How does the time period (T) of a simple pendulum depend on its length (l) ? Draw a graph showing the variation of T² with l. How will you use this graph to determine the value of g (acceleration due to gravity)?

Time period of a simple pendulum is directly proportional to the square root of its effective length.

T ∝ √l

Graph showing the variation of T² with l is given below:

In order to find the acceleration due to gravity with the help of the above graph, we follow the following steps —

The slope of the straight line obtained in the T² vs l graph, as shown in fig, can be obtained by taking two points P and Q on the straight line and drawing normals from these points on the X and Y axes. Then, note the value of T², say T₁² and T₂² at a and b respectively, and also the value of l say l₁ and l₂ respectively at c and d.

Then,

Slope = PR\over QR = ab\over cd = T_1^2 - T_2^2\over l_1-l_2

This slope is found to be a constant at a place and,

Slope = 4 \pi ^2\over g

where, g = acceleration due to gravity at that place.

Thus, g can be determined at a place from the graph using the following relation,

g = 4 \pi ^2\over slope\ of\ T^2\ vs\ l\ graph