Experiment: To find the time period of a simple pendulum for small amplitudes and draw the graph of length of pendulum against square of the time period. Use the graph to find the length of the second’s pendulum.
A simple pendulum is a small heavy bob B hanging by a light and inextensible string S. In equilibrium position string is vertical. While oscillating, the amplitude of oscillation is the maximum angle that thread makes with the vertical (or sometimes the maximum horizontal displacement of the bob). Its time period T, i.e., time taken for one oscillation depends on its length i.e. distance from point of suspension to C.G. of bob B.
T ∝ √l
T2 ∝ l
Graph between T2 versus l is a straight line passing through the origin. T also increases if amplitude is large, but for small amplitudes it is constant.
Second’s pendulum is one which takes one second to move from one end of the swing to other. Thus, its time period is 2 s.
A spherical bob; stop watch (with least count of 0.1 second or less), tall laboratory stand with clamp, split cork, fine thread, two small wooden blocks, metre scale
1. Measure diameter of the bob with help of the metre scale and the two wooden blocks. Then tie one end of thread in the hook of the bob.
2. Pass the other end of the thread between two pieces of the split cork and clamp it in the clamp of the stand. The point P, where the thread comes out of the cork is thus a sharp point of suspension, whose position does not change as the pendulum swings. To ensure this, check up that two pieces of the split-cork have sharp lower edges at P.
3. Make a length of about 125 cm of this pendulum for the first set of readings. Measure the length from foot of the hook H to point of suspension P. Add to it half the diameter of the, bob to obtain l, the length of the pendulum. Length PH must be measured with bob suspended, as the thread may have some elastic extension by the weight of the bob.
4. Adjust position of stand to bring this pendulum close to edge E of the table. On a white strip of paper stuck at the vertical end face of the table, mark a vertical line. The thread coincides with this line in its vertical position, when you see it from the front.
5. Pull the bob to one side and release so that it oscillates with an amplitudes of less than 4°. If height of P above table is about 60 cm, then maximum displacement of thread from central mark is not more than about 4 cm.
6. With the help of stop watch, measure time of 20 oscillation. You should start the watch when thread crosses the central mark in a given direction and count ‘zero’. At the count ‘twenty’ when thread crosses the central mark in the same direction, stop the watch. Take three consistent readings, lest there is an error in counting. Then calculate time of one oscillation T.
7. Repeat steps (3) to (6) making shorter lengths of the pendulum up to about 20 cm.
8. For each length calculate T2 and plot a graph between T2 versus l and from this graph find the value of I for T2 = 4 s2.