Question

Match the following physical quantities with their dimensional formulae: \[ \begin{array}{ll} \text{A) Gravitational potential} & \text{I) } LT^{-2} \\ \text{B) Gravitational potential energy} & \text{II) } L^{2}T^{-2} \\ \text{C) Gravitational constant} & \text{III) } ML^{2}T^{-2} \\ \text{D) Gravitational intensity} & \text{IV) } M^{-1}L^{3}T^{-2} \end{array} \] The correct match is:

• A. A-II, B-III, C-IV, D-I
• B. A-II, B-III, C-I, D-IV
• C. A-I, B-II, C-IV, D-III
• D. A-I, B-IV, C-III, D-II

Hint:

Gravitational intensity shares identical dimensions with acceleration due to gravity (\( g \)), making its unit immediately tracking to \( \text{m/s}^2 \), which is simply written as \( LT^{-2} \).

Answer 1

The Correct Option is A

Concept: Dimensional formulas can be systematically derived from fundamental physics definitions linking work, mass, force, and length intervals.

Step 1: Deriving Gravitational Potential (A).
Gravitational potential \( (V) \) is defined as the work done per unit mass: \[ V = \frac{W}{M} = \frac{ML^2T^{-2}}{M} = L^2T^{-2} \quad \cdots \text{(Matches II)} \]

Step 2: Deriving Gravitational Potential Energy (B).
Potential energy represents stored work capacity, sharing identical dimensions with mechanical energy: \[ E = \text{Force} \times \text{Distance} = (MLT^{-2})(L) = ML^2T^{-2} \quad \cdots \text{(Matches III)} \]

Step 3: Deriving Gravitational Constant (C).
From Newton’s universal law of gravitation, \( F = \frac{G \cdot M_1 \cdot M_2}{R^2} \): \[ G = \frac{F \cdot R^2}{M^2} = \frac{(MLT^{-2})(L^2)}{M^2} = M^{-1}L^3T^{-2} \quad \cdots \text{(Matches IV)} \]

Step 4: Deriving Gravitational Intensity (D).
Gravitational field intensity represents gravitational pull force experienced per unit mass point: \[ I = \frac{F}{M} = \frac{MLT^{-2}}{M} = LT^{-2} \quad \cdots \text{(Matches I)} \] This configuration corresponds to combination option (A).