91Ó°ÊÓ

In physics, a moment of inertia is a quantitative measure of a body's rotational inertia—that is, the body's resistance to having its speed of rotation about an axis changed by the application of a torque (turning force). The axis might be internal or exterior, and it can be fixed or not.

The moment of inertia of a uniform square about the axis perpendicular to it and passing through its vertex is –

$$\begin{gathered} I=\frac{2}{3}M{L}^2 \\ dl=\frac{2}{3}\mathrm{ÒÏ}{L}^{4}dz \end{gathered}$$

Now, consider a line L parallel to Z-axis at a distance d from the z-axis as shown in the figure given below.

The cube can be thought of as being built up from many of those squares, stacked one atop another –

$$\begin{gathered} I=∫dl \\ ={∫}_0^L\frac{2}{3}{L}^4\mathrm{ÒÏ}dz \\ =\frac{2}{3}\mathrm{ÒÏ}{L}^5 \\ =\frac{2}{3}M{L}^2 \end{gathered}$$

Where mass of the individual square is$\mathrm{ÒÏ}{L}^{2}dz$, and mass of the cube is$\mathrm{ÒÏ}{L}^3$.

Hence, the moment of inertia (I) is obtained as$\frac{2}{3}M{L}^2$ .