To study the simple pendulum. Measurement of 9
For experimental purposes, a simple pendulum is made by attaching a length of thread to a small brass or lead sphere called the bob. The thread is held firmly between two small pieces of wood (or a split cork) held by a clamp and stand. One complete to and fro movement of the pendulum is called an oscillation or vibration. The time taken for one complete oscillation is called the periodic time. The length of the pendulum is defined as the distance from the point of suspension to the
centre of gravity* of the bob.
As the pendulum swings to’ and fro, the maximum displacement of the bob from its rest position is called the amplitude, Alternatively, we speak of the angular amplitude of the pendulum or the angle between the extreme and the rest positions of
Provided the amplitude is small, i.e., not more than a few degrees, the periodic time depends only on the length of the pendulum and the acceleration due to gravity. Experiments carried out using bobs of different sizes show that the periodic
time does not depend on the mass or material of the bob. In more advanced books it is shown that the periodic time, T, of a simple pendulum is given by,
where I = length in m;
g = acceleration due to gravity in m/s2.
centre of gravity is explained on page 65, in this case it is the centre of the bob.
In order to verify this equation experimentally it will be found convenient to rearrange it so that all the constant factors are on one side only.
The object of the experiment now to be described is to verify this equation by measuring the periodic time T, for a series of different values of I. If the equation is true, the results should show that ~2 = constant. Fig. 3.7 shows how the length of the pendulum is measured by means of a metre rule and a set-square. If a set-square is not available a small rectangular card may be used instead. The zero end of the rule is held in contact with the under-surface of the blocks and the set-square reading x noted when it is just touching the bottom of the bob. The radius of the bob is now subtracted from x to give the required length, I. The observer should sit in front of the pendulum and note the rest position of the lower part of the string against some convenient mark. The pendulum is then set swinging with a small amplitude.
Timing of the oscillations is begun by counting “bought” as the bob passes
through its rest position and simultaneously a stop-clock is started. Counting is continued, “one, two, three, etc.” each time the bob passes through its rest position in the same direction. If a stop-clock is not available the time may be measured by an ordinary clock or watch which has a seconds hand. In this way, the time of 50 oscillations is found for at least half a dozen different lengths varying from about 0.30 to 1.0 m.
In each case the timing should be repeated as a check on the previous reading and the results tabulated as shown.
The square of the periodic time is proportional to the length of the pendulum. This may also be demonstrated by plotting a graph of T? against I. A straight line through the origin should be obtained. Calculation of the acceleration due to gravity from the results
Hence a value for g may be calculated either by dividing 4n2 by the mean value Of obtained from the last column of the table, or from the gradient of the graph of T?against I.