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blow; what was the distance between us, allowing 70 pulses to a minute? Ans. 1 mile and 198 yards. EXAM. 5. How far off was the cloud from which thunder issued, whose report was 5 pulsations after the flash of lightning; counting 75 to a minute? Ans. 1523 yards. EXAM. 6. If I see the flash of a cannon, fired by a ship in distress at sea, and hear the report 33 seconds after, how far is she off? Ans. 7
PRACTICAL EXERCISES IN MECHANICS, STATICS, HYDROSTATICS, SOUND, MOTION, GRAVITY, PROJECTILES, AND OTHER BRANCHES OF NATURAL PHILOSOPHY.
QUESTION 1. REQUIRED the weight of a cast iron ball of 3 inches diameter, supposing the weight of a cubic inch of the metal to be 0.258lb avoirdupois ? Ans. 3.64739lb. QUEST. 2. To determine the weight of a hollow spherical iron shell, 5 inches in diameter, the thickness of the metal being one inch? Ans. 13.2387lb.
QUEST. 3. Being one day ordered to observe how far a battery of cannon was from me, I counted, by my watch, 17 seconds between the time of seeing the flash and hearing the report; what then was the distance? Ans. 3 miles.
QUEST. 4. It is proposed to determine the proportional quantities of matter in the earth and moon; the density of the former being to that of the latter, as 10 to 7, and their diameters as 7930 to 2160. Ans. as 71 to 1. nearly.
QUEST. 5. What difference is there, in point of weight, between a block of marble, containing 1 cubic foot and a half, and another of brass of the same dimensions?
Ans. 496lb 14oz.
QUEST. 6. In the walls of Balbeck in Turkey, the ancient Heliopolis, there are three stones laid end to end, now in sight, that measure in length 61 yards; one of which in particular is 21 yards or 63 feet long, 12 feet thick, and 12 feet broad: now if this block be marble, what power would balance it, so as to prepare it for moving?
tons, the burden of an East-India ship. QUEST. 7. The battering-ram of Vespasian weighed, suppose 10,000 pounds; and was moved, let us admit, with
such a velocity, by strength of hand, as to pass through 20 feet in one second of time; and this was found sufficient to demolish the walls of Jerusalem. The question is, with what velocity a 32lb ball must move, to do the same execution? Ans. 6250 feet.
QUEST. 8. There are two bodies, of which the one contains 25 times the matter of the other, or is 25 times heavier; but the less moves with 1000 times the velocity of the greater in what proportion then are the momenta, or forces, with which they moved?
Ans. the less moves with a force 40 times greater. QUEST. 9. A body, weighing 20lb, is impelled by such a force, as to send it through 100 feet in a second; with what velocity then would a body of 8lb weight move, if it were impelled by the same force? Ans. 250 feet per second.
QUEST. 10. There are two bodies, the one of which weighs 100lb, the other 60; but the less body is impelled by a force 8 times greater than the other; the proportion of the velocities, with which these bodies move, is required? Ans. the velocity of the greater to that of the less, as 3 to 40. QUEST. 11. There are two bodies, the greater contains 8 times the quantity of matter in the less, and is moved with a force 48 times greater: the ratio of the velocities of these two bodies is required?
Ans. the greater is to the less, as 6 to 1. QUEST. 12. There are two bodies, one of which moves 40 times swifter than the other; but the swifter body has moved only one minute, whereas the other has been in motion 2 hours: the ratio of the spaces described by these two bodies is required?
Ans. the swifter is to the slower, as 1 to 3.
QUEST. 13. Supposing one body to move 30 times swifter than another, as also the swifter to move 12 minutes, the other only 1: what difference will there be between the spaces described by them, supposing the last has moved 5 feet? Ans. 1795 feet.
QUEST. 14. There are two bodies, the one of which has passed over 50 miles, the other only 5; and the first had moved with 5 times the celerity of the second; what is the ratio of the times they have been in describing those spaces? Ans. as 2 to 1.
QUEST. 15. If a lever, 40 effective inches long, will, by a certain power thrown successively on it, in 13 hours, raise a weight 104 feet; in what time will two other levers,
each 18 effective inches long, raise an equal weight 1 feet?
Ans. 10 hours 8 minutes.
QUEST. 16. What weight will a man be able to raise, who presses with the force of a hundred and a half, on the end of an equipoised handspike, 100 inches long, meeting with a convenient prop exactly 74 inches from the lower end of the machine? Ans. 2072lb. QUEST. 17. A weight of 141b, laid on the shoulder of a man, is no greater burden to him than its absolute weight, or 24 ounces: what difference will he feel, between the said weight applied near his elbow, at 12 inches from the shoulder, and in the palm of his hand, 28 inches from the same; and how much more must his muscles then draw, to support it at right angles, that is, having his arm stretched right out? Ans. 24lb avoirdupois.
QUEST. 18. What weight hung on at 70 inches from the centre of motion of a steel-yard, will balance a small gun of 9cwt, freely suspended at 2 inches distance from the said centre on the contrary side? Ans. 30 lb.
QUEST. 19. It is proposed to divide the beam of a steelyard, or to find the points of division where the weights of 1, 2, 3, 4, &c, lb, on the one side, will just balance a constant weight of 95lb at the distance of 2 inches on the other side of the fulcrum; the weight of the beam being 10lb, and its whole length 36 inches?
Ans. 30, 15, 10, 71, 6, 5, 42, 34, 3, 3, 2, 2%, &c. QUEST. 20. Two men carrying a burden of 200lb weight between them, hung on a pole, the ends of which rest on their shoulders; how much of this load is borne by each man, the weight hanging 6 inches from the middle, and the whole length of the pole being 4 feet?
Ans. 125lb and 75lb.
QUEST. 21. If, in a pair of scales, a body weigh 90lb in one scale, and only 40lb in the other; required its true weight, and the proportion of the lengths of the two arms of the balance beam, on each side of the point of suspension ?
Ans. the weight 60lb, and the proportion 3 to 2.
QUEST. 22. To find the weight of a beam of timber, or other body, by means of a man's own weight, or any other weight. For instance, a piece of tapering timber, 24 feet long, being laid over a prop, or the edge of another beam, is found to balance itself when the prop is 13 feet from the less end; but removing the prop a foot nearer to the said end, it takes a man's weight of 210lb, standing on the less
end, to hold it in equilibrium. Required the weight of the tree? Ans. 2520lb.
QUEST. 23. If AB be a cane or walking-stick, 40 inches long, suspended by a string SD fastened to the middle point D: now a body being hung on at E, 6 inches distance from D, is balanced by a weight of 2lb, hung on at the larger end A; but removing the body to F, one inch nearer to D, the 2lb weight on the other side is moved to G, within 8 inches of D, before the cane will rest in equilibrio. Required the weight of the body? Ans. 24lb.
QUEST. 24. If AB, BC be two inclined planes, of the lengths of 30 and 40 inches, and moveable about the joint at B: what will be the ratio of two weights P, Q, in equi librio on the planes, in all positions of them: and what will be the altitude BD of the angle в above the horizontal plane AC, when this is 50 inches long?
Ans. ED = 24; and p to Q as AB to BC, or as 3 to 4. QUEST. 25. A lever, of 6 feet long, is fixed at right angles in a screw, whose threads are one inch asunder, so that the lever turns just once round in raising or depressing the screw one inch. If then this lever be urged by a weight or force of 50lb, with what force will the screw press?
QUEST. 26. If a man can draw a weight of 150lb up the side of a perpendicular wall, of 20 feet high; what weight will he be able to raise along a smooth plank of 30 feet long, laid aslope from the top of the wall? Ans. 225lb.
QUEST. 27. If a force of 150lb be applied on the head of a rectangular wedge, its thickness being 2 inches, and the length of its side 12 inches; what weight will it raise or balance perpendicular to its side? Ans. 900lb.
QUEST. 28. If a round pillar of 30 feet diameter be raised on a plane, inclined to the horizon in an angle of 75o, or the shaft inclining 15 degrees out of the perpendicular; what length will it bear before it overset ?
Ans. 30 (23) or 111.9615 feet.
QUEST. 29. If the greatest angle at which a bank of na tural earth will stand, be 45°; it is proposed to determine what thickness an upright wall of stone must be made throughout, just to support a bank of 12 feet high; the specific gravity of the stone being to that of earth, as 5 to 4.
Ans. 4, or 4 29325 feet.
QUEST. 30. If the stone wall be made like a wedge, or having its upright section a triangle, tapering to a point at
top, but its side next the bank of earth perpendicular to the horizon; what is its thickness at the bottom, so as to support the same bank ? Ans. 12, or 5.36656 feet.
QUEST. 31. But if the earth will only stand at an angle of 30 degrees to the horizontal line; it is required to determine the thickness of wall in both the preceding cases?
Ans. the breadth of the rectangle 12, or 5.36656, but the base of the triangular bank 12, or 6.53667. QUEST. 32. To find the thickness of an upright rectangular wall, necessary to support a body of water; the water being 10 feet deep, and the wall 12 feet high; also the specific gravity of the wall to that of the water, as 11 to 7.
Ans. 4.204374 feet. QUEST. 33. To determine the thickness of the wall at the bottom, when the section of it is triangular, and the altitudes as before. Ans. 5 1492866 feet.
QUEST. 34. Supposing the distance of the earth from the sun to be 95 millions of miles; I would know at what distance from him another body must be placed, so as to receive light and heat quadruple to that of the earth.
Ans. at half the distance, or 474 millions. QUEST. 35. If the mean distance of the sun from us be 106 of his diameters; how much hotter is it at the surface of the sun, than under our equator?
QUEST. 36. The distance between the earth and the sun being accounted 95 millions of miles, and between Jupiter and the sun 495 millions; the degree of light and heat received. by Jupiter, compared with that of the earth, is required?
Ans., or nearly of the earth's light and heat. QUEST. 37. A certain body on the surface of the earth. weighs a cwt, or 112lb; the question is whither this body must be carried, that it may weigh only 10lb?
Ans. either at 3.3466 semi-diameters, or of a semidiameter, from the centre. QUEST. 38. If a body weigh 1 pound, or 16 ounces, on the surface of the earth; what will its weight be at 50 miles above it, taking the earth's diameter at 7930 miles?
Ans. 15oz. 9 dr. nearly.
QUEST. 39. Whereabouts, in the line between the earth and moon, is their common centre of gravity; supposing the earth's diameter to be 7930 miles, and the moon's 2160; also