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OF THE WEIGHT AND DIMENSIONS OF BALLS AND SHELLS.

THE weight and dimensions of Balls and Shells might be found from the problems given under the head of specific gravity. But they may be found still easier by means of the experimental weight of a ball of a given size, from the known proportion of similar figures, namely, as the cubes of their diameters, or like linear dimensions.

PROBLEM I.

To find the weight of an iron ball from its diameter.

An iron ball of 4 inches diameter weighs 9lb. and the weights being as the cubes of the diameters, it will be, as 64 (which is the cube of 4) is to 9 its weight, so is the cube of the diameter of any other ball, to its weight. Or, take of the cube of the diameter, for the weight. Or, take of the cube of the diameter, and of that again, and add the two together, for the weight. Or, d = 3/3w+} 3/3w.

Ex. 1. The diameter of an iron shot being 6-7 inches, required its weight? Ans. 42.294lb.

Ex. 2. What is the weight of an iron ball, whose diameter is 5'54 inches? Ans. 24lb. nearly.

PROBLEM II.

To find the weight of a leaden ball.

A leaden ball of 1 inch diameter weighs of a pound; therefore as the cube of 1 is to, or as 14 is to 3, so is the cube of the diameter of a leaden ball, to its weight. Or, take of the cube of the diameter, for the weight, nearly. Ex. 1. Required the weight of a leaden ball of 6.6 inches diameter ?

Ans. 61.606lb.

Ex. 2. What is the weight of a leaden ball of 5.30 inches diameter ?

Ans. 32lb. nearly.

Ex. 3. How many shot, each of an inch diameter, may be made out of 10lb. of lead? Ans. 2986667.

PROBLEM III.

To find the diameter of an iron ball.

Multiply the weight by 73, and the cube root of the product will be the diameter.

Ex. 1. Required the diameter of a 421b. iron ball?

Ex. 2. What is the diameter of a 24lb. iron ball?

PROBLEM IV.

To find the diameter of a leaden ball.

Ans. 6.685 inches.

Ans. 5.54 inches.

Multiply the weight by 14, and divide the product by 3; then the cube root of the quotient will be the diameter.

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Ex. 1. Required the diameter of a 64lb. leaden ball?

Ans. 6·684 inches.

Ans. 3.343 inches.

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Take of the difference of the cubes of the external and internal diameter, for the weight of the shell.

That is, from the cube of the external diameter, take the cube of the internal diameter, multiply the remainder by 9, and divide the product by 64.

Ex. 1. The outside diameter of an iron shell being 12.8, and the inside diameter 9.1 inches; required its weight? Ans. 188 941lb. Ex. 2. What is the weight of an iron shell, whose external and internal diameters are 9.8 and 7 inches? Ans. 84lb.

PROBLEM VI.

To find how much powder will fill a shell.

Divide the cube of the internal diameter, in inches, by 57.3, for the lbs. of powder *.

Ex. 1. How much powder will fill a shell whose internal diameter is 9'1 inches ? Ans. 13 lb. nearly.

Ex. 2. How much powder will fill a shell whose internal diameter is 7 inches? Ans. 6lb.

PROBLEM VII.

To find how much powder will fill a rectangular box.

Find the content of the box in inches, by multiplying the length, breadth, and depth altogether. Then divide by 30 for the pounds of powder.

Ex. 1. Required what quantity of powder will fill a box, the length being 15 inches, the breadth 12, and the depth 10 inches?

Ans. 60lb.

Ex. 2. How much powder will fill a cubical box whose side is 12 inches?

Ans. 573lb.

PROBLEM VIII.

To find how much powder will fill a cylinder.

Multiply the square of the diameter by the length, then divide by 38.2 for the pounds of powder.

Ex. 1. How much powder will the cylinder hold, whose diameter is 10 inches, and length 20 inches? Ans. 52 lb. nearly.

Ex. 2. How much powder can be contained in the cylinder whose diameter is 4 inches, and length 12 inches? Ans. 5,5lb.

* This and the following are only approximative rules, founded upon the supposition that, at a medium, 30 cubic inches of gunpowder weigh a pound. Of 18 different kinds of gunpowder used in the Royal Laboratory, Woolwich, the weights vary from 58lb. loz. to 491b. 13oz. per cubic foot, and the specific gravities, consequently, from 929 to 727. The specific gravity of French gunpowder usually lies between narrower limits; viz. those of 944 and 897.

PROBLEM IX.

To find the size of a shell to contain a given weight of powder.

Multiply the pounds of powder by 57.3, and the cube root of the product will be the diameter in inches.

Ex. 1. What is the diameter of a shell that will hold 13lb. of powder ?

Ans. 9.1 inches.

Ex. 2. What is the diameter of a shell to contain 6lb. of powder ?

Ans. 7 inches.

PROBLEM X.

To find the size of a cubical box, to contain a given weight of powder. Multiply the weight in pounds by 30, and the cube root of the product will be the size of the box in inches.

Ex. 1. Required the side of a cubical box, to hold 50lb. of gunpowder ?

Ans. 11.44 inches.

Ex. 2. Required the side of a cubical box, to hold 400lb. of gunpowder ?
Ans. 22.89 inches.

PROBLEM XI.

To find what length of a cylinder will be filled by a given weight of gunpowder. Multiply the weight in pounds by 38.2, and divide the product by the square of the diameter in inches for the length.

Ex. 1. What length of a 36-pounder gun, of 63 inches diameter, will be filled with 12lb. of gunpowder ? Ans. 10.314 inches. Ex. 2. What length of a cylinder, of 8 inches diameter, may be filled with 20lb. of powder ? Ans. 11 inches.

OF DISTANCES BY THE VELOCITY OF SOUND.

FROM various experiments recently made, with great care, by the editor of this volume, it has been found that sound flies through the air uniformly at the rate of about 1110 feet per second, when the air is quiescent, and at a medium temperature. At the temperature of freezing, or a little below, the velocity is 1100 feet; at the temperature of 75°, on Fahrenheit's thermometer, the velocity is about 1120. The approximate velocity under different temperatures may be found, by adding to 1100, half a foot, for every degree, on Fahrenheit's thermometer, above the freezing point. The mean velocity may be taken at 370 yards per second; or a mile in 43 seconds.

Hence, multiplying any time employed by sound in moving, by 370, will give the corresponding space in yards. Or, dividing any space in yards by 370, will give the time which sound will occupy in passing uniformly over that space.

If the wind blow briskly, as at the rate of from 20 to 60 feet per second, in the

direction in which the sound moves, the velocity of the sound will be proportionably augmented: if the direction of the wind is opposed to that of the sound, the difference of their velocities must be employed.

Note. The time for the passage of sound in the interval between seeing the flash of a gun, or lightning, and hearing the report, may be observed by a watch, or a small pendulum. Or, it may be observed by the beats of the pulse in the wrist, counting, on an average, about 70 to a minute for persons in moderate health, or 53 pulsations to a mile; and more or less according to circumstances. Ex. 1. After observing a flash of lightning, it was 12 seconds before the thunder was heard; required the distance of the cloud from whence it came? Ans. 2.52 miles. Ex. 2. How long, after firing the Tower guns, may the report be heard at Shooter's-Hill, supposing the distance to be 8 miles in a straight line?

Ans. 383 seconds.

Ex. 3. After observing the firing of a large cannon at a distance, it was 7 seconds before the report was heard; what was its distance? Ans. 1.47 mile. Ex, 4. Perceiving a man at a distance hewing down a tree with an axe, I remarked that four of my pulsations passed between seeing him strike and hearing the report of the blow; what was the distance between us, allowing 70 pulses to a minute ?

Ex. 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?

Ex. 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?

A FEW EXERCISES IN MECHANICS, STATICS, 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 0.2581b. 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. 13.78lb.

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. 35 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? Ans. 683 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 321b. 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 move? Ans. the less moves with a force 40 times greater.

Quest. 9. A body, weighing 201b. is impelled by such a force, as to send it through 100 feet in a second; with what velocity then would a body of 81b. 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. It is proposed to divide the beam of a steel-yard, 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, 38, 31, 3, 21, 21, &c.

Quest. 16. 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. 17. 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

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