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Ella. Please explain your work for our benefit, and not come to the conclusion so suddenly.
8. John. The weight 500, multiplied by its distance 2, is 1000. The product of the power by its distance must be equal to 1000. But the distance of the power is 5, hence the other factor, or the power, will be found by dividing 1000 by 5, which will give 200.
Ella. Are the calculations for all kinds of levers made so easily?
9. Mr. M. I am most happy to assure you that not only are all calculations pertaining to the lever thus simple, but also all calculations of the other simple mechanical powers. Do you understand this expression, P× Pd=W ×Wd?*
10. George. I think it must mean that the product of power by power's distance from the fulcrum is equal to the product of the weight by the weight's distance from the fulcrum.
11. Mr. M. That is the law for equilibrium; but to produce motion the power must exceed that necessary for equilibrium or balancing. Universally, the product of the power by the distance it moves is always equal to the product of the weight by the distance it moves in a vertical direction. Whenever you have any difficulty in solving questions in mechanical powers, think of this principle.
12. John. Does not the weight of the long end of the lever interfere with this rule? I saw some engineers once weighing the lever of a safety-valve, and heard them say the rule for calculating levers would not do for them.
Mr. M. Very true; the weight of the lever is a part of the power, and should be so calculated. In the formulas I have given you the lever is considered as without weight.
13. John. As all levers do really have weight, will you please show us how to estimate that weight in practice?
Mr. M. Have you not been able to find the information you seek in any school-books or mechanics' manuals ??
John. No, sir. I have searched diligently even in college text-books in your library.
14. Mr. M. I really can not point you to the book where
* This should be read, "P multiplied by Pd equals W multiplied by Wd."
you will find what you wish, and what is so important, but I think it can be made very plain. We will use this diagram for our illustration.
15. Suppose the lever to be a bar of iron sixteen inches long, every inch of which weighs one pound, and that the fulcrum, F, is six inches from the weight, W. The centre of gravity of the short arm will be three inches from the fulcrum, where the weight will be six pounds. The centre of the long arm will be five inches from the fulcrum, where its weight will be ten pounds. Now we have only to calculate the short end as an additional weight of six pounds three inches from the fulcrum, and the weight of the long arm as a power of ten pounds five inches distant, and combine these with the theoretical10 calculation.
16. John. I think I can now accomplish what I have heard many mechanics wish themselves able to do. The problem does not seem to be a very difficult one.
Mr. M. Will you tell me, then, with such a lever, what power at P will balance 100 pounds at W?
17. John. If we multiply 100 by 6 (six inches), we have 600. Then 6 pounds, the weight of the short arm of the lever, multiplied by 3 (three inches), will give 18, which, added to 600, will make 618, for the products of the weights by their distances. Then, for the long arm of the lever, we multiply the weight 10 by its distance 5, and take the product, 50, from 618, and this will leave 568 pounds to be balanced by a weight at P; but, as P is ten inches from the fulcrum, we divide 568 by 10, and this gives us 56 pounds and eight tenths of a pound.
18. Mr. M. You are correct in your answer. Fifty-six pounds and eight tenths of a pound at P will balance one hundred pounds at W. Can you tell me what would have been the theoretical answer?
Ida. I have already made the calculation, and I find, if we
suppose the bar or lever not to have any weight, 60 pounds at P will balance 100 at W.
19. Mr. M. Thus, you see, there is a difference of over three pounds. If the lever is not a straight and uniform bar, the distance of the centres of gravity of its arms must be calculated by means we can not introduce here.
Ida. I used to learn about three kinds of levers. Can the power of all of them be calculated in the same way?
20. Mr. M. Yes. Their parts are essentially the same; viz., the power and its distance, and the weight and its distance from the centre of motion; and the formula I gave will solve them all. Can you tell me what constitutes a lever of the second kind?
21. George. The second kind of lever is that in which the weight and the power are on the same side of the fulcrum, and the power is furthest from the fulcrum.
Thus, if a mason desires to move forward a large piece of stone, instead of bearing down upon the lever to raise it up a little, he sticks his crowbar into the ground, and pushing upward, moves the stone little by little onward, the ground being the fulcrum.
22. John. Is not a common wheelbarrow a kind of lever of this kind?
Mr. M. It is a lever on a rolling fulcrum. So, also, is the oar of a boat, the water being the fulcrum, the person who rows the power, and the boat itself the resistance.
23. Frank. It seems to me that the masts of a ship are levers.
Mr. M. So they are; and also the rudders by which ships are steered. Can the young ladies give me some examples of levers either of the first or second kind?
Ida. Nut-crackers and lemon-squeezers are levers of the
Ella. Scissors, forceps, and snuffers are double levers of the first kind.
24. Mr. M. Well said; for when you readily state the kind of levers, I think you understand what is the fulcrum, power, and weight. The scale-beam used in weighing is a simple O
you give me a description of the third kind of lever? 25. Frank. The third kind of lever is that in which the fulcrum is at one end, the weight at the other, and the power placed between them. A man raising a ladder which rests on one end is an example; so, also, are fire-tongs. I have read that at one time this was called the losing lever, because the power had to be greater than the weight; but the advantages of it are that a small power causes the extreme point of a long arm to move over a great space.
26. Mr. M. Yes; and it is one of those wonderful adaptations of the Divine Being in the construction of the limbs of animals. This arrangement is seen in all its beauty in the wings of birds, whose muscles are sometimes very powerful, sustaining the weight of their bodies while they travel unrested for days amid the tempests of the heavens.
27. Ella. We have had examples of this kind of lever in our reading-lessons in Ornithology, in the instances of the longsustained flights of the stormy petrel and the albatross, and of the wild or passenger pigeon; and I see now that a knowledge of the lever makes those cases all the more interesting.
Mr. M. And perhaps some one of you can find an example of the same kind of lever in your reading-lessons on Human Physiology.
lever. The arms, a a, on each side, are made of equal length, and suspended over the centre of gravity. The axis11 or pivot, b, which is the point of suspension, is sharpened to a very thin edge, that the beam may easily turn with as little friction as possible when weights are applied in the scales. Can
28. Ida. Oh yes, here it is, on the twenty-second
Mr. M. You are right; and from the principles already learned you will perceive that if the weigh in the hand be fifty pounds, and be raised twenty inches while the muscles springing from the shoulder contract one inch, the force exerted by the muscles must be equal to one thousand pounds. John. I believe a horse draws a cart on the principle of the lever.
Mr. M. Yes; the weight of the horse is the power, especially in drawing up hill, and his hind feet constitute the fulcrum.
29. George. I understand now why, when one of our horses could not draw a cart up a short steep hill, the driver got on the forward part of the cart, and the horse drew up the man and cart, when he could not draw the cart alone. The weight of the man bearing on the back of the horse added to the power.
30. Mr. M. Is it not delightful to trace the causes of things? Gates, doors, and chests furnish us
illustrations of the principles we have been considering. You will find that a thousand common things are accomplished on the principle of the lever. But we must proceed to the pulley; so what can you say of this power?
31. Frank. It is a small grooved wheel, with a cord passing over it.
Mr. M. What is gained by such a pulley?
Frank. Nothing but change of direction. A man, by pulling down, can raise a weight up.
John. Yes, and several men can join their strength at the same time to raise a very large weight.
32. George. By putting another pulley near the ground, horses and oxen can walk off horizontally and raise weights vertically. When I went to Washington last summer, I saw horses raising large blocks of marble, and iron pillars, to a great height on the Capitol.