pression for $ T $ as follows:

Given:

Time period of revolution is $ T $.

Radius of circular orbit is $ R $.

We use the relation:

$ T \propto G^a M^b R^c $

This can be expressed as:

$ T = K G^a M^b R^c $

where $ K $ is a constant of proportionality.

The dimensional expressions for the relevant quantities are:

$ [T] = [T] $

$ [G] = [M^{-1}L^3T^{-2}] $

$ [M] = [M] $

$ [R] = [L] $

Thus, the dimensional equation becomes:

$ [T] = [M^{-a}L^{3a+c}T^{2a}] $

Comparing dimensions on both sides, we obtain the following equations:

$ -a + b = 0 $

$ 3a + c = 0 $

$ 2a = -1 $

Solving these equations:

From $ 2a = -1 $, we find $ a = -\frac{1}{2} $.

Using $ -a + b = 0 $, we substitute for $ a $ to get $ b = -\frac{1}{2} $.

From $ 3a + c = 0 $, substituting for $ a $ gives $ c = \frac{3}{2} $.

Substituting these values back into the relation for $ T $:

$ T = K G^{-\frac{1}{2}} M^{-\frac{1}{2}} R^{\frac{3}{2}} $

This simplifies to:

$ T = K \frac{R^{\frac{3}{2}}}{G^{\frac{1}{2}} M^{\frac{1}{2}}} $

Therefore:

$ T = K \sqrt{\frac{R^3}{G M}} $

Gravitation

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