Introduction

In physics, understanding the dimensional formula of moment of force helps verify equations and avoid conceptual mistakes in rotational mechanics. Moment of force, also called torque, is a rotational analogue of force and is defined as the product of force and perpendicular distance from the pivot. Using dimensional analysis, this quantity can be expressed in terms of the fundamental dimensions mass (M), length (L), and time (T) to obtain a unique and testable expression.

Step‑by‑Step Solution for the Correct Option

Moment of force (torque) is defined as:

τ = Force × Perpendicular distance

Dimension of Force

From Newton’s second law, F = ma.

• Mass m → M
• Acceleration a is change of velocity per unit time, so its dimension is LT⁻².

Hence, the dimension of force is:

[F] = M × LT⁻² = MLT⁻².

Dimension of Distance (Lever Arm)

Distance is a length, so [r] = L.

Dimension of Moment of Force

[τ] = [F][r] = (MLT⁻²)(L) = ML²T⁻²

Therefore, the dimensional formula of the moment of force is ML²T⁻², which matches option (D).

Detailed Explanation of Each Option

Option (A) — MLT⁻¹

This option has:

• Mass: M¹
• Length: L¹
• Time: T⁻¹

Such a combination typically corresponds to quantities like angular momentum per unit length, but not to moment of force. It misses one power of length and has an incorrect time exponent, so it cannot represent the moment of force.

Option (B) — MLT⁻²

This is exactly the dimensional formula of force, not torque. It arises directly from F = ma with one power of length and time power −2. Moment of force must further include multiplication by distance, which adds an extra L, giving ML²T⁻². Therefore, option (B) represents only force, not torque.

Option (C) — ML⁻¹T⁻¹

Here, length has power −1 and time has power −1, typical of viscosity-related or other complex quantities, but not of torque. For torque, the dimension of length must be +2, not negative, because torque increases with increasing lever arm. Hence, option (C) is dimensionally incompatible with moment of force.

Option (D) — ML²T⁻² (Correct)

This option correctly combines:

• Mass: M¹
• Length: L² (one from force and one from distance)
• Time: T⁻²

It matches both the standard reference for torque and the derivation from τ = F × r, so it is the correct dimensional formula for the moment of force.

Key Takeaways for Exams

• Moment of force (torque) is given by τ = F × r, where F is force and r is the perpendicular distance from the axis.
• Force has dimension MLT⁻² and distance has dimension L; multiplying them gives ML²T⁻².
• Whenever a multiple‑choice question mixes MLT⁻² and ML²T⁻², remember that the extra power of length distinguishes torque from linear force.