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

Step 1: Understand what is asked.
We are given four gravity related quantities (A, B, C, D) and four dimensional formulas (I, II, III, IV). We must match each quantity to its correct formula. The trick is to write a short formula for each quantity and then read off its dimensions. We use the base building blocks Mass $M$, Length $L$ and Time $T$.
Step 2: Recall the energy block.
Many gravity quantities are built from work or energy. Work is force times distance. Force has dimensions $MLT^{-2}$ and distance is $L$, so energy has dimensions: \[ [E] = MLT^{-2}\times L = ML^{2}T^{-2} \] We will reuse this block again and again.
Step 3: Find A, gravitational potential.
Gravitational potential is energy (or work) carried by each unit of mass. So we divide energy by mass: \[ [V] = \frac{ML^{2}T^{-2}}{M} = L^{2}T^{-2} \] This matches II. The mass cancels, which is why A pairs with II.
Step 4: Find B, gravitational potential energy.
Potential energy is just a type of energy, so it uses the energy block directly: \[ [E] = ML^{2}T^{-2} \] This matches III.
Step 5: Find C, gravitational constant.
From Newton's law $F = \dfrac{G m_1 m_2}{r^{2}}$, solve for $G$: \[ G = \frac{F r^{2}}{m_1 m_2} = \frac{(MLT^{-2})(L^{2})}{M^{2}} = M^{-1}L^{3}T^{-2} \] This matches IV.
Step 6: Find D, gravitational intensity, and conclude.
Field intensity is force felt by each unit of mass, so divide force by mass: \[ [I] = \frac{MLT^{-2}}{M} = LT^{-2} \] This matches I. So the full match is A-II, B-III, C-IV, D-I. \[ \boxed{\text{A-II, B-III, C-IV, D-I}} \]