such methods as those of Art. 307; and then the components into which it is resolved can be found: thus the truth of the statement in Art. 303 can be tested,

310. Suppose a man to run round a circle of which the radius is 20 feet, at the rate of 8 feet in a second. Then the resultant force which acts on him is directed towards 64 the centre of the circle, and is equal to

1

32 × 20

of his

weight, that is to of his weight. This resultant force

10

1

10

of

must be produced by a combination of the man's weight and the action of the ground. Hence the action of the ground must not be entirely vertical but oblique; the vertical component of it must just balance the man's weight, and the horizontal component of it must be equal to the weight. In order that the action of the ground may pass through the man's centre of gravity, which is necessary in order that it may combine with the weight to form the horizontal force, the man must lean inwards towards the centre of the circle: the amount of this leaning must be at the rate of 1 inch horizontal to 10 inches vertical.

311. We find in Astronomy some of the best illustrations of the motion of a body under the influence of a force which has its direction always changing but always passing through a fixed point. For instance the Earth moves round the Sun under the action of the Sun's attraction. The Earth does not describe a circle, and so does not furnish exactly a case of the motion considered in the present Chapter; but still the path in which the Earth moves is very nearly a circle, and the amount of the Sun's force is not much different from that assigned by Art. 303. So also the Moon relatively to the Earth describes a path which is very nearly a circle. The distance of the Moon from the Earth is about 240,000 miles; thus the circumference of the circle which the Moon describes round the Earth is about

The time in which this circle is described is about 274 days, that is about 274 × 24 × 60 × 60 seconds. Hence we obtain the velocity by dividing the former number by the latter. Then by the statement of Art. 303 we can compare the force which the Earth exerts on a body moving like the Moon moves, with the weight of the body; that is in fact, we compare the force which the Earth exerts on a body moving like the Moon moves, with the force which the Earth would exert on the body if it were close to the Earth's surface. This comparison was the foundation of Newton's system of Astronomy; the result is that the force on a body in the situation of the Moon is about of 3600 the force on the same body if it were at the Earth's surface: see Art. 301.

XXII. SIMPLE PENDULUM.

1

312. Let one end of a fine string be fastened to a fixed point, and the other end to a small heavy particle. In the position of equilibrium the string will be vertical. Let the particle be displaced from its position of equilibrium, the string being kept stretched, and then allowed to move. The particle will go backwards and forwards; this is called oscillating. The particle thus describes arcs of a circle; owing to friction and the resistance of the air the arcs described become gradually less and less, until at last the particle comes to rest. The string and particle together constitute what is called a simple pendulum.

313. The forces which act on the heavy particle are its own weight and the tension of the string. The former force acts always vertically downwards, and is always of the same amount; so that it is constant in direction and magnitude. The latter force perpetually changes its direction, though the direction always passes through a fixed point. The weight acting vertically may be supposed at any instant to be resolved into two components, one along the string at that instant, and the other at right angles to it. The former produces no motion, being resisted by the string; the latter urges the heavy particle along the circular arc

towards the lowest point. The motion is found to be of the following kind: the particle being at one of the extreme points of an arc starts, as if from rest, and the velocity continually increases until the particle reaches the lowest point of the arc; then as it goes up through the rest of the arc the velocity diminishes until the particle reaches its highest point at the other end of the arc. The time of moving from the starting point to the lowest point is the same as that of moving from the lowest point to the other end of the arc; and when the arc is very small it is found that this time does not sensibly change as the arc becomes smaller and smaller. The time of passing from one end of an arc to the other is called the time of oscillation; it may be found according to theory, by the following rule: Take the length of the string in feet, divide by 32, and extract

22

7

the square root of the result; then multiply by and the product will be the time in seconds.

22

314. The important point to notice with respect to the preceding rule is that it supposes the arc through which the particle moves to be very small; but then it is true without taking into account the greater or less extent of this small arc. The rule may be made more accurate by using instead of the number 3'1416: see Art. 28. Also for the sake of extreme precision we should have instead of 32 to put a slightly different number, different for different places: see Art. 98. The length of a simple pendulum which oscillates in a second at the latitude of London is 39 1393 inches. This is about 994 of the metre, the French standard of length.

7

315. We have said that the rule in Art. 313 supposes the arc of oscillation to be very small; and therefore it will be proper to give some notion of the correction which must be made when the arc is not very small. The time found by the rule must then be increased by a small fraction of itself, and this fraction may be found with sufficient accuracy in the following manner: the numerator is the square of the number of degrees in the angle between the extreme position of the pendulum and the position of

equilibrium, and the denominator is 50000. Thus, for example, suppose the pendulum oscillates through an angle of 10 degrees altogether, then there are 5 degrees in the angle with which we are concerned; the square of 5 is 25, and Therefore the time found by

1

25
50000 2000

=

1

the rule of Art. 313 must be increased by of itself;

2000

so that if that rule gives one second for the time when the arc is very small the correct time when the arc corresponds to 10 degrees will be 10 seconds.

316. The time of oscillation does not depend on the nature of the substance of which the heavy particle is composed; this corresponds with the fact that, setting aside the resistance of the air, all bodies fall to the ground from the same height in the same time. The word oscillation is used by some writers to denote the time taken by the heavy particle in passing from one end of an arc to the same point again; this amounts to twice the time which we assign to an oscillation. Also the word vibration is sometimes used instead of oscillation.

317. Instead of compelling a particle to describe an arc of a circle by means of a string we might have a fine tube made in the form of an arc of a circle, and fixed in a vertical plane; and then the particle might be placed within the tube so as to slide up and down. Theory shews that the motion is of the same kind as the other, provided the tube is smooth internally. The resistance of the tube in this case takes the place of the tension of the string in the other.

318. We may also have other cases of motion by supposing a fine smooth tube, as in the preceding Article, not in the form of an arc of a circle but in that of an arc of any other curve. One interesting result obtained by theory then is that whatever be the form of this curvo the velocity of the heavy particle at any point is just the same as if it had fallen freely through a vertical spaco equal to the depth of this point vertically below the starting point. We have already remarked in Art. 289 that this is the case when the tube is in the form of a straight line.

T. P.

9.

319. Two very curious results in connexion with this subject may be noticed. Suppose the fine smooth tube made in the form of half a particular curve which mathematicians call a cycloid, and let it be placed as they would say with its base horizontal and its vertex downwards; denote the highest point of the curve by A, and the lowest point by B. Then the heavy particle would slide from A to B down this curve in less time than down any other curve from A to B. And if C denote any point of the curve between A and B the particle would slide down the portion of the curve from Ĉ to B, starting at C, in the same time as down the whole curve from A to B. The two statements can be well demonstrated experimentally by constructing tubes or troughs on a large scale. In particular the truth of the second statement can be very effectively shewn; a man takes a ball in each hand, and by stretching out his arms he can put one ball at a point of the trough far above the point at which he puts the other, and let both start at the same instant; then the upper ball just overtakes the lower ball at the bottom of the curve.

320. It is easy to give a notion of the curve which we call a cycloid. It is the curve which a point in the circumference of a carriage wheel would trace out as the wheel turns once round in rolling along the ground; the point being supposed the lowest point of the wheel at the beginning and at the end of the turning. The curve thus formed will bear some resemblance to the outline of a very flat arch of a bridge. The curve must be supposed turned upside down and half of it taken when used in the manner of Art. 319.

XXIII. FRICTION.

321. WE have hitherto supposed that all bodies are smooth, but practically this is not the case, and we must now examine the results which follow from the roughness of bodies.