from a rigid support, so that it can oscillate about its position of equilibrium.
In the simple, or mathematical, pendulum the bob is assumed to be a material particle, and to be suspended by a thread without weight. If the bob be stationary and acted on by gravity alone, the line of the thread will be the direction of the force. If the bob be withdrawn from the position of equilibrium (Fig. 25), it will be acted on by a force at right angles to the thread, in a direction opposite that of the displacement, expressed by , where is the angle between the perpendicular and the new position of the thread.
The force acting upon the bob at any point in the circle of which the thread is radius, if it be released and allowed to swing in that circle, varies as the sine of the angle between the perpendicular and the radius drawn to that point. If we make the oscillation so small that the arc may be substituted for its sine without sensible error, the force acting on the bob varies as the displacement of the bob from the point of equilibrium.
A body acted on by a force varying as the displacement of the body from a fixed point will have a simple harmonic motion about its position of equilibrium (§ 21).
Hence it follows that the oscillations of the pendulum are symmetrical about the position of equilibrium. The bob will have an amplitude on the one side of the vertical equal to that which it has on the other, and the oscillation, once set up, will continue forever unless modified by outside forces.
The importance of the pendulum as a means of determining the value of consists in this: that, instead of observing the space traversed by the bob in one second, we may observe the number of oscillations made in any period of time, and determine the time of one oscillation; from this, and the length of the pendulum, we can calculate the value of . The errors in the necessary observations and measurements are very slight in comparison with those of any other method.
