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:
\[ [M] = W^1 L^{-2} T^{-2} \]

Answer 2

To solve the problem, we need to determine the dimensional formula for mass when work done (W), length (L), and time (T) are considered as the fundamental quantities.

1. Understanding the Relation for Work Done:
Work (W) is defined as the product of force and displacement. The dimensional formula for work is given by: \[ W = F \times L = [M L T^{-2}] \times L = M L^2 T^{-2} \] where M is mass, L is length, and T is time.

2. Finding the Dimensional Formula for Mass:
Now, we are given the work done (W) and need to find the dimensional formula for mass (M). From the equation for work: \[ W = M L^2 T^{-2} \] Rearranging for M: \[ M = W L^{-2} T^2 \] Thus, the dimensional formula for mass is: \[ [M] = [W L^{-2} T^2] \]

Final Answer:
The correct answer is Option A: \([W L^{-2} T^2]\).