The period of a pendulum T is assumed to depend only on the mass m, the length of the pendulum `, the acceleration due to gravity g, and the angle of swing θ. By means of dimensional analysis, simplify this problem and express this dependence in non dimensional terms.

The expression is shown in the explanation below:

Explanation:

Thinking process:

Let the time period of a simple pendulum be given by the expression:

Let the fundamental units be mass= M, time = t, length = L

Then the equation will be in the form

where k is the constant of proportionality.

Now putting the dimensional formula:

Equating the powers gives:

a = 0

b + c = 0

2c = 1, c = -1/2

b = 1/2

so;

a = 0 , b = 1/2 , c = -1/2

Therefore:

T =

where k =