Question

If force $[ F ]$, acceleration $[ A ]$ and time $[ T ]$ are chosen as the fundamental physical quantities. Find the dimensions of energy.

• A. $[ F ][ A ][ T ]$
• B. $[ F ][ A ]\left[ T ^{2}\right]$
• C. $[ F ][ A ]\left[ T ^{-1}\right]$
• D. $[ F ]\left[ A ^{-1}\right][ T ]$

Answer 1

The Correct Option is B

To find the dimensions of energy using the chosen fundamental quantities of force \([F]\), acceleration \([A]\), and time \([T]\), we must express the dimensional formula for energy in terms of these quantities.

Energy in terms of fundamental dimensions is given by:

\[ \text{Energy} = \text{Force} \times \text{Distance} \]

The dimensional formula for force \([F]\) is \([M][L][T]^{-2}\), and since we are choosing force as a fundamental quantity, its dimension is simply \([F]\).

Acceleration \([A]\) has the dimensional formula \([L][T]^{-2}\), and it is treated as \([A]\).

Since time \([T]\) is a fundamental quantity, its dimensional representation stays as \([T]\).

We need to express distance in terms of acceleration \([A]\) and time \([T]\). The formula connecting these is:

\[ \text{Distance} = \frac{1}{2} \times \text{Acceleration} \times \text{Time}^2 \]

The dimensional formula for distance \([L]\) when expressed in terms of \([A]\) and \([T]\) is:

\[ [L] = [A][T]^2 \]

Now, substituting the expression for distance in the expression for energy gives:

\[ \text{Energy} = [F] \times ([A][T]^2) \]

This simplifies to:

\[ \text{Energy} = [F][A][T]^2 \]

Therefore, the dimensions of energy in terms of the given fundamental quantities are \([F][A][T]^2\).

Thus, the correct answer is:

\([F][A][T]^2\)