Chs 5 & 6

Chs 5 & 6 - nated by turbulence, however small the...

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Unformatted text preview: nated by turbulence, however small the ratio B/A may be. The same consideration guarantees that at sufficiently low speeds the resistance will be dominated by the viscous term, directly propor- tional to u [see Fig. 5—9(b)]. CONCLUDING REMARKS PROBLEMS In this chapter we have given a brief account of the three major types of physical interactions and have indicated the general areas in which they are dominant. To recapitulate: Nuclear forces are significant only for nuclear distances, the gravitational force is important only if objects of astronomical scale are in- volved, and nearly everything else ultimately depends on electro- magnetic interactions. The study of physics is essentially the attempt to understand these interactions and all their con- sequences. In mechanics we have, for the most part, the more modest goal of taking the forces as given and considering various dynamical situations in which they enter. We shall, however, be discussing two classic cases—gravitation and alpha-particle scattering—in which Newtonian mechanics provided the key to the basic laws of force. The present chapter has provided a kind of preview, because it summarizes the state of our current knowl- edge without entering into any detailed discussion of how we have come to know it. The real work lies ahead! 5—] At what distance from the earth, on the line from the earth to the sun, do the gravitational forces exerted on a mass by the earth and the sun become equal and opposite? Compare the result with the radius of the moon’s orbit around the earth. 5—2 By what angle, in seconds of arc, will a plumbline be pulled out of its normal vertical direction by the gravitational attraction of a 10-ton truck that parks 20 ft away? Do you think that this effect could be detected? 5-3 In a Cavendish—type apparatus (see the figure) the large spheres are each 2 kg, the small spheres each 20 g. The length of the'arm con- necting the small spheres is 20 cm, and the distance between the centers of a small sphere and the'big sphere close to it is 5 cm. The torsion constant of the suspending fiber is 5 X 10‘8 m-N/rad. The angular deflection of the suspended system is deduced from the displacement ver small the ratio B/A may be. The tees that at sufficiently low speeds the i by the viscous term, directly propor- |. en a brief account of the three major ons and have indicated the general ominant. To recapitulate: Nuclear )r nuclear distances, the gravitational abjects of astronomical scale are in- I g else ultimately depends on electro— : study of physics is essentially the :se interactions and all their con- e have, for the most part, the more rces as given and considering various :h they enter. We shall, however, be res—gravitation and alpha-particle nian mechanics provided the key to present chapter has provided a kind irizes the state of our current knowl— any detailed discussion of how we real work lies ahead! 1e earth, on the line from the earth to 'orces exerted on a mass by the earth 1 opposite? Compare the result with tround the earth. . of arc, will a plumbline be pulled out I by the gravitational attraction of a way? Do you think that this effect ratus (see the figure) the large spheres each 20 g. The length of the arm con- n, and the distance between the centers phere close to it is 5 cm. The torsion ' is 5 X 10‘8 m-N/rad. The angular em is deduced from the displacement an“; aw. "Am 155 Mk .Ztli'i’lii L \20 g of a reflected Spot of light on a scale 5 m away. (Remember that the change in direction of the reflected light is twice the angle through which the mirror is turned.) It is observed that when the large spheres are moved from their initial positions to equivalent positions on the other side (dashed lines) the mean position of the spot of light is shifted by 8 cm. (a) Deduce the value of G according to these data, ignoring the effect on each small sphere of the force due to the more distant of the larger Spheres. _ (b) Estimate the percentage correction on the result of (a) re- quired to allow for the effect of the more distant spheres. 5—4 The original Cavendish experiment was done with a large-size apparatus, as is natural if one wants to make the gravitational forces and torques as big as possible. However, this requires a strong, stiff wire to support the suspended masses. Much later (1895) C. V. Boys made a miniature apparatus, using thin fibers of fused quartz for the suspensions. It is an interesting exercise to see how the attainable sensitivity of the apparatus depends on its size. Imagine two versions of the Cavendish apparatus, A and B, both using solid lead spheres, in which the radii and separations of all the masses in B, together with the length of the torsion fiber, are scaled down by a certain factor L with respect to A. We then design for maximum sensitivity in each apparatus by using the thinnest possible torsion fiber that will take the weight of the suspended masses without breaking. Now for a torsion fiber of given material and of circular cross section, the maxi- mum supportable load is proportional to (P, where d is its diameter, and its torsion constant is proportional to 114/], where l is its length. Using this information, compare the maximum angular deflections obtainable with the two different sizes of apparatus. (Remember, the lengths of the torsion fibers also differ by the scaling factor L.) 5-5 The radius of the hydrogen atom according to the original Bohr theory is 0.5 A. (a) What is the Coulomb force between the proton and the electron at this distance? What is the gravitational force? (b) How far apart must the proton and electron be for the Coulomb force to be equal to the value that the gravitational attrac- tion has at 0.5 A? What familiar astronomical distance is this com- parable to? 5—6 Suppose electrons could be added to earth and moon until the Coulomb repulsion thus developed was of just the size to balance the gravitational attraction. What would be the smallest total mass of electrons that would achieve this? 5—7 For a person living at 45° latitude, what is the approximate fractional difference, during the day, between the maximum and minimum gravitational forces due to the moon—the change resulting from the fact that the earth’s rotation causes the person’s distance from the moon to vary? What is a manifestation of this kind of force effect in nature? 5-8 You know that the Coulomb force and the gravitational force both obey an inverse-square law. Suppose that it were put to you that the origin of the gravitational force is a minute difference be- tween the natural unit of positive charge, as carried by a proton, and the natural unit of negative charge, as carried by an electron. Thus “neutral” matter, containing equal numbers of protons and electrons, would not be quite neutral in fact. (a) What fractional difference between the positive and negative elementary charges would lead to “gravitational” forces of the right magnitude between lumps of ordinary “neutral” matter? How could such a difference be looked for by laboratory experiments? (b) Is the theory tenable? 5—9 (a) The text (p. 148) quotes a value of the nuclear force for two nucleons close together but also suggests that to describe the nuclear interactions in terms of forces is not very practical. Can you suggest any way in which a nuclear force as such could be measured? (b) According to one of the earliest and simplest theoretical descriptions of the nuclear interaction (by H. Yukawa) the force of attraction between two nucleons at large separation would be given by m) = _ A I" where the distance ro is about lO—lsm and the constant A is about 10‘11 N-m. At about what separation between a proton and a neutron would the nuclear force be equal to the gravitational force between these two particles ? 5-10 Can you think of any systems or processes in which gravita— mb force between the proton and the at is the gravitational force? st the proton and electron be for the > the value that the gravitational attrac. iiliar astronomical distance is this com- be added to earth and moon until the oped was of just the size to balance the it would be the smallest total mass of his? 45° latitude, what is the approximate the day, between the maximum and due to the moon—the change resulting i rotation causes the person’s distance L is a manifestation of this kind of force lomb force and the gravitational force aw. Suppose that_it were put to you tional force is a minute difference be- ive charge, as carried by a proton, and large, as carried by an electron. Thus qual numbers of protons and electrons, act. ‘ence between the positive and negative ! to “gravitational” forces of the right rdinary “neutral” matter? How could by laboratory experiments ? l lets a value of the nuclear force for two 0 suggests that to describe the nuclear .s not very practical. Can you suggest :e as such could be measured ‘1 the earliest and simplest theoretical traction (by H. Yukawa) the force of ‘. at large separation would be given by 10*15 m and the constant A is about .ration between a proton and a neutron ml to the gravitational force between stems or processes in which gravita- mus. a tional, electromagnetic, and nuclear forces all play an important role? 5—11 As mentioned in the text, the attractive (long-range) part of the force between neutral molecules varies as l/r7. For a number of molecules, the order of magnitude of this van der Waals force is well represented by the equation mm x —1o‘7“/r7 where F yw is in newtons and r in meters. Compare the magnitude of the van der Waals force with the Coulomb force between two ele- mentary charges [Eq. (5—2), with q, = q2 = e = 1.6 x 10"” C]: (a) For r = 4 A. (This is a distance about equal to the diameter of a molecule of oxygen or nitrogen and hence barely exceeding the closest approach of the centers of two such molecules in a collision.) (b) For a value of r correSponding to the mean distance between molecules in a gas at STP. ' 5—12 One of the seemingly weakest forms of contact force is the surface tension of a liquid film. One of the seemingly strongest is the tensile force of a stretched metal wire. However, when expressed in terms of a force between individual atoms in contact, they do not look so different. Use the following data to evaluate them in these terms: (a) If a water film is formed between a rectangular wire frame, 3 in. wide, and a freely sliding transverse wire (see the figure), it takes the weight of about 1 g to prevent the film from contracting. This contractile force can be ascribed to the contact of the atoms lying within a monomolecular layer along each side of the film. Supposing that the molecules are 3 A across and closely packed, calculate the force per molecule. (b) A copper wire of 0.025-in. diameter was found to break when a weight of about 10 kg was hung from its lower end. First, calculate this breaking force in tons per square inch. If the fracture is assumed to involve the rupturing of the contacts between the atoms on the upper and lower sides of a horizontal section right across the wire, calculate the force per atom, assuming an atomic diameter of about 3 A. 5-13 A time-honored trick method for approximately locating the midpoint of a long uniform rod or bar is to support it horizontally at any two arbitrary points on one’s index fingers and then move the fingers together. (Of course, just finding its balance point on one finger alone works very well, too!) Explain the workings of the trick method, using your knowledge of the basic principles of static equi— librium and a property of frictional forces: that they have a maximum value equal to a constant [I (the coefficient of friction) times the com- ponent of force normal to the surface of contact between two objects. 5—14 (a) A string in tension is in contact with a circular rod (radius r) over an arc sintending a small angle A0 (see the figure). Show that the force with which the string presses radially inward on the pulley (and hence the normal force AN With which the pulley pushes on the string) is equal to TAB. I (b) Hence show that the normal force per unit length is equal to T/r. This is a sort of pressure which, for a given value of T, gets bigger as r decreases. (This helps to explain why, when a string is tightly tied around a package, it cuts into the package most deeply as it passes around corners, where r is least.) (c) If the contact is not perfectly smooth, the values of the tension at the two ends of the arc can differ by a certain amount AT before slipping occurs. The value of AT is equal to uAN, where u is the coefficient of friction between string and rod. Deduce from this the exponential relation m9) = Toe” where To is the tension applied at one end of an arbitrary are (0) of string and T(0) is the tension at~the other end. (d) The above result expresses the possibility of withstanding a large tension T in a rope by wrapping the rope around a cylinder, a phenomenon that has been exploited since time immemorial by sailors. Suppose, for example, that the value ofp in the contact between a rope and a bollard on a dock is 0.2. For To = 100 lb, calculate the values of T corresponding to one, two, three and four complete turns of rope around the bollard. (It is interesting to note that T is proportional to To. This allows sailors to produce a big pull or not, at will, by having a rope passing around a continuously rotating motor-driven drum. The arrangement can be described as a force amplifier.) 5—15 In a very delicate torsion-balance experiment, such as the Cavendish experiment, the stray forces due to the fluid friction of slow air currents pushing on the suspended system may be quite significant. To make this quantitative, consider the gravity torsion-balance ex- periment described in Problem 5—3. For suspended spheres of the size stated (r z 0.8 cm), the force due to a flow of air of speed 0 is given approximately by the formula [Eg. (5—4)] R (newtons) = 2.5 X 10—6!) + 5 X 10—5172 where v is in m/sec. Calculate the value of u that would cause a force due to air currents that equaled the gravitational force exerted on the sphere in this experiment (i.e., the force exerted on a 20-g sphere by a 2-kg sphere with their centers 5 cm apart). (Hint: Do not bother to )ntact With 8 Circular rod (radius r) i V I solve a quadratic equation for v. Just find the values of v for which gle A0 (see the figure)- ShOW that 5' the contributions to R, taken separately, would equal the gravitational 35565 radially inward 0“ the puuey .v j - - force. The smaller of the two values ofv so obtained is clearly already 'th WhiCh the puney pUShes 0“ the - enough to spoil the experiment.) mal force per unit length is equal which, for a given value of T, gets to explain why, when a string is uts into the package most deeply ' is least.) rfectly smooth, the values of the :an differ by a certain amount AT )f AT is equal to uAN, where [.l. is tring and rod. Deduce from this one end of an arbitrary arc (0) of other end. ; the possibility of withstanding a ing the rope around a cylinder, a : since time immemorial by sailors. of u in the contact between a rope To = 100 lb, calculate the values e and four complete turns of rope proportional to To. This allows at will, by having a rope passing )r—driven drum. The arrangement ~) alance experiment, such as the es due to the fluid friction of slow d system may be quite significant. r the gravity torsion-balance ex- For suspended spheres of the lue to a flow of air of speed 0 is Big. (5-4)] 5 x 10-502 alue of u that would cause a force gravitational force exerted on the rce exerted on a 20-g sphere by a apart). (Hint: Do not bother to 159 §_32'<>i‘3§em3 ,ments gathered themselves together into n jumped up onto the table, this would ‘nature doesn’t act like that. Yet a ividual atomic encounters at every stage >e perfectly time-reversible. vith a puzzle: Newton’s law implies that cal behavior of an individual particle is when one takes a system of very large parently the behavior ceases to be time- n of this mystery is found in the detailed .ny-particle systems—the subject known As long as we are dealing, as we shall of only a few particles, the problems :rsal do not arise and we shall not con- :ome apparent to you during the course : foundation of classical mechanics, as : second law, is a complex and in many The precise content of the law is still a ly three centuries after Newton stated n a fine discussion entitled “The Origin ; Laws of Motion,” one author (Brian Newton’s second law of motion? What llS law? Is it a definition of force? Of cal preposition relating force, mass, and gues that it is something of all of these: an’s second law is actually used. In some tably true that Newton’s second law is used 'orce. How else, for example, can we mea- gravitational forces? But it is also un- at Newton’s second law is sometimes used nass. Consider, for example, the use of the And in yet other fields, where force, mass, : all easily and independently measurable, v of motion functions as an empirical cor- se three quantities. Consider, for example, :ssays, Beyond the Edge of Certain! y (Robert G. Englewood Cliffs, N.J., 1965. PROBLEMS 181 the application of Newton’s second law in ballistics and rock- etry . .. To suppose that Newton’s second law of motion, or any law for that matter, must have a unique role that we can describe generally and call the logical status is an unfounded and unjustifiable supposition. Since force and mass are both abstract concepts and not objective realities, we might conceive of a description of nature in which we dispensed with both of them. But, as one physicist (D. H. Frisch) has remarked, “Whatever we think about ultimate reality it is convenient to follow Newton and split the description of our observations into ‘forces,’ which are what make masses accelerate, and ‘masses,’ which are what forces make accelerate. This would be just tautology were it not that the observed phe- nomena can best be classified as the result of dlflerent forces acting on the same’set of masses.” Ellis spells out this same idea in more detail: Now there are, in fact, many and various procedures by which the magnitudes of the individual forces acting on a given system may be determined—electrostatic forces by charge and distance measurements, elastic forces by measurement of strain, magnetic forces by current and distance determinations, gravitational forces by mass and distance measurements, and so on. And it is an empirical fact that when all such force measurements are made and the magnitude of the resultant force determined, then the rate of change of momentum of the system under considera- tion is found to be proportional to the magnitude of this re- sultant force. And so it is that we obtain an immensely fruitful and accurate description of a very large part of our whole experience of objects in motion, through the simple and compact statement of Newton’s second law.1 6—1 Make a graphical analysis of the data represented by the strobe- scopic photograph of Fig. 6—2(a) to test whether this is indeed ac- celeration under a constant force. 1For further discussion of these questions, the American Journal of Physics is a perennial source. See, for example, the following articles: I... Eisenbud, “On the Classical Laws of Motion," Am. J. Phys, 26, 144 ( 1958); N. Austern, “Presentation of Newtonian Mechanics,” Am. J. Phys., 29, 617 (196]); R. Weinstock, “Laws of Classical Motion: What's F? What‘s m? What's 0?", Am. J. Phys., 29, 698 (1961). ’5' ..,- i. .,»..--t t); §.?i;>§;f:§£;:: 6—2 A cabin cruiser of mass 15 metric tons drifts in toward a dock at a Speed of 0.3 m/sec after its engines have been cut. (A metric ton is 103 kg.) A man on the dock is able to touch the boat when it is l m from the dock, and thereafter he pushes on it with a force of 700 N to try to stop it. Can he bring the boat to rest before it touches the dock? 6—3 (a) A man of mass 80 kg jumps down to a concrete patio from a window ledge only 0.5 m above the ground. He neglects to bend his knees on landing, so that his motion is arrested in a distance of about 2 cm. With what average force does this jar his bone structure? (b) If the man jumps from a ledge 1.5 m above the ground but bends his knees so that his center of gravity descends an additional distance 11 after his feet touch the ground, what must h be so that the average force exerted on him by the ground is only three times his normal weight? 6—4 An object of mass 2 kg is acted upon by the following combina- tion of forces in the xy plane: 5 N at 0 = 0, 10 N at 0 = 7r/4, and 20 N at 0 -—- 47r/3. The direction 0 = 0 corresponds to the +x direc- tion. At! = 0 the object is at the pointx = —6mandy = 3 m and has velocity components v1 = 2m/sec and 1),, = 4m/sec. Find the object’s velocity and position at t = 2 sec. 6~5 The graphs shown give information regarding the motion in the xy plane of three different particles. In diagrams (a) and (b) the small y Note! Parabolic \u (a) t . x I 0 (Vertical lines 0 .v equally spaced) .v Direction of motion X () (C) I )f mass 15 metric tons drifts in toward a dock : after its engines have been cut. (A metric ton he dock is able to touch the boat when it is l m reafter he pushes on it with a force of 700 N to ring the boat to rest before it touches the dock? ss 80 kg jumps down to a concrete patio from .5 m above the ground. He neglects to bend 0 that his motion is arrested in a distance of average force does this jar his bone structure? nps from a ledge 1.5 m above the ground but his center of gravity descends an additional touch the ground, what must I: be so that the n him by the ground is only three times his 2 kg is acted upon by the following combina- plane: 5N at 0 = 0,10N at 0 = 7r/4, and direction 6 = 0 corresponds to the +x direc- sat thepointx = n6mandy == 3 m and has = 2m/sec and Dy = 4m/sec. Find the tion at t = 2 sec. give information regarding the motion in the. t particles. In diagrams (a) and (b) the small Note! abolic \‘u, L .r 0 r l . . .V Direction 31‘ motion \ X 0 ’ (C) dots indicate the positions at equal intervals of time. For each case, write equations that describe the force components F,n and Fy. 6—6 An observer first measures the velocity of an approaching object to be 10—2 m/sec and then, lsec later, to be 2 X 10‘2 m/sec. No intermediate readings are possible because the observer’s instruments take a full second to determine a velocity. If the object has a mass of 5 g, what conclusions can the observer make about (a) The size of the force that had been acting? (b) The impulse supplied by the force? (c) The work done by the force? 6—7 A particle of mass 2 kg oscillates along the x axis according to the equation . 1r‘ —— 0.2 sm (5! -— E) where x is in meters and t in seconds. (a) What is the force acting on the particle at t = 0? (b) What is the maximum force that acts on the particle? 6—8 A car of mass 103 kg is traveling at 28 m/sec (a little over 60 mph) along a horizontal straight road when the driver suddenly sees a fallen tree blocking the road 100 m ahead. The driver applies the brakes as soon as his reaction time (0.75 sec) allows and comes to rest 9 m short of the tree. (a) Assuming constant deceleration caused by the brakes, what is the decelerating force? What fraction is it of the weight of the car (take g = 9.8 m/sec2)? (b) If the car had been on a downward grade of sin"(715) with the brakes supplying the same decelerating force as before, with what speed would the car have hit the tree? 6—9 A particle of mass m follows a path in the xy plane that is de- scribed by the following equations: x = A(at —— sin at) y = A(l — cosa!) (a) Sketch this path. (b) Find the time-dependent force vector that causes this motion. Can you suggest a way of producing such a situation in practice? 6'10 A piece of string of length l, which can support a maximum tension T, is used to whirl a particle of mass m in a circular path. What is the maximum Speed with which the particle may be whirled if the circle is (a) horizontal; (b) vertical? ...
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Chs 5 &amp;amp; 6 - nated by turbulence, however small the...

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