Question

Dimension of work done is
\[\begin{align}
  & \text{A}\text{. }[{{L}^{2}}{{M}^{1}}{{T}^{-2}}] \\
 & \text{B}\text{. }[{{L}^{2}}{{M}^{2}}{{T}^{2}}] \\
 & \text{C}\text{. }[{{L}^{2}}{{M}^{0}}{{T}^{0}}] \\
 & \text{D}\text{. }[{{L}^{2}}{{M}^{2}}{{T}^{-3}}] \\
\end{align}\]

Answer 1

Hint: Dimension of length, mass, time is L, M, T respectively. Convert the mathematical formula for work done into dimensions of length, mass, time. Work done can be find out from the equation,
\[W=F.d\] , where F is the force and d is the displacement.

Complete step by step answer:
Dimensional formula is the representation of a physical quantity in terms of fundamental quantities.
Work is the dot product of two vectors, i.e. force and displacement. Dot product of two vectors will make a scalar quantity. It means, we can understand the physical quantity in terms of only magnitude and it does not need any direction to understand the work done. We can get the dimensional formula of work done from the formula of work done.
We can also write its SI unit as joule and it can be even represented as J.

To express in dimension,
For \[W=F.d\]
Unit for work done is \[N.m\]or \[\dfrac{kg.{{m}^{2}}}{{{s}^{2}}}\].
Dimension of d is \[[L]\].
For dimension of force,
\[F=m.a\]
Unit of force is \[\dfrac{kg.m}{{{s}^{2}}}\] or \[N\].
Dimension for force is \[[LM{{T}^{-2}}]\].
Now we combine the dimension of force and distance according to the formula of work done.
i.e.,
\[W=[ML{{T}^{-2}}]\times [L]\]

\[W=[M{{L}^{2}}{{T}^{-2}}]\].
So, the dimension for the work done is \[[{{L}^{2}}M{{T}^{-2}}]\]. Hence, the correct option is A.

Additional Information:
SI units for work done is joule and it is represented as J and its CGS unit is erg.

Note: Students should try to remember the basic formula such as force, work done, power etc. Then it will be easier for the conversion of these formulas into dimension formula or SI units. Students should remember all the fundamental physical quantities, its symbol and its dimension.