Question

Principle of homogeneity and its applications

Answer 1

The principle of homogeneity is that the dimensions of each the terms of a dimensiional equation on both sides are the same .

Any equation or formula  involving dimensions (like  mass, length, time , temperature electricity) have the terms with same dimensions. This helps us, therefore, to convert the  units in one sytem to another system.

This also helps us to check a formula or the involvement of the dimensions in a formula.

Example : It is conjectured that time of the period of the oscillation of a pendulum  is dependent on its mass,  length   and the acceleration  due to gravity.

So time = some constant K*(mass of the pendulum)^a*(length l of the the pendulum)^b* (acceleration due to gravity g)^c.Or

T = K*m^a*l^b*g^c. Dimensionally this is like:

[T] = [M]^a* [L]^b*[L*T^-2]^c.

Comparing the powers of each dimensions on both sides,(K being dimensionles), we get:

T: 1 = -2c. Therefore, c =-1/2

M: 0 = a. Therefore, a =0

L:  0 = b+2c. Therefore, b =-2c = -1. So the formula for the period T of the pendulum is :

T = K* m^0*L^(1/2)* g^(-1/2) = K*(l/g)^(1/2).

Answer 2

A pendulum basically is a load or mass that is suspended to a stringaucha that when the load is displaced at an amplitude from the Me a position or the equilibrium position I.ei x=0, then the pendulum acts under a restoring force which brings back it to the Mean position and then the this periodic motion contionous to be happen thus this motion is example of a oscillating peroidic motion......