Question

Match the column according to dimensions. \[ \begin{array}{c|c|c} \text{Column A} & & \text{Column B} \\ \hline (A) & \sqrt{\dfrac{GM}{C}} & (1)\; M^{-1}L^{2}T^{-3/2} \\ (B) & \sqrt{\dfrac{h^{2}}{GC}} & (2)\; M^{0}L^{2}T^{-3/2} \\ (C) & \sqrt{GMC} & (3)\; M^{3/2}L^{0}T^{0} \\ (D) & \sqrt{\dfrac{GC}{M}} & (4)\; LT^{-1/2} \\ \end{array} \]

• A. A-1, B-2, C-3, D-4
• B. A-4, B-3, C-1, D-2
• C. A-4, B-3, C-2, D-1
• D. A-1, B-3, C-4, D-2

Hint:

When solving dimensional matching problems, first write dimensions of constants such as: \[ [G],\; [h],\; [c] \] Then simplify algebraically before taking square roots.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
We need to determine the dimensional formula of each expression in Column-I by substituting the dimensional formulae of fundamental constants.
Step 2: Key Formula or Approach:
The standard dimensional formulae are:
$[G] = \text{M}^{-1}\text{L}^3\text{T}^{-2}$
$[C] = \text{L}\text{T}^{-1}$ (Speed of light)
$[M] = \text{M}$
$[h] = \text{M}\text{L}^2\text{T}^{-1}$
Step 3: Detailed Explanation:
For (A): $\sqrt{\frac{GM}{C}}$
\[ \left[ \frac{GM}{C} \right] = \frac{(\text{M}^{-1}\text{L}^3\text{T}^{-2}) (\text{M})}{(\text{L}\text{T}^{-1})} = \frac{\text{L}^3\text{T}^{-2}}{\text{L}\text{T}^{-1}} = \text{L}^2\text{T}^{-1} \]
Taking the square root:
\[ \left[\sqrt{\frac{GM}{C}}\right] = (\text{L}^2\text{T}^{-1})^{1/2} = \text{L}\text{T}^{-1/2} \]
Matches with (4).
For (B): $\sqrt{\frac{h^2}{GC}} = \frac{h}{\sqrt{GC}}$
\[ [GC] = (\text{M}^{-1}\text{L}^3\text{T}^{-2})(\text{L}\text{T}^{-1}) = \text{M}^{-1}\text{L}^4\text{T}^{-3} \]
\[ \sqrt{[GC]} = (\text{M}^{-1}\text{L}^4\text{T}^{-3})^{1/2} = \text{M}^{-1/2}\text{L}^2\text{T}^{-3/2} \]
\[ \left[ \frac{h}{\sqrt{GC}} \right] = \frac{\text{M}\text{L}^2\text{T}^{-1}}{\text{M}^{-1/2}\text{L}^2\text{T}^{-3/2}} = \text{M}^{3/2}\text{L}^0\text{T}^{1/2} \]
Matches with (3).
For (C): $\sqrt{GMC}$
\[ [GMC] = (\text{M}^{-1}\text{L}^3\text{T}^{-2})(\text{M})(\text{L}\text{T}^{-1}) = \text{M}^0\text{L}^4\text{T}^{-3} \]
Taking the square root:
\[ \left[\sqrt{GMC}\right] = (\text{M}^0\text{L}^4\text{T}^{-3})^{1/2} = \text{M}^0\text{L}^2\text{T}^{-3/2} \]
Matches with (2).
For (D): $\sqrt{\frac{GC}{M}}$
\[ \left[ \frac{GC}{M} \right] = \frac{\text{M}^{-1}\text{L}^4\text{T}^{-3}}{\text{M}} = \text{M}^{-2}\text{L}^4\text{T}^{-3} \]
Taking the square root:
\[ \left[\sqrt{\frac{GC}{M}}\right] = (\text{M}^{-2}\text{L}^4\text{T}^{-3})^{1/2} = \text{M}^{-1}\text{L}^2\text{T}^{-3/2} \]
Matches with (1).
Step 4: Final Answer:
The proper matching is A-4, B-3, C-2, D-1.