34. Mr. M. I believe you correctly understand the theory. The operations of every-day life afford numerous examples of the motion resulting from a composition of forces. If we attempt to row a boat directly across a rapid river, the action of the oars and the action of the current will result in a diagonal motion down the stream. In the science of projectiles, or of gunnery, it is necessary to take into consideration not only the force exerted by the powder, but of gravity, or the earth's attraction, also; for the cannon ball must take the direction of what is called the resultant of these two forces. This, however, brings us to the consideration of our next subject, which is GRAVITY AND FALLING BODIES; and on that you may prepare yourselves for our next conversation. 35. Here Master John remarked that Natural Philosophy was a most delightful study, because it led the mind to think about almost every thing. "And to think with some satisfaction, too," said George, "because it puts one in the way of learning real truths about things; and I think nothing is so satisfactory as truth." "I would like," said Ella, "to learn the truth about every thing in nature." "That is a very large wish," said Frank, "for it seems to me to be a wish to know every thing." "And that," said John, "is what Deity alone can know." 36. This was leading to quite a long discussion upon the nature of truth, when Mr. Maynard suggested that it might be well to postpone16 the consideration of that subject until they came to the departments of Mental and Moral Philosophy, which they would find treated in their Sixth Reader. The class then separated, and the several members proceeded to make preparations for their afternoon's rambles, which ing on a point. Two of the forces are first taken, and their resultant found. This resultant is combined with the third force, and a second resultant found. This again is combined with the fourth force, and so on, until the forces are exhausted. The final resultant represents the conjoint action of all. Thus, let there be three forces applied to the point α, represented in intensity and direction by the lines a b, a c, a d, respectively. If a b and a c be combined, they give as their resultant a e; and if this resultant, a e, be combined with the third force, a d, the resultant will be af, which, therefore, represents the common action of all three forces. d were so planned by their teacher as to have in view the acquisition of new truths in some of the departments of Natural History. 1 IN-TER'-RO-GA-TORS, those who ask ques-9 FRIC'-TION, the act of rubbing the surface tions. 2 AC-CEL-ER-A-TED, quickened in motion. 4 MO-MENT'-UM, the quantity of motion in a 5 PROD'-UCT, here means the result obtained by multiplying the quantity of matter by the velocity. 6 AB'-SO-LUTE, positive; real; not relative. 7 PAR-A-DOX, something seemingly absurd, but true in fact. 8 EX-TERN'-AL, outward; exterior. of one body against that of another. 11 IM-PRESS'ED, exerted; made to act. 14 OB-LIQ'-UI-TY, deviation from a perpen- 15 IN-TENS'-I-TY, degree of violence, energy, or power. 16 POST-PONE', put off; defer. LESSON VI. GRAVITY AND FALLING BODIES. 1. WHILE the class were on their way to the library, Miss Ida remarked that it was so pleasant out of doors that morning she wished Mr. Maynard would give them their lesson under the old oak-tree on the lawn. This suggestion was very favorably received by the class, and on arriving at the library, and making known their wishes to Mr. Maynard that he would give them an out-door lesson, he very cheerfully complied with their request. So they all proceeded to the oak-tree, where Mr. Maynard took his seat in a chair which Frank had brought for him, and the others on the rustic benches which were placed there. 2. Mr. M. On the ground you observe acorns which have fallen from the tree above us. Will you tell me why they very appropriately suggest the consideration of the subject of gravity? Ella. Because in falling from the tree to the earth they have illustrated the great law of falling bodies; and if there had been no such law as gravity, they would have been just as likely to go upward as downward. 3. Frank. I have another reason to give. I have seen it stated that while Newton was sitting alone in his garden, the falling of an apple from a tree suggested the inquiry, “Why did the apple fall?" and that this trifling circumstance led to his great discovery of the laws of gravity, and of their application to the motions of the heavenly bodies. I think these acorns very naturally suggest a similar inquiry. Mr. M. Reminded, by these evidences around us, of the constant operation of the laws of gravity, we will begin our lesson. In our last conversation John gave the example of a stone thrown upward as an illustration of retarded and accelerated motion. Can he now explain the cause of the increase of velocity in descending? 4. John. I have read that every body or mass of matter in the universe attracts every other mass, and that as the bodies approach each other the attraction is increased. I think this increase of attraction between the stone and earth, as they come nearer each other, is the cause of the accelerated motion of the stone in falling. 5. Mr. M. You have mistaken the cause of its accelerated motion; for, though it is true that the force of gravity increases as a body approaches the earth, the difference is so trifling at small distances from the surface as not to be perceptible. When a stone falls from a height, the impulse1 which it receives from gravitation in the first instant of its fall would be sufficient to bring it to the ground with a uniform velocity, even if the force of gravity were then taken away; but as the force of gravity is exerted during the next instant also, the stone then receives an additional impulse downward, and so during each succeeding instant, and thus the motion is uniformly accelerated. 6. John. It is perfectly plain that, while the first impulse continues, gravity is constantly acting, and thus the velocity of a falling body is increased. I would like to ask if a stone occupies the same time in going up as in coming down. 7. Mr. M. It does; for in going up its force is constantly diminished by gravity. Can you tell me how many feet a body will fall in one second of time? Ida. It is found by experiment that a body falls sixteen feet during the first second. Mr. M. How far the next second? 8. Ida. Forty-eight feet, making sixty-four feet in the two seconds. Ella. It appears to me, then, by the principles just stated, that out of the forty-eight feet which the body falls during the second second, sixteen must be owing to gravity, the same as in the first second, and the remaining thirty-two feet to the velocity which the body had acquired in falling the first sixteen feet. Is not that so? 9. Mr. M. That is the correct explanation. The laws of falling bodies may be shown by triangles, as in this diagram,2 6 5 2 1 Fig. 6. 2 3 4 C 6 Figure 6. The acceleration of a falling mass may be represented by the divergence of the two sides, a, b, and a, c. If we divide the large triangle into smaller ones by the lines 1, 1, 2, 2, etc., which may represent seconds, the bases of the triangles which we thus make will show the acceleration at any required time, and the areas of the several smaller triangles will represent the space fallen through. 5 10. In such case the area of each smaller triangle must be considered 16, the number of feet which the body falls by the force of gravity alone; and the base must be called 32, the velocity which a body attains in falling 16 feet by the force of gravity alone. You see, by the figure, that in the first second there is one triangle, or the body falls 16 feet, and has a velocity at the end of the time of 32 feet. In the next second it passes over the space of three triangles, or 48 feet, and has a velocity of 64 feet. In the third second we have five triangles, or 80 feet, and a velocity of 96 feet. In the same manner the velocity and spaces for any subsequent second in the fall of the body may be shown. 11. Frank. I thought the number of feet described5 during any portion of time was the product of the square of the time in seconds multiplied by 16, but I do not see how that follows from the figure. 12. George. I think I understand it. You have only to count the number of triangles above any line, and you will have the square of the time represented by that line. Thus, above the line 3 there are 9 triangles, and as each one is 16, the space above the line 3 is 9 times 16, or 144, which is the space a body falls in three seconds, though it only falls 80 feet in the third second. Do you not see 5 triangles between lines 3 and 2? 13. Frank. Is it possible that we can extend that figure as far as we please, and work any problems' of falling bodies by it? Mr. M. Let us try a problem. How high is a flag-staff, if an arrow, thrown as high as its top, is six seconds in the air? 14. George. It will be as long in going up as in coming down; hence it will be falling three seconds. Three squared is nine, and nine times sixteen are one hundred and fortyfour, the height of the staff. Mr. M. Correctly solved; but with what velocity was the arrow shot upward? 15. George. I see in the figure three bases of the small triangles, and as each one represents 32 feet, the three will be 96 feet, which the arrow acquired in falling; that must equal the velocity which was destroyed by gravity in its ascent. 16. Mr. M. Do you observe from the figure that when the bases are doubled, the whole number of triangles above such bases is quadrupled ?8 Thus, above the line 3 there are 9 triangles, and above the line 6 there are 4 times 9, or 36. George. Can we not prove from that, that if a person doubles his charge of powder he can shoot four times as far upward? 17. John. Yes; and if I start to run up stairs with double the velocity of another boy, I can go up with one half the muscular exertion. Frank. How can that be? John. Do you not see that I can go four times as far by doubling the velocity, but to double the speed I must use twice as much force? 18. Mr. M. I have listened with interest to your discussion, and am pleased to see you so readily apprehend the doctrine of falling bodies; and I recommend you to construct and study such figures as the one I have described to you. Ella. Do not heavy bodies fall swifter than light ones? |