Moment of Inertia of a Cube Explained for Students

The moment of inertia of a cube is a fundamental concept in rotational dynamics. It quantifies a cube’s resistance to angular acceleration about a specified axis and depends on the cube’s mass distribution and axis location. This property is necessary for solving problems involving rotational motion in competitive exams such as JEE Main.

Definition and Physical Significance

The moment of inertia of a body about an axis represents how mass is distributed relative to that axis. For a cube, the moment of inertia varies based on whether the axis passes through its center, an edge, a face, or along a diagonal. The SI unit is $\mathrm{kg \cdot m^2}$.

Moment of Inertia of a Cube About Its Center

When the axis of rotation passes through the center and is perpendicular to a face of the cube, the calculation simplifies due to symmetry. For a solid cube of mass $m$ and side $a$, the moment of inertia about this axis is given by:

$ I_{\text{center}} = \dfrac{1}{6}ma^2 $

This formula is derived by integrating the mass distribution using rectangular coordinates, considering area density and the contribution of each infinitesimal element. The process is similar to finding the moment of inertia of a square lamina through its center.

For an overview of general moment of inertia principles, refer to Moment of Inertia Overview.

Moment of Inertia of a Cube About Its Edge

For rotation about an axis passing through one of the edges of the cube (usually taken along the z-axis), the moment of inertia increases as the mass elements are farther from the axis. For a cube of mass $m$ and side $a$, the formula is:

$ I_{\text{edge}} = \dfrac{2}{3}ma^2 $

This result is obtained by integrating the squares of the perpendicular distances from the axis to infinitesimal mass elements throughout the cube’s volume. The density is constant, and the limits of integration run from $0$ to $a$ for each axis.

Derivation: Moment of Inertia About Center and Edge

For the axis through the center, consider a mass element $dm = \rho dxdy$ where $\rho$ is the area density. The elemental moment is $\rho(x^2 + y^2)dxdy$. Integrating over the square lamina yields:

$ I_{\text{center}} = \int_{-a/2}^{a/2} \int_{-a/2}^{a/2} \rho (x^2 + y^2) dx dy = \dfrac{\rho a^4}{6} $

Substituting $\rho = \dfrac{m}{a^2}$ gives $I = \dfrac{ma^2}{6}$.

For the edge, an infinitesimal element $dm = \rho\, dxdydz$ at $(x, y, z)$ is at distance $r = \sqrt{x^2 + y^2}$ from the edge. Integrating:

$ I_{\text{edge}} = \int_{0}^{a} \int_{0}^{a} \int_{0}^{a} (x^2 + y^2) \rho dxdydz = \dfrac{2}{3}\rho a^5 $

Substituting $m = \rho a^3$ yields $I = \dfrac{2}{3}ma^2$. For further reading on discs and their inertia, see Moment of Inertia of a Disc.

Moment of Inertia of a Cube About Its Body Diagonal

When the axis passes through the center and along a body diagonal, the calculation adjusts due to the varying perpendicular distances. The moment of inertia about the body diagonal for a uniform solid cube is:

$ I_{\text{diagonal}} = \dfrac{1}{6}ma^2 $

This result is derived using perpendicular axes and symmetry considerations. Related calculations for planar figures can be referred from Moment of Inertia of a Triangle.

Table: Moments of Inertia of a Cube about Common Axes

Factors Affecting Moment of Inertia of a Cube

The moment of inertia of a cube depends on its mass, the square of its side length, and the choice of the axis of rotation. For minimum value, the axis must pass through the cube’s center of mass.

Applications and Related Concepts

Understanding the moment of inertia of a cube is important in solving rotational dynamics questions. Concepts such as radius of gyration, torque, and conservation of angular momentum often involve these formulas. Similar derivations can be studied for other shapes like circles and hollow spheres using links such as Moment of Inertia of a Circle and Moment of Inertia of Hollow Sphere.

Key Formulas: Summary

• About center (perp. to face): $I = \dfrac{1}{6}ma^2$
• About edge: $I = \dfrac{2}{3}ma^2$
• About body diagonal: $I = \dfrac{1}{6}ma^2$

Further Study and Reference

Comprehensive understanding of rotational inertia requires studying different forms and axes. For additional details on moments of inertia for solids and their rotational axes, refer to topics such as Moment of Inertia of Hollow Cone.