Question

If the work done \(W\), length \(L\) and time \(T\) are considered as the fundamental quantities, the dimensional formula for mass is:

• A. \([W^1L^{-2}T^{-2}]\)
• B. \([W^{-1}L^2T^2]\)
• C. \([W^1L^2T^{-2}]\)
• D. \([W^{-1} L^{-2} T^{-2}]\)

Hint:

When using dimensional analysis, ensure each unit is correctly accounted for and inverted if necessary to isolate the desired quantity.

Answer 1

The Correct Option is A

Step 1: Understanding the relation between work and force.
We know that work (\( W \)) is defined as the product of force and distance moved in the direction of the force. Mathematically,
\[ W = F \cdot L \] where:
\( W \) is the work done,
\( F \) is the force, and
\( L \) is the distance moved.
Step 2: The dimensional formula for force.
Force is related to mass, length, and time by the second law of motion:
\[ F = M L T^{-2} \] where:
\( M \) is the mass,
\( L \) is the length, and
\( T \) is the time.
Step 3: Substituting the dimensional formula of force into the formula for work.
Now, substitute the dimensional formula of force into the equation for work:
\[ W = (M L T^{-2}) \cdot L = M L^2 T^{-2} \] Thus, the dimensional formula of work is:
\[ [W] = M L^2 T^{-2} \] Step 4: Solving for the dimensional formula of mass.
We are asked to find the dimensional formula for mass. From the above equation, we see that:
\[ M = W L^{-2} T^2 \] This means the dimensional formula for mass is:
\[ [M] = W^1 L^{-2} T^{-2} \]