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Of The PILING OF BALLS AND SHELLS. IRON Balls and Shells are commonly piled by horizontal courses, either in a pyramidical or in a wedge-like form; the base being either an equilateral triangle, or a square, or a rectangle. In the triangle and square, the pile finishes in a single ball; but in the rectangle, it finishes in a single row of balls, like an edge.
In triangular and square piles, the number of horizontal rows, or courses, is always equal to the number of balls in one side of the bottom row. And in rectangular piles, the number of rows is equal to the number of balls in the breadth of the bottom row. Also, the number in the top row, or edge, is one more than the difference between the length and breadth of the bottom row.
To find the Number of Balls in a Triangular Pile. MULTIPLY continually together the number of balls in one side of the bottom row, and that number increased by 1, also the same number increased by 2; then of the last product will be the answer.
n+1.1 + 2 That is,
is the number or sum, where
6 n is the number in the bottom row.
Exam. 1. Required the number of balls in a triangular pile, each side of the base containing 30 balls ? Ans. 4960,
EXAM. 2. How many balls are in the triangular pile, each side of the base containing 20 ?
To find the Number of Balls in a Square Pile. Multiple continually together the number in one side of the bottom course, that number increased by 1, and double the same number increased by 1; then of the last product will be the answer.
n+1. 2n + 1 That is,
is the number. 6
Exam. 1. How many balls are in a square pile of 30 rows?
Ans. 9455. Exam. 2. How many balls are in a square pile of 20 rows ? ·
To find the Number of Balls in a Rectangular Pile. FROM 3- times the number in the length of the base row, subtract one less than the breadth of the same, multiply the remainder by the same breadth, and the product by one more than the same; and divide by 6 for the answer.
b.b+1.31 – b + 1 That is,
is the number; where l is
6 the length, and b the breadth of the lowest course.
Note. In all the piles the breadth of the bottom is equal to the number of courses. And in the oblong or rectangular pile, the top row is one more than the difference between the length and breadth of the bottom.
Exam. 1. Required the number of balls in a rectangular pile, the length and breadth of the base row being 46 and 15?
Ans. 4960. Exam. 2. How many shot are in a rectangular complete pile, the length of the bottom course being 59, and its breadth 20?
To find the Number of Balls in an Incomplete Pile. From the number in the whole pile, considered as complete, subtract the number in the upper pile which is wanting at the top, both computed by the rule for their proper form; and the remainder will be the number in the frustum, or incomplete pile.
Exam. 1. To find the number of shot in the incomplete triangular pile, one side of the bottom course being 40, and the top course 20?
Exam. 2. How many shot are in the incomplete triangular pile, the side of the base being 24, and of the top 8?
Ans. 2516. Exam. 3. How many balls are in the incomplete square pile, the side of the base being 24, and of the top 8 ?
Ans. 4760. EXAM. 4. How many shot are in the incomplete rectangular pile, of 12 courses, the length and breadth of the base being 40 and 20 ?
OF DISTANCES BY THE VELOCITY OF SOUND.
By various experiments it has been found, that sound flies, through the air, uniformly at the rate of about 1142 feet in 1 second of time, or a mile in 4 or 4 seconds. . And therefore, by proportion, any distance may be found corresponding to any given time; namely, multiplying the given time, in seconds, by 1142, for the corresponding distance in feet; or taking it of the given time for the distance in miles. Or dividing any given distance by these numbers, to find the corresponding time,
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 51 pulsations to a mile; and more or less aecording to circumstances.
Exam. 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. 24 miles. Exam. 2. How long, after firing the Tower guns, may the report be heard at Shooter's-Hil, supposing the distance to be 8 miles in a straight line ?
Ans. 374 seconds. Exam. 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 ] mile. Exam. 4. Perceiving a man at a distance hewing down a tree with an axe, I remarked that 6 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?
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. 77. miles.
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 0.2581b avoirdupois ? Ans. 3•647391b.
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. 13.23871b. 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?
Ans, 68316 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 l. 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,