how it is known to be true. It is in fact capable of demonstration by strict mathematical reasoning, but it may also be verified by trial. There is however no very obvious example which immediately presents itself; a boy's sling is sometimes mentioned, but this does not strictly fulfil all the conditions. For the loaded end of the sling in general does not describe an exact circle; the hand at the other end shifts its position perpetually, and keeps urging the loaded end to increased speed. If the hand remains quite still, so that the loaded end describes an exact circle, still this circle is not described with uniform velocity, when, as is usually the case, the sling moves in a vertical plane; the velocity is greater at the lower than at the upper points of the circle. 307. In the absence of appropriate spontaneous examples we must contrive experiments. Put a ball on a smooth table, fasten it by a string to a fixed point in the table, and start the ball so as to describe a circle round the fixed point as centre. If the table is smooth the ball will move for some time pretty uniformly, and the velocity with which it moves can be observed. It is easy to devise means for measuring the force which acts on the ball and tends towards the centre of the circle; this is in fact the tension of the string. We may have the string formed of some material which will stretch, and observe carefully the length of the string when the ball describes a circle uniformly. Then stop the motion, take away the ball from the string, fix one end of the string and hang a weight at the other end just heavy enough to stretch the string to the length it had in the case of motion: then the tension of the string in both cases is equal to this weight. Thus we know the velocity with which the ball moved, the radius of the circle, and the force acting towards the centre of the circle; and accordingly we can test the truth of the statement in Art. 303 as to the relation between these quantities. Or instead of fastening one end of the string to a fixed point in the table, when the ball at the other end is describing a circle, we may pass the string through a small hole in the table and hang a weight at the other end. When this weight remains at rest it measures the tension of the string, and therefore the force which is directed towards the small hole as centre, and acts on the body describing a circle round that centre. 308. In this experiment of the ball on the smooth table the weight of the ball vertically downwards is just balanced by the resistance of the table upwards, and thus these two forces do not affect the motion of the ball on the table. Setting aside the weight and the resistance of the table the only force which acts on the ball during the motion is the force towards the centre of the circle. Suppose we stop the action of this force at any instant, which we may do by cutting the string, then the ball will continue to move uniformly in the direction in which it is moving at that instant; this direction is at right angles to the straight line drawn from the ball to the centre of the circle at that instant, or in the language of Geometry it is the tangent to the circle at the point which the ball occupies at the instant. Thus it must be remembered that while a body describes a circle with uniform velocity the resultant of all the forces which act on it is a single force towards the centre of the circle. We say the resultant of all the forces, because there may be forces which just balance each other as in the case of the ball on the smooth table, where the weight and the resistance balance each other. 309. There is still another mode of making the experiment which may be noticed. Let one end of a string be fastened to a weight and the other end to a fixed point. Let the string be drawn aside from the vertical direction, and let a velocity be given to the weight in a horizontal direction. By trial it will be found possible to get the weight to move for some time in a horizontal plane and to describe a circle. The tension of the string may be supposed to be resolved into two components, one vertical and the other horizontal: see Art. 156. The vertical component will balance the weight of the body, so that the body goes neither up nor down. The horizontal component constitutes the force towards the centre of the circle which makes the weight describe the circle. It is easy to determine the value of the tension of the string, by 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 10 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 of 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 27 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 |