Dimensions Of Physical Quantities

To determine the dimensions of the constant in the expression for the displacement of a particle, we need to analyze the dimensional equation. Assuming the displacement of a particle is given by an equation of the form:

where and are given with their respective dimensions, and we need to find the dimensions of .

The general approach involves the following steps:

1. x represents displacement, typically having the dimensional formula of [M^0L^1T^0] .
2. t represents time, having the dimension [T] .
3. is a constant, which contributes to the displacement dimensions; since it is constant, we typically consider it as dimensionless or with known dimensions based on context.
4. The unknown dimension we seek is for .

Without loss of generality, let's assume the displacement is expressed as a function including terms involving both and , such as:

Here, k is some power associated with to make the overall dimensional equation consistent with dimensions of length [L] for displacement .

Given the options, the only dimension can logically hold to ensure dimensional consistency is if it is combined with time for a term that reflects fundamental displacement properties. Thus:

Assuming is dimensionally compatible or null (non-dimensionality effects), solve for:

This leads us to the conclusion that for the given expression must have dimensions of [T^{-1/2}] so that in combination with , the equation satisfies dimensional consistency for displacement .

Therefore, the correct dimension of is:

This matches the given correct answer .