Out of the following four dimensional quantities, which one qualifies to be called a dimensional constant?

The correct option is The velocity of light in vacuum.

To understand why "The velocity of light in vacuum" qualifies as a dimensional constant, let us analyze the terms and the options step-by-step.

1. Understanding Dimensional Constants:
A quantity is called a dimensional constant if it satisfies two conditions:
- It has dimensions (i.e., it is not a dimensionless number).
- Its value remains constant under all conditions and does not change with location, temperature, time, or other physical environment factors.

2. Analyzing the Options:

Option 1: Acceleration due to gravity (g)
- Dimensional Formula:

$g=\left[{\mathrm{LT}}^{-2}\right]$

- Nature: It has dimensions. However, its value is not constant. The acceleration due to gravity varies from place to place on the Earth's surface (higher at the poles, lower at the equator), decreases with altitude and depth, and is completely different on other celestial bodies (like the Moon or Mars). Thus, it is a dimensional variable, not a constant.

Option 2: Surface tension of water
- Dimensional Formula:

$S=\left[{\mathrm{MT}}^{-2}\right]$

- Nature: It has dimensions. However, the surface tension of water changes with temperature (it decreases as temperature increases) and the presence of impurities. Therefore, it is a dimensional variable.

Option 3: Weight of a standard kilogram mass
- Dimensional Formula: Weight is a force, so its dimensional formula is:

$W=\left[{\mathrm{MLT}}^{-2}\right]$

- Nature: Weight depends on the acceleration due to gravity ($W=mg$). Since gravity varies depending on the location, the weight of a standard kilogram mass also changes depending on where it is measured. Hence, it is a dimensional variable.

Option 4: The velocity of light in vacuum (c)
- Dimensional Formula: Velocity of light has the dimensions of speed:

$c=\left[{\mathrm{LT}}^{-1}\right]$

- Nature: According to the postulates of Einstein's special theory of relativity and experimental physics, the speed of light in a vacuum is a universal physical constant. Its value is exactly:

$299,792,458{\text{m s}}^{-1}$

Since it has dimensions and its value is absolutely constant throughout the universe, it qualifies as a dimensional constant.