Chs 7 & 8

Chs 7 & 8 - PROBLEMS nitudes Taking the x...

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Unformatted text preview: PROBLEMS nitudes.) Taking the x components alone, we have 2 mu .7 9 = __ __ 9 cos A cos ll F: 9 U ~- 0 ACOS ll (I: Thus the x component of the complete vector equation, F = ma is F: = max, with the values of F1 and a, stated‘ above. In order to display the dynamical identity of this component motion with SHM, we can take the expressions for F, and aI separately, introducing the angular velocity w and putting u = «2/1. We then have 3 FI = —mw2A c050 = ——mw2X aI = ——ng c050 = ——w‘~’x The first of these equations defines a restoring force proportional to diSplacement, exactly in accord with our initial statement of Hooke’s law [Eq. (7—38)]. The second corresponds exactly to theequation [Eq. (7~40)] that was our starting point for the kinematic analysis of the problem: ‘ 2 dx 2 ———=—wx (1/2 Thus we see that the dynamical correspondence is complete in every respect. It tells us, moreover, that we could, if we wished, go the other way and treat a uniform circular motion as a super- position of two simple harmonic motions at right angles. This is, in fact, an extremely important and useful procedure in some contexts, although we shall not take time to follow it up here and now. 7—) Two identical gliders, each of mass m, are being towed thr0ugh the air in tandem, as shown. Initially they are traveling at a constant speed and the tension in the tow rope A is T0. The tow plane then begins to accelerate with an acceleration a. What are the tensions in A and B immediately after this acceleration begins? alone, we have ete vector equation, F = ma and a; stated' above. :al identity of this component 1e expressions for FI and a, velocity to and putting v = a l restoring force proportional with our initial statement of cond corresponds exactly to 5 our starting point for the Irrespondence is complete in that we could, if we wished, n circular motion as a super- .otions at right angles. This nd useful procedure in some :e time to follow it up here iss m, are being towed through .hey are traveling at a constant A is To. The tow plane then In a. What are the tensions in tion begins? "I m, m._, 235 7—2 Two blocks, of masses M = 3 kg and m = 2 kg, are in contact on a horizontal table. A constant horizontal force F = 5 N is applied to block M as shown. There is a constant frictional force of 2 N be- tween the table and the block m but no frictional force between the table and the first block M. (a) Calculate the acceleration of the two blocks. (b) Calculate the force of contact between the blocks. 7—3 A sled of mass m is pulled by a force of magnitude P at angle 0 to the horizontal (see the figure). The sled slides over a horizontal surface of snow. It experiences a tangential resistive force equal to )1 times the perpendicular force N exerted on the sled by the snow. (a) Draw an isolation diagram showing all the forces exerted on the sled. - (b) Write the equations corresponding to F = ma for the horizontal and vertical components of the motion. (c) Obtain an expression for the horizontal acceleration in terms of P, 0, m, u, and g. (d) For a given magnitude of P, find what value of 0 gives the biggest acceleration. 7—4 A block of mass In; rests on a frictionless horizontal surface; it is connected by a massless string, passing through a frictionless eyelet, to a second block o'fmass m2 that rests on a frictionless incline (see the figure). (a) Draw isolation diagrams for the masses and write the equa- tion of motion for each one separately. (b) Find the tension in the string and the acceleration of mg. (c) Verify that, for 0 = 7r/2, your answers reduce to the ex- pected results. 7—5 In the figure, P is a pulley of negligible mass. An external force F acts on it as indicated. (21) Find the relation between the tensions on the right-hand and left-hand sides of the pulley. Find also the relation between F and the tensions. (b) What relation among the motions of m, M, and P is pro- vided by the presence of the string? (c) Use the above results and Newton's law as applied to each block to find the accelerations of m, M, and P in terms of m, M, g, and F. Check 'that the results make sense for various specialized or simplified situations. 7—6 A man is raising himself and the platform on which he stands with a uniform acceleration of 5 m/sec2 by means of the r0pe-and- pulley arrangement shown. The man has mass 100 kg and the plat- form is 50 kg. Assume that the pulley and rope are massless and move without friction, and neglect any tilting effects of the platform. Assume g = 10m/sec2. (a) What are the tensions in the ropes A, B, and C? (b) Draw isolation diagrams for the man and the platform and draw a separate force diagram for each, showing all the forces acting on them. Label each force and clearly indicate its direction. (C) What is the force of contact exerted on the man by the platform? 7—7 In an equal-arm arrangement, a mass 51m) is balanced by the masses 3/710 and 2mg, which are connected by a string over a pulley of negligible mass and prevented from moving by the string A (see the figure). Analyze what happens if the string A is suddenly severed, e.g., by means ofa lighted match. 2/11., 7—8 A prisoner in jail decides to escape by sliding to freedom down a rope provided by an accomplice. He attaches the top end of the rope to a hook outside his window; the bottom end of the rope hangs clear of the ground. The rope has a mass of 10 kg, and the prisoner has a mass of 70 kg. The hook can stand a pull of 600 N without giving way. If the prisoner‘s window is 15 m above the ground, what is the least velocity with which he can reach the ground, starting from rest at the top end of the rope? 7—9 (a) Suppose that a uniform rope of length L, resting on a fric- tionless horizontal surface, is accelerated along the direction of its length by means ofa force F pulling it at one end. Derive an expression for the tension 7‘ in the rope as a function of position along its length. How is the expression for T changed if the rope is accelerated vertically in a constant gravitational field? (b) A mass M is accelerated by the rope in part (21). Assuming the mass of the rope to be m, calculate the tension for the horizontal and vertical cases. 7—10 In 1931 F. Kirchner performed an experiment to determine the charge-to-mass ratio, e/m, for electrons. An electron gun (see the figure) produced a beam of electrons that passed through two “gates,” each gate consisting of a pair of parallel plates with the upper plates connected to an alternating voltage source. Electrons could pass B, and C? and the platform and lg all the forces acting ts direction. 1 on the man by the 10 is balanced by the a string over a pulley by the string A (see 4 is suddenly severed, 2111., 1g to freedom down the top end of the ld of the rope hangs 1g, and the prisoner ll of 600 N without re the ground, what ound, starting from ., resting on a fric— he direction of its Derive an expression )n along its length. :celerated vertically art (a). Assuming for the horizontal t to determine the tron gun (see the :mgh two “gates,” 1 the upper plates :trons could pass 237 straight through a gate only if the voltage on the upper plate were momentarily zerc. With the gates separated by a distance I equal to 50.35 cm, and with a gate voltage varying sinusoidally at a frequency f equal to 2.449 X '107 Hz (1 Hz = 1 cycle/sec), Kirchner found that electrons could pass completely undeflected through both gates when the initial accelerating voltage (V0) was set at 1735 V. Under these conditions the flight-time between the gates corresponded to one half-cycle of the alternating voltage. (a) What was the electron speed, deduced directly from land f? (b) What value of e/m is implied by the data? (c) Were corrections due to special relativity significant? [For Kirchner’s original paper, see Ann. Physik, 8, 975 (1931).] 7—11 A certain loaded car is known to have its center of gravity half- way between the front and rear axles. It is found that the drive wheels (at the rear) start slipping when the car is driven up a 20° incline. How far back must the load (weighing a quarter the weight of the empty car) be shifted for the car to get up a 25° slope '3 (The distance between the axles is 10ft.) 7—12 A child sleds down a snowy hillside, starting from rest. The hill has a 15° slope, with a long stretch of level field at the foot. The child starts 50ft up the slope and continues for 100 ft on the level field before coming to a complete stop. Find the coefficient of friction between the sled and the snow, assuming that it is constant throughout the ride. Neglect air resistance. 7—13 A beam of electrons from an electron gun passes between two parallel plates, 3 mm apart and 2 cm long. After leaving the plates the electrons travel to form a spot on a fluorescent screen 25cm farther on. It is desired to get the spot to deflect vertically through 3 cm when 100 V are applied to the deflector plates. What must be the accelerating voltage V0 on the electron gun? (Show first, in general, that if the linear displacement caused by the deflector plates can be neglected, the required voltage is given by V0 = V(/D/2 Yd), where Y is the linear displacement of the spot on the screen. The notation is that used on p. 197.) 7—14 A ball of mass m is attached to one end of a string of length I. It is known that the string will break if pulled with a force equal to nine times the weight of the ball. The ball, supported by a frictionless table, is made to travel a horizontal circular path, the other end of the string being attached to a fixed point 0. What is the largest number of revolutions per unit time that the mass can make without breaking the string ? 7—15 A mass of 100g is attached to one end ofa very light rigid rod 20 cm long. The other end of the rod is attached to the shaft of a motor so that the rod and the mass are caused to rotate in a vertical circle with a constant angular velocity of 7 rad/sec. (a) Draw a force diagram showing all the forces acting on the mass for an arbitrary angle 00f the rod'to the downward vertical. (b) What are the magnitude and the direction of the force exerted by the rod on the mass when'the rod points in a horizontal direction, i.e., at 0 = 90°? 7—16 You are flying along in your Sopwith Camel at 60 mph and 2000-ft altitude in the vicinity of Saint Michel when suddenly you notice that the Red Baron is just 300 ft behind you flying at 90 mph. Recalling from captured medical data that the Red Baron can with- stand only 4 g's of acceleration before blacking but, whereas you can withstand 5 g’s, you decide on the following plan. You dive straight down at full power, then level out by flying in a circular arc that comes out horizontally just above the ground. Assume that your speed is constant after you start to pull out and that the acceleration you experience in the arc is 5 g's. Since you know that the Red Baron will follow you, you are assured he will black out and crash. Assuming that both planes dive with 2 g’s acceleration from the same initial point (but with initial speeds given above), to what altitude must you descend so that the Red Baron, in trying to follow your subsequent arc, must either crash or black out? Assuming that the Red Baron is a poor shot and must get within 100 ft of your plane to shoot you down, will your plan succeed? After starting down you recall reading that the wings of your plane will fall off if you exceed 300 mph. ls ‘ your plan sound in view of this limitation on your plane? 7—17 A curve of 300 m radius on a level road is banked for a speed of 25 m/sec (z 55 mph) so that the force exerted on a car by the road is normal to the surface at this speed. (a) What is the angle of bank? (b) The friction between tires and road can provide a maximum tangential force equal to 0.4 of the force normal to the road surface. What is the highest speed at which the car can take this curve without skidding? 7—18 A large mass M hangs (stationary) at the end of a string that l of a string of length I. :d with a force equal to pported by a frictionless ath, the other end of the at is the largest number make without breaking of a very light rigid rod ached to the shaft of a :d to rotate in a vertical l/sec. the forces acting on the : downward vertical. :tion of the force exerted n a horizontal direction, Camel at 60 mph and be] when suddenly you .d you flying at 90 mph. Ie Red Baron can with- ig out, whereas you can plan. You dive straight 1 circular arc that comes .ume that your speed is at the acceleration you ow that the Red Baron ut and crash. Assuming I from the same initial what altitude must you follow your subsequent ing that the Red Baron (our plane to shoot you down you recall reading ou exceed 300 mph. Is your plane? 1 is banked for a speed ted on a car by the road :an provide a maximum nal to the road surface. I take this curve without the end of a string that 239 passes through a smooth tube to a small mass m that whirls around in a circular path of radius [sin 0, where l is the length of the string from m to the top end of the tube (see the figure). Write down the dynamical equations that apply to each mass and show that m must complete one orbit in a time of 21r(lm/gM)”2. Consider whether there is any restriction on the value of the angle 0 in this motion. 7—19 A model rocket rests on a frictionless horizontal surface and is joined by a string of length l to a fixed point so that the rocket moves in a horizontal circular path of radius [. The string will break if its tension exceeds a value T. The rocket engine provides a thrust F of constant magnitude along the rocket’s direction of motion. The rocket has a mass m that does not decrease appreciably with time. (a) Starting from rest at t = 0, at what later time I; is the rocket traveling so fast that the string breaks? Ignore air resistance. (b) What'Was 'the magnitude of the rocket’s instantaneous net acceleration at time t1/2? Obtain the answer in terms of F, T, and m. (c) What distance does the rocket travel between the time ti when the string breaks and the time 2!. ? The rocket engine continues to operate after the string breaks. 7—20 It has been suggested thatgthe biggest nuclear accelerator we are likely to make will be an evacuated pipe running around the earth’s equator. The strength of the earth‘s magnetic field at the equator is about 0.3 G or 3 x .10-5 MKS units (N—sec/C-m). With what speed would an atom of lead (at. wt. 207), singly ionized (i.e., carrying one elementary charge), have to move around such an orbit so that the magnetic force provided the correct centripetal acceleration? (e = 1.6 X 10—19 C.) Through what voltage would a singly ionized lead atom have to be accelerated to give it this correct orbital speed? 7—21 A trick cyclist rides his bike around a “wall of death" in the form of a vertical cylinder (see the figure). The maximum frictional force parallel to the surface of the cylinder is equal to a fraction 1:. of the normal force exerted on the bike by the wall. (a) At what speed must the cyclist go to avoid slipping down? go '—————r I I /, Axis of ' cylinder $13 3 .L (b) At what angle (go) to the horizontal must he be inclined? (c) If u z 0.6 (typical of rubber tires on dry roads) and the radius of the cylinder is 5 m, at what minimum speed must the cyclist ride, and what angle does he make with the horizontal? 7—22 The following expression gives the resistive force exerted on a sphere of radius r moving at speed 0 through air. It is valid over a very wide range of speeds. W) = 3.1 x 10-% + 0.87%2 where R is in N, r in m, and u in mf/sec. Consider water drops falling ' under their own weight and reaching a terminal speed. (a) For what range of values of small r is the terminal speed determined within 1% by the first term alone in the expression for R(u)? (b) For what range of values. of larger r is the terminal speed determined within 1% by the second term alone? (c) Calculate the terminal speed of a raindrop of radius 2 mm. If there were no air resistance, from what height would it fall from rest before reaching this speed? 7—23 An experiment is performed with the Millikan oil—drop ap- paratus. The plates are 8 mm apart. The experiment is done with oil droplets of density 896 kg/m“. The droplets are timed between two horizontal lines that are 2.58 mm apart. With the plates un~ charged, a droplet is observed to take 23.6 sec to fall from one line to the other. When the upper plate is made 1100 V positive with respect to the lower, the droplet rises and takes 22.0 sec to cover the same distance. Assume that the resistive force is 3.1 X 10T4rv (MKS units). (a) What is the radius of the droplet? (b) What is its net charge, measured as a number of elementary charges? (e = 1.6 x 10-190) (c) What voltage would hold the droplet stationary? [Use the precise value of the charge deduced in part (b)]. 7—24 Two solid plastic spheres of the same material but of different radii, R and 2R, are used in a Millikan experiment. The spheres carry equal charges q. The larger sphere is observed to reach terminal speeds as follows: (1) plates uncharged: terminal speed no (down- ward), and (2) plates charged: terminal speed = u; (upward). As- ll 7suming that the resistive force on a sphere of radius r at speed 0 is c150, find, in terms of Do and ur, the corresponding terminal speeds for the smaller sphere. 7—25 Analyze in retrospect the legendary Galilean experiment that took place at the leaning tower of Pisa. Imagine such an experiment done with two iron spheres, of radii 2 and 10 cm, respectively, dropped simultaneously from a height of about 15 m. Make calculations to Lontal must he be inclined? r tires on dry roads) and the linimum speed must the cyclist the horizontal? te resistive force exerted on a hrough air. It is valid over a Consider water drops falling erminal speed. small r is the terminal speed n alone in the expression for .arger r is the terminal speed rn alone? f a raindrop of radius 2 mm. fiat height would it fall from "t the Millikan oil-drop ap- The experiment is done with : droplets are timed between apart. With the plates un- .6 sec to fall from one line to : 1100 V positive with respect 5 22.0 sec to cover the same s 3.1 x 10—4ru (MKS units). :t? d as a number of elementary troplet stationary? [Use the rt (b)]. me material but of different periment. The spheres carry observed to reach terminal terminal speed = D() (down- speed = U} (upward). As- :re of radius r at Speed 0 is rresponding terminal speeds y Galilean experiment that Imagine such an experiment 10 cm, respectively, dropped Sm. Make calculations to 11p: 241 determine, approximately, the difference in the times at which they hit the ground. Do you think this could be detected without special measuring devices? (Density of iron z 7500 kg/m“.) 7—26 Estimate the terminal Speed of fall (in air) of an air-tilled toy balloon, with a diameter of 30 cm and a mass (not counting the air inside) of about 0.5 g. About how long would it take for the balloon to come to within a few percentof this terminal speed? Try making some real observations of balloons inflated to different sizes. 7—27 A spring that obeys Hooke’s law in both extension and com- pression is extended by 10cm when a mass of 2 kg is hung from it. (a) What is the spring constant k? (b) The spring and the 2-kg mass are placed on a smooth table. The mass is pulled so as to extend the spring by 5 cm and is then released at I = 0. What is the equation of the ensuing motion ? (c) If, instead of being released from rest, the mass were started off at x = 5 cm with a speed of 1 m/sec in the direction of increasing x, what would be the equation of motion? 7—28 When the mass is doubled in diagram (a), the end of the spring descends an additional distance Ii. What is the frequency of oscillation for the arrangement in diagram (b)? All individual springs shown are identical. 7—29 Any object, partially or wholly submerged in a fluid, is buoyed up by a force equal to the weight of the displaced fluid. A uniform cylinder of density p and length l is floating with its axis vertical, in a fluid of density pm. What is the frequency of small-amplitude vertical oscillations of the cylinder ? 7—30 (a) A small bead of mass m is attached to the midpoint of a string (itself of negligible mass). The string is of length L and is under constant tension T. Find the frequency of the SHM that the mass describes when given a slight transverse displacement. (b) Find the frequency in the case where the mass is attached at a distance D from one end instead of the midpoint. 7~3l A block rests on a tabletop that is undergoing-simple harmonic motion of amplitude A and period T. (a) If the oscillation is vertical, what is the maximum value of A that will allow the block to remain always in contact with the table? (b) If the oscillatior. is horizontal, and the coefficient of friction between block and tabletop is [1, what is the maximum value of A that will allow the block to remain on the surface without slipping? 7—32 The springs of a car of mass 1200 kg give it a period when empty of 0.5 sec for small vertical oscillations. (at) How far does the car sink down when a driver and three passengers, each of mass 75 kg, get into the car? 4352;; q..- (b) The car with its passengers is traveling along a horizontal road when it suddenly runs onto a piece of new road surface, raised 2in. above the old surface. Assume that this suddenly raises the wheels and the bottom ends of the springs through 2 in. before the body of the car begins to move upward. In the ensuing rebound, are the passengers thrown clear of their seats? Consider the maximum acceleration of the resulting simple harmonic motion. ed not in terms I "curvature of :t and mathe- :ometries. ‘avitation gives )unds for pre~ than practical. ions there is a This is in the try. The phe- s‘ distinctly el— .ses in its own erent direction this precession 'itury) can be 3 other planets rere remains a ldS of are per n theory—for ide Mercury's other facts of 1’s theory, on c parameters, exactly with :nce ofa very nce than the .y in which a )recess is dis- ttions of the the inverse- :d his theory, r led to false )ry, however. ing term was of the planet try, with its irth is actually JOUS change in .luinoxes — see PROBLEMS 8—1 Given a knowledge of Kepler’s third law as it applies to the solar system, together with the knowledge that the disk of the sun subtends an angle of about §° at the earth, deduce the period of a hypothetical planet in a circular orbit that skims the surface of the sun. 8—2 It is well known that the gap between the four inner planets and the five outer planets is occupied by the asteroid belt instead of by a tenth planet. This asteroid belt extends over a range of orbital radii from about 2.5 to 3.0 AU. Calculate the corresponding range of periods, expressed as multiples of the earth’s year. 8—3 It is proposed to put up an earth satellite in a circular orbit with a period of 2 hr. (a) How high above the earth's surface would it have to be? (b) If its orbit were in the plane of the earth’s equator and in the same direction as the earth’s rotation, for how long would it be continuously visible from a given place on the equator at sea level? 8—4 A satellite is to be placed in synchronous circular orbit around the planet Jupiter to study the famous “red spot” in Jupiter’s lower atmosphere. How high above the surface of Jupiter will the satellite be? The rotation period of Jupiter is 9.9 hr, its mass M; is about 320 times the earth’s mass, and its radius RJ is about 11 times that of the earth. You may find it convenient to calculate first the gravita- tional acceleration g; at Jupiter’s surface as a multiple of g, using the above values of M] and RJ, and then use a relationship analogous to that developed in the text for earth satellites [Eq. (8—16) or (847)]. 8—5 A satellite is to be placed in a circular orbit 10km above the surface of the moon. What must be its orbital speed and what is the period of revolution? 8—6 A satellite is to be placed in synchronous circular earth orbit. The satellite’s power supply is expected to last 10 years. If the maxi- mum aceeptable eastward or westward drift in the longitude of the satellite durin its lifetime is 10°, what is the margin of error in the radius of its 0%3it? 8—7 The sprihgs found in retractable ballpoint pens have a relaxed length of about 3cm and a spring constant of perhaps 0.05 N/mm. Imagine that two lead spheres, each of 10,000 kg, are placed on a frictionless surface so that one of these springs just fits, in its un- compressed state, between their nearest points. (a) How much would the spring be compressed by the mutual gravitational attraction of the two spheres? The density of lead is about 11,000 kg/m”. I . ,.r’\_/_Z? (b) Let the system be rotated in the horizontal plane. At what frequency of rotation would the presence of the spring become ir- relevant to the separation of the masses? 8—8 During the eighteenth century, an ingenious attempt to find the mass of the earth was made by the British Astronomer Royal, Nevil Maskelynef He observed the extent to which a plumbline was pulled out of true by the gravitational attraction of a mountain. The figure illustrates the principle of the method. The change of direction of the plumbline was measured between the two sides of the mountain. (This was done by sighting on stars.) After allowance had been made for the change in direction of the local vertical because of the curvature of the earth, the residual angular difference a was given by 2FM/FE, where :bFM is the horizontal force on the plumb-bob due to the moun- tain, and FE = GMEm/Ryz. (m is the mass of the plumb~bob.) The value of a is about 10 seconds of arc for measurements on opposite sides of the base ofa mountain about 2000 m high. Suppose that the mountain can be approximated by a cone of rock (of density 2.5 times that of water) whose radius at the base is equal to the height and whose mass can be considered to be concentrated at the center of the base. Deduce an approximate value of the earth’s mass from these figures. (The true answer is about 6 X 1024 kg.) Compare the gravitational deflection a to the change of direction associated with the earth‘s curvature in this experiment. 8—9 Imagine that in a certain region of the ocean floor there is a roughly cone—shaped mound of granite 250 m high and 1 km in diam- eter. The surrounding floor is relatively flat for tens of kilometers in all directions. The ocean depth in the region is 5 km and the density of the granite is 3000 kg/m”. Could the mound’s presence be de- tected by a surface vessel equipped with a gravity meter that can detect a change in g of 0.1 mgal?. (Hint: Assume that the field produced by the mound at the surface can be approximated by the field of a mass point of the same total mass located at the level of the surrounding floor. Note that in calculating the change in g you must keep in mind that the mound has displaced its own volume of water. The density of water, even at such depths, can be taken as about equal to its surface value of about 1000 kg/m“. horizontal plane. At what of the spring become ir- ;enious attempt to find the Astronomer Royal, Nevil ch a plumbline was pulled )f a mountain. The figure change of direction of the 0 sides of the mountain. 0/2 \; ——>l allowance had been made al because of the curvature a was given by ZFM/Fg, imb-bob due to the moun- ; of the plumb—bob.) ’ arc for measurements on )ut 2000 m high. Suppose a cone of rock (of density base is equal to the height :oncentrated at the center of the earth’s mass from < 1024 kg.) Compare the direction associated with he ocean floor there is a 11 high and 1 km in diam- L for tens of kilometers in n is 5 km and the density mound’s presence be de- 1 gravity meter that can :d by the mound at the a mass point of the same riding floor. Note that in in mind that the mound density of water, even at its surface value of about 8~10 Show that the period of a particle that moves in a circular orbit close to the surface of a sphere depends only on G and the mean density of the sphere. Deduce what this period would be for any sphere having a mean density equal to the density of water. (Jupiter almost corresponds to this case.) 8-11 Calculate the mean density of the sun, given a knowledge of G, the length of the earth’s year, and the fact that the sun’s diameter subtends an angle of about 0.55° at the earth. 8-12 An astronaut who can lift 50 kg on earth is exploring a planetoid (roughly spherical) of 10 km diameter and density 3500 kg/m3. (a) How large a rock can he pick up from the planetoid’s surface, assuming that he finds a well-placed handle? (b) The astronaut observes a rock falling from a cliff. The rock’s radius is only 1 m and as it approaches the surface its velocity is l m/sec. Should he try to catch it? (This is obviously a fanciful problem. One would not expect a planetoid to have clifl‘s or loose rocks, even if an astronaut were to get there in the first place.) 8—13 It is pointed out in the text that a person can properly be termed “weightless” when he is in a satellite circling the earth. The moon is a satellite, yet it is noted in many discussions that we would weigh % of our normal weight there. Is there a contradiction here? 8—14 A dedicated scientist performs the following experiment. After installing a huge spring at the bottom of a 20-story-high elevator shaft, he takes the elevator to the top, positions himself on a bathroom scale inside the airtight car with a stopwatch and with pad and pencil to record the scale reading, and directs an assistant to cut the car’s support cable at t = O. Presuming that the scientist survives the first encounter with the spring, sketch a graph of his measured weight versus time from t = 0 up to the beginning of the second bounce. (Note: Twenty stories is ample distance for the elevator to aCQuire terminal velocity.) 8—15 A planet of mass M and a single satellite of mass M/10 revolve in circular orbits about their stationary center of mass, being held together by their gravitational attraction. The distance between their centers is D. (a) What is the period of this orbital motion? (b) What fraction of the total kinetic energy resides in the satellite? (Ignore any spin of planet and satellite about their own axes.) 8-16 We have considered the problem of the moon’s orbit around the earth as if the earth’s center represented a fixed point about which the motion takes place. In fact, however, the earth and the moon revolve about their common center of mass. ' (a) Calculate the position of the center of mass, given that the earth’s mass is 81 times that of the moon and that the distance be— tween their centers is 60 earth radii. (b) How much longer would the month be if the moon were of negligible mass compared to the earth '? 8—17 The sun appears to be moving at a speed of about 250 km/sec in a circular orbit of radius about 25,000 light-years around the center of our galaxy. (One light-year 2 10H5 m.) The earth takes 1 year to describe an almost circular orbit of radius about 1.5 X 10” m around the sun. What do these facts imply about the total mass responsible for keeping the sun in its orbit? Obtain this mass as a multiple of the sun’s mass M. (Note that you do not need to introduce the numerical value ’of G torobtain the answer.) 8—18 (A good problem for discussion.) In 1747 Georges Louis Lesage explained the inverse-square law of gravitation by postulating that vast numbers of invisible particles were flying through space in all directions at high speeds. Objects like the sun and planets block these particles, leading to a shadowing effect that has the same quantitative result as a gravitational attraction. Consider the arguments for and against this theory. (Suggestion: First consider a theory in which opaque objects block the particles completely. This proposal is fairly easy to refute. Next consider a theory in which the attenuation of the particles by objects is incomplete or even very small. This theory is much harder to dismiss.) 8—19 The continuous output of energy by the sun corresponds (through Einstein‘s relation E = Meg) to a steady decrease in its mass M, at the rate of about 4 X 106 tons/sec. This implies a progressive in- crease in the orbital periods of the planets, because for an orbit of a given radius we have T~ M‘“2 [cf Eq. (8—23)]. A precise analysis of the effect must take into account the fact that as M decreases the orbital radius itself increases the planets gradually spiral away from the sun. However, one can get an order- of-magnitude estimate of the size of the effect, albeit a little bit on the low side, by assuming that r remains constant. (See Problem 13~21 for a more rigorous treatment.) Using the simplifying assumption of constant r, estimate the approximate increase in the length of the year resulting from the sun‘s decrease in mass over the time span of accurate astronomical ob- servations—about 2500 years. 8—20 It is mentioned at the end of the chapter how Einstein’s theory of gravitationreads to a small correction term on top of the basic Newtonian to tee of gravitation. For a planet of mass m, traveling at speed 1: in a e rcular orbit of radius r, the gravitational force becomes of mass, given that the d that the distance be- be if the moon were of i of about 250 km/sec 'ears around the center e earth takes 1 year to 1.5 X 10" m around total mass responsible lSS as a multiple of the (troduce the numerical Georges Louis Lesage 1 by postulating that through space in all 1d planets block these the same quantitative Ie arguments for and 'hich opaque objects fairly easy to refute. 1 of the particles by Ieory is much harder :orresponds (through se in its mass M, at es a progressive in- ise for an orbit of a no account the fact :reases———the planets e can get an order- :it a little bit on the :See Problem 13—21 int r, estimate the [ting from the sun’s : astronomical ob- w Einstein’s theory n top of the basic lass m, traveling at mal force becomes 305 ark / in effect the following: 2 p: 61"?"(1 +693) r C” where c is the speed of light. (Correction terms of the order of 03/02 are typical of relativistic effects.) (a) Show that, if the period under a pure Newtonian force GMm/r2 is denoted by To, the modified period T is given approxi- mately by 0 9 1211"?" T_ To<1 — 237,02) (Treat the relativistic correction as representing, in effect, a small fractional increase in the value of G, and use the value of u corre- sponding to the Newtonian orbit.) (b) Hence show that, in each revolution, a planet in a circular orbit would travel through an angle greater by about 241r3r3/c3Tu‘3 than under the pure Newtonian force, and that this is also expressible as 67rGM/cgr, where M is the mass of the sun. (c) Apply these results to the planet Mercury and verify that the accumulated advance in angle amounts to about 43 seconds of are per century. This corresponds to what is called the precession of its orbit. ...
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Chs 7 &amp; 8 - PROBLEMS nitudes Taking the x...

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