tions of a given pendulum are performed in very nearly the same time, whether it moves through longer or shorter arcs. Thus, in Fig. 54, if the pendulum A B were raised only to E, it would be as long in swinging from E to F as from C to D. The shorter the arc, therefore, the slower its motion. It is on this principle that a swing, when first set in motion, goes very slowly, but increases in velocity as it is pushed higher and higher. 141. Second Law.-The vibrations of pendulums of different length are performed in different times; and their lengths are proportioned to the squares of their times of vibration. One pendulum vibrates in 2 seconds, another in 4. Then the latter will be four times as long as the former; because they will be to each other as the square of 2 is to the square of 4,-that is, as 4 is to 16. Hence, to have its time of vibration doubled, a pendulum must be made 4 times as long; to have it tripled, 9 times as long; to have it quadrupled, 16 times as long, &c. A pendulum, to vibrate only once in a minute, would have to be 60 times 60, that is 3,600, times as long as one that vibrates once in a second,over 2 miles. -or a little Conversely, the times in which different pendulums vibrate are to each other as the square roots of their length. If one pendulum be 16 feet long and another 4, the former will be twice as long in vibrating as the latter; for their times of vibration are to each other as the square root of 16 is to the square root of 4,- -or as 4 to 2. 142. Third Law.-The vibrations of the same pendulum are not performed in the same time at all parts of the earth's surface; but, being caused by gravity, differ slightly, like gravity, according to the distance from the earth's centre. On the top of a mountain five miles high, for instance, a pendulum vibrating seconds would make 10 less vibrations in an hour than at the level of the sea, because it would be farther from the earth's centre. At either pole, a second-pendulum would make 13 more vibrations in an hour than at the equator, because it is nearer the centre, the earth being flattened at the poles. Hence the vibrations of the pendulum afford a means of measuring heights. 140. What is the first law relating to the pendulum? Illustrate this with Fig. 56. 141. What is the second law? Apply this law in an example. When the lengths of different pendulums are known, how can we find the relative times of vibration? If we have two pendulums, 16 and 4 feet long, how will their times of vibration compare? 142. What is the third law? What is the difference in the number of vibrations in a second-pendulum at the level of the sea and at an elevation of five miles? How would the number of vibrations at the pole compare with those at the equator? They also confirm what we have learned, that the polar diameter of the earth is 26 miles shorter than its equatorial diameter. In the latitude of New York, a pendulum, to vibrate seconds, must be about 39/10 inches long; whereas at Spitzbergen, in the far North, it must be a little over 391/5, and at the equator exactly 39 inches. 143. APPLICATION OF THE PENDULUM TO CLOCK-WORK. -Galileo, to whom science owes so much, was the first to think of turning the pendulum to a practical use. Observing that a chandelier suspended from the ceiling of a church in Pisa, when moved by the wind, vibrated in exactly the same time, whether carried to a greater or less distance, he at once saw that a similar instrument might be employed in measuring small intervals of time in astronomical observations. To adapt it to this use, it was necessary to invent some way of counterbalancing the constant loss of motion caused by friction and the air's resistance. This was done by the Dutch astronomer Huygens [hi'-genz], who in the year 1656 first applied the pendulum to clock-work. To this great invention modern astronomy owes its precision of observation, and consequently much of the progress it has made. 144. As a pendulum vibrating seconds, which is over 39 inches long, would be inconvenient in clocks, it is customary to use one that vibrates half-seconds; which, according to the principles laid down in § 141, is one-fourth as long, or a little less than 10 inches. 145. At the same distance from the equator, the same elevation above the sea, and the same temperature, a pendulum of given length will always vibrate in exactly the same time, and a clock regulated by a pendulum will keep uniform time. If taken from the equator towards the poles, the pendulum will vibrate more rapidly, and the clock What is the length of a second-pendulum at New York? At Spitzbergen? At the equator? 143. Who first thought of turning the pendulum to a practical use? Relate the circumstance that led him to do so. To enable it to measure small intervals of time, what was first necessary? Who did this, and thus first applied the pendulum to clock-work? 144. What is the length of the pendulums generally used in clocks? 145. Under what circumstances will a pendulum always vibrate in the same will go too fast. If taken up a mountain, the pendulum will vibrate less rapidly, and the clock will go too slow. If expanded by the heat of summer (for such we shall hereafter learn is the effect of heat), the pendulum will also vibrate less rapidly, and the clock will go too slow. 146. THE GRIDIRON PENDULUM.-To prevent a clock from being affected by heat and cold, the Compensation Pendulum is used. Fig. 55. GRIDIRON PENDULUM. One form of the Compensation Pendulum, known as the Gridiron Pendulum, is represented in Fig. 55. It consists of a frame of nine bars, alternately of steel and brass. These are so arranged that the steel bars, being fastened at the top, have to expand downward; while the brass ones, fastened at the bottom, expand upward. The expansive power of brass is to that of steel as 100 to 61; therefore, if the length of the steel bars is made 10061 the length of the brass bars, the expansion of the one metal counterbalances that of the other, and the pendulum always remains of the same length. The steel bars in the figure are represented by heavy black lines; the brass ones, by close parallel lines. 147. A clock is regulated by lengthening or shortening its pendulum. This is done by screwing the ball up or down on the rod. The ball is lowered when the clock goes too fast, and raised when it goes too slow. EXAMPLES FOR PRACTICE. 1. (See Fig. 45, and §§ 107, 109.) What would be the weight (that is, the measure of the earth's attraction) of an iceberg containing 40,000 tons of ice, if raised to a height of 1,000 miles above the earth's surface? What would it weigh 1,000 miles beneath the earth's surface? 2. A horse at the earth's surface weighs 1,200 pounds; what would he weigh 4,000 miles above the surface? How far beneath the surface would he have to be sunk, to have the same weight? 3. A Turkish porter will carry 800 pounds; how many such pounds could he carry, if he were placed half way between the surface and the centre of the earth, and retained the same strength ?—Ans. 1,600. How many such pounds could he carry, if elevated 4,000 miles above the surface with the same strength? time? What will cause it to vibrate more rapidly, and what less? 146. To prevent a clock from being affected by heat and cold, what is used? Describe the Gridiron Pendulum. 147. How is a clock regulated? 4. What would a body weighing 100 pounds at the earth's surface weigh 1,000 miles above the surface? What would it weigh 1,000 miles below the surface? 5. Would an 18-pound cannon-ball weigh more or less, 2,000 miles above the earth's surface, than 2,000 miles below it,-and how much? 6. At the centre of the earth, what would be the difference of weight between a man weighing 200 pounds at the surface and one weighing 100 pounds Four thousand miles above the surface, what would be the difference in their weight? 7. (See Rule 1, § 121.—In the examples that follow, no allowance is made for the resistance of the air.) A man falls from a church steeple; how many feet will he pass through in the third second of his descent? 8. How far will a stone fall in the twelfth second of its descent? 9. (See Rule 2, § 121.) How great a velocity does a falling stone attain in 7 seconds? 10. A hail-stone has been falling one-third of a minute; what is its velocity? 11. (See Rule 3, § 121.) How far will a stone fall in 10 seconds? 12. How far will a hail-stone fall in one-third of a minute? 13. I drop a pebble into an empty well, and hear it strike the bottom in exactly two seconds. How deep is the well? How many feet does the pebble fall in the first second of its descent? How many, in the second? What velocity has the pebble at the moment of striking? 14. A musket-ball dropped from a balloon continues falling half a minute before it reaches the earth; how high is the balloon, and what is the velocity of the ball when it reaches the earth? 15. What is the velocity of a stone dropped into a mine, after it has been falling 7 seconds, and how far has it descended?. 16. (See 122.) What would be the velocity of the same stone at the end of the seventh second, if thrown into the mine with a velocity of 20 feet in a second, and how far would it have descended? 17. An arrow falls from a balloon for 9 seconds. How far does it fall altogether, how far in the last second, and what velocity does it attain? What would these three answers be, if the arrow were discharged from the balloon with a velocity of 10 feet per second? 18. (See § 125.) How long will a ball projected upwards with a velocity of 1282/3 feet per second, continue to ascend? How great a height will it attain? What will be its velocity, after it has been ascending one second? After two seconds? After three seconds? 19. How many seconds will a musket-ball, shot upward with a velocity of 225 feet in a second, continue to ascend? How many feet will it rise? 20. A stone thrown up into the air rises two seconds; with what velocity was it thrown? 21. (See § 141.) How many times longer must a pendulum be, to vibrate only once in a second, than to vibrate three times in a second? 22. Two pendulums at the Cape of Good Hope vibrate respectively in 40 seconds and 10 seconds; how many times longer is the one than the other? 23. Two pendulums at New Orleans vibrate in 40 seconds and 10 seconds; how many times longer is one than the other? 24. In the latitude of New York, a pendulum vibrating seconds is 39/10 inches in length; how long must one be, to vibrate once in 10 seconds? -Ans. 3,910 inches. How long must one be, to vibrate 4 times in a second at the same place? -Ans. 271/160 inches. 25. At the equator, a pendulum 39 inches long vibrates once in a second; how long must a pendulum be, to vibrate once in half an hour at the same place? How long must one be, to vibrate 10 times in a second? 26. At Trinidad, a seconds-pendulum must be about 391/50 inches long; what would be the length of one that would vibrate 3 times in a second? What would be the length of one that would vibrate 3 times in a minute? What would be the length of one that would vibrate 3 times in an hour? CHAPTER VI. MECHANICS (CONTINUED). CENTRE OF GRAVITY. 148. THE Centre of Gravity of a body is that point about which all its parts are balanced. The centre of gravity is nothing more than the centre of weight. Cut a body of uniform density in two, by a plane passed in any direction through its centre of gravity, and the parts thus formed will weigh exactly the same. The whole weight of a body may be regarded as concentrated in its centre of gravity. 149. The Centre of Gravity must be carefully distinguished from the Centre of Magnitude and the Centre of Motion. 148. What is the Centre of Gravity? How may we divide a body of uniform density into two parts of equal weight? Where may we regard the whole weight of a body as concentrated? 149. From what must the centre of gravity be carefully |