Match the LIST-I with LIST-II Choose the correct answer from the options given below: LIST-I LIST-II A. Gravitational constant I. $[LT^{-2}]$ B. Gravitational potential energy II. $[L^2T^{-2}]$ C. Gravitational potential III. $[ML^2T^{-2}]$ D. Acceleration due to gravity IV. $[M^{-1}L^3T^{-2}]$
A-IV, B-III, C-II, D-I
Dimensional Analysis of Physical Quantities
This solution matches physical quantities from LIST-I with their correct dimensional formulas from LIST-II.
A. Gravitational Constant Dimensions
Newton's law of gravitation is $F = G \frac{m_1 m_2}{r^2}$. Solving for G yields $G = \frac{F r^2}{m_1 m_2}$.
- Dimensions of Force (F): $[MLT^{-2}]$
- Dimensions of distance (r): $[L]$
- Dimensions of mass (m): $[M]$
- Dimensions of G: $\frac{[MLT^{-2}][L]^2}{[M][M]} = \frac{[ML^3T^{-2}]}{[M^2]} = [M^{-1}L^3T^{-2}]$
- This matches LIST-II item IV.
B. Gravitational Potential Energy Dimensions
Gravitational potential energy is a form of energy. Energy has dimensions equivalent to work (Force × Distance).
- Dimensions of Force (F): $[MLT^{-2}]$
- Dimensions of Distance: $[L]$
- Dimensions of Gravitational Potential Energy: $[MLT^{-2}] \times [L] = [ML^2T^{-2}]$
- This matches LIST-II item III.
C. Gravitational Potential Dimensions
Gravitational potential is defined as potential energy per unit mass ($V = U/m$).
- Dimensions of Potential Energy (U): $[ML^2T^{-2}]$
- Dimensions of Mass (m): $[M]$
- Dimensions of Gravitational Potential: $\frac{[ML^2T^{-2}]}{[M]} = [L^2T^{-2}]$
- This matches LIST-II item II.
D. Acceleration due to Gravity Dimensions
Acceleration due to gravity is a measure of acceleration.
- Dimensions of Acceleration: $[LT^{-2}]$
- This matches LIST-II item I.
Final Matching and Answer
The correct pairings are:
- A. Gravitational constant ⇒ IV. $[M^{-1}L^3T^{-2}]$
- B. Gravitational potential energy ⇒ III. $[ML^2T^{-2}]$
- C. Gravitational potential ⇒ II. $[L^2T^{-2}]$
- D. Acceleration due to gravity ⇒ I. $[LT^{-2}]$
This combination corresponds to A-IV, B-III, C-II, D-I.
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