Out of the following pairs, which one does not have the same...

Out of the following pairs, which one does not have the same dimensions?
(A) Angular momentum and Planck's constant.
(B) Impulse and momentum.
(C) Moment of inertia and moment of force.
(D) Work and torque.

Key Concepts

Dimensional Analysis

Dimensional analysis is a powerful tool in physics used to check the consistency of equations and relationships by comparing the units or dimensions of the physical quantities involved. It helps to ascertain whether two quantities can be directly equated or compared based on their underlying physical dimensions, such as mass (M), length (L), and time (T).

Physical Dimensions and Units

Every physical quantity can be represented in terms of fundamental dimensions, such as mass, length, and time. By breaking down derived units into these basic dimensions, one can verify the dimensional consistency of equations and ensure that quantities being equated or compared share the same dimensional makeup.

Angular Momentum and Related Quantities

Angular momentum is a measure of the rotational motion of an object and carries dimensions of mass, length squared, and inverse time (M L² T?¹). It is conceptually similar to Planck's constant in that both share the same dimensional form, illustrating the fundamental nature of angular momentum in both classical and quantum mechanics.

Impulse and Momentum

Impulse and momentum are closely related physical quantities in mechanics, with both sharing the same dimensions of mass times velocity (M L T?¹). Impulse, defined as the change in momentum, directly reflects the dimensional attributes of momentum, ensuring consistency in their use within mechanical equations.

Moment of Inertia versus Moment of Force

Moment of inertia and moment of force (commonly known as torque) are distinct in their dimensional makeup. Moment of inertia, which measures an object's resistance to rotational acceleration, has dimensions of mass times length squared (M L²). On the other hand, moment of force has dimensions of force times distance (M L² T?²). This difference is critical in understanding why these two quantities, despite their similar nomenclature, are not dimensionally the same.

Work and Torque

Work and torque, although conceptually different—with work being a measure of energy transfer and torque representing a rotational force—share the same dimensional representation of energy (M L² T?²). This similarity in dimensions highlights how different physical phenomena can be related through their underlying units while still representing distinct physical processes.

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