Question

For a uniform cube, find Iabout one edge.

Short answer

The moment of inertia (I) about one edge for a uniform cube is $\frac{2}{3}M{L}^2$.

Step-by-step solution

Definition of moment of Inertia

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.

Plotting the diagram

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

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

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

Calculation of Moment of Inertia

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

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

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

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