Question

The $[M{L^2}{T^{ - 2}}]$ are the dimensions of
A. Force
B. Moment of force
C. Momentum
D. Power

Answer 1

Hint: Dimensions are the powers to which a base quantity is raised to get the units of a particular physical quantity. When this is enclosed in square brackets it becomes the dimensional formula for the physical quantity. A dimensional formula would satisfy any physical relation satisfied by that quantity. In the given question we find out the dimensions for each of the given physical quantities and then compare with the dimensional formula given in the question. The dimensional formula follows multiplication and division like normal variables but does not show addition or subtraction.

Complete step by step answer:
We know that the dimensions for a physical quantity are the powers to which a base quantity is raised to get the units of that particular physical quantity.For force, we have
$[F] = [ML{T^{ - 2}}]$
For the moment of force,
$m = r \times F$
$\Rightarrow [m] = [r][F]$
Substituting the dimensions in the above formula,
\[[m] = [L][ML{T^{ - 2}}]\]
$ \Rightarrow [m] = [M{L^2}{T^{ - 2}}]$

For momentum,
$p = Ft$
$\Rightarrow [p] = [t][F]$
Substituting the dimensions in the above formula,
\[[p] = [T][ML{T^{ - 2}}]\]
$ \Rightarrow [p] = [ML{T^{ - 1}}]$
For Power,
$P = vF$
$\Rightarrow [P] = [v][F]$
Substituting the dimensions in the above formula,
\[[P] = [L{T^{ - 1}}][ML{T^{ - 2}}]\]
$ \Rightarrow [P] = [M{L^2}{T^{ - 3}}]$
As we can clearly see that, $[m] = [M{L^2}{T^{ - 2}}]$

Hence, the correct answer is option B.

Note: The dimensional formula is only expressed in terms of base quantities. For a physical relation between two or more quantities to be correct, their dimensional formula should also satisfy the same relation. It is better to learn the dimensional formula for force since force is a very common quantity and used in many relations. So, many quantities can be directly derived from force. And hence its dimensional formula helps to derive the dimensional formula for many quantities.