Question

What is the dimensional formula for the torque?

Answer 1

Definition

Torque would be the force that might also cause an item to revolve around an axis. It is also known as the rotational comparable of linear force. This same axis of rotation would be the point at which the item rotates.

Formula: $\mathrm{Torque}\left(\mathrm{T}\right)=\mathrm{Moment}\mathrm{of}\mathrm{Inertia}\times \mathrm{Angular}\mathrm{Acceleration}$

Dimension formula

We know that,

Moment of Inertia:

$\mathrm{M}.\mathrm{O}.\mathrm{I}=\mathrm{Radius}\mathrm{of}{\mathrm{Gyration}}^2\times \mathrm{Mass}$

The dimensional formula of M.O.I = ${\mathrm{M}}^1{\mathrm{L}}^2{\mathrm{T}}^0$ …(equation $1$)

$\mathrm{Angular}\mathrm{Acceleration}=\mathrm{Angular}\mathrm{velocity}\times {\mathrm{Time}}^{-1}$

∴ The dimensional formula of Angular Acceleration = ${\mathrm{M}}^0{\mathrm{L}}^0{\mathrm{T}}^{-2}$ …(equation $2$)

On substituting the value of both equations in the formula of torque, we get

$$\begin{gathered} \mathrm{Torque}=\mathrm{Moment}\mathrm{of}\mathrm{Inertia}\times \mathrm{Angular}\mathrm{Acceleration} \\ =\left[{\mathrm{M}}^1{\mathrm{L}}^2{\mathrm{T}}^0\right]\times \left[{\mathrm{M}}^0{\mathrm{L}}^0{\mathrm{T}}^{-2}\right] \\ =\left[{\mathrm{ML}}^2{\mathrm{T}}^{-2}\right] \end{gathered}$$

Therefore, the dimension formula for the torque is $\left[{\mathrm{M}}^1{\mathrm{L}}^2{\mathrm{T}}^{-2}\right]$.