frictionless bearings and vqx+a8:g q35o) qm-gngaays/has a moment of inertia of 1700 $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground w;nmkx y*pl53xjj87pl qz ( j7oith the end of the rope lying on the top edge of the spool. A person grabs7lp3 po8(y*lq;j x7 xzj5jnmk 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 mass move?

  The Moon orbits the Earth such that the swyg8y2-s7 8r nzy ezpqc2l,v8 ame side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angularyn2zqplscgw8v y7 8 r z2y,e-8 momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from resty6mr;-n dhu0vw,vc ce4zs7+r7:j w dq at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a 1 dt)/ tvun2kouniform 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 torque delivered by the)dton v/ku21 t motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 kg an.7apdm1jozyxyuai ,6yi(a:bd 4p y*y*s *e j.d 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 b (.ysj4yo1:uayi,d7pa* e 6yai*jymzp.x d*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 wheelu e4i j6tx;ca. ibnal 12vb7w2 ($\omega_R$) 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 3ta-5py2 pwme 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 uniformm2wp y35epat- 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 y1/ mkw .a9r,1kqpyjz is for it to use energy stored in a heavy rotatinwk ,jqyky/.1 1prz m9ag flywheel. Suppose such a car has a total mass of 1400 kg, uses a uniform cylindrical flywheel of diameter 1.50 m and mass 240 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 aq2m iz2y+-sm-*pmeaw n $H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) is rolling on k3rblwp.b qm m3p9,y )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 lf 9fos. bs;n*q82qdsccq mv lr;;8g)qength L can pivot freely (i.e., we ignore friction) about a hinge attached to a wall, as in Fig. 8–54. sdqcg fc; qnr2 )fqmlo*8;s .9vbs;8q 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 rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radiu y7nqyi.eu )0d (n0upps R. It is standing vertically on the floor, and we want to exert a horizontal force F at its axle so that it will clim(uen0pp0.iqn 7dy )u yb a step against which it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is making a -1irv 0qx7nowturn with a radius The forces acting on t7qwx -0nvoir1 he 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 evnjktdl jyh 1 ;:blc *gk);jkv5x(1,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 rate of 120 rpm after 5.0 s. What is the torque required to achieve this y:;ekvj1ctdjkb;(l)v hn1 j, 5gk* xlfeat, and where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cyvg: pio6*gfbx v+m5ox6 d,4gllinder and her arms as thin rods, making reasonable estimates for the dimensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightg m+fobx66,i5:4 pvl vxog g*dly against her body.   

  You are designing a cluxcf ns:52v)hlrv* d:btch assembly which consists of two cylindrical plates, of marvvb x l:2)hs:d* 5cnfss $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the looped ro7odxp4 06brnsugh track of Fig. 8–58. What is the minimum value of the vertical height h thod4s70b rxnp 6at the marble must drop if it is to reach the highest point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not ashc jmr;5) kss ilaa6bl46;m k;e.*ftpsume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as thy7uxqv b*9emge(i jw) zza 9:::* a6idi9pekhe car reduces its speed uniformly from 90km/h to 60km/h The tires have a diameter of 0.90 m.e(j*x : uiz:7:9i9epd9vi6aq*g ywbmhkeaz ) (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