Question

If E and G respectively denote energy and gravitational constant, then \(\frac{E}{G}\) has the dimensions

• A. [M²] [L⁻²] [T⁻¹]
• B. [M²] [L⁻¹] [T⁰]
• C. [M] [L⁻¹] [T⁻¹]
• D. [M] [L⁰] [T⁰]

Answer 1

The Correct Option is B

To determine the dimensions of the expression \(\frac{E}{G}\), where \(E\) is energy and \(G\) is the gravitational constant, we follow these steps:

1. Identify the dimensions of energy \((E)\):

Energy is defined in terms of force (Newton) and displacement (meter). The dimensional formula for energy \((E)\) is:

\([E] = [M^1][L^2][T^{-2}]\)

2. Identify the dimensions of the gravitational constant \((G)\):

The gravitational constant is used in Newton's law of universal gravitation. The dimensional formula for \(G\) is derived from the formula \(F = \frac{G M_1 M_2}{r^2}\), where \(F\) is force:

\([G] = [M^{-1}][L^3][T^{-2}]\)

3. Calculate the dimensions of \(\frac{E}{G}\):

To find the dimensions of \(\frac{E}{G}\), we divide the dimensional formula of energy by that of the gravitational constant:

\[\frac{E}{G} = \frac{[M^1][L^2][T^{-2}]}{[M^{-1}][L^3][T^{-2}]}\]

Simplifying the division by cancelling the common terms:

\[\frac{E}{G} = [M^{1-(-1)}][L^{2-3}][T^{-2-(-2)}]\]

\[\frac{E}{G} = [M^{2}][L^{-1}][T^{0}]\]

4. Select the correct answer:

Comparing the obtained dimensional formula \([M^2][L^{-1}][T^0]\) with the provided options, we find that the correct answer is:

[M²][L⁻¹][T⁰]

This matches the given correct answer option.