Theorem of Perpendicular Axes

The Theorem of Perpendicular Axes is an important concept in the field of rotational motion in physics. It is particularly relevant when dealing with the moment of inertia of planar bodies (bodies with a flat shape) that are rigid and lie in a plane.

Understanding the Theorem

The theorem states that for a rigid body lying in the XY plane, the moment of inertia about an axis perpendicular to the plane (the Z-axis) is equal to the sum of the moments of inertia about two mutually perpendicular axes (the X and Y axes) lying in the plane.

Mathematically, the theorem can be expressed as:

$$ I_z = I_x + I_y $$

where:

• $I_z$ is the moment of inertia about the Z-axis (perpendicular to the plane),
• $I_x$ is the moment of inertia about the X-axis (in the plane), and
• $I_y$ is the moment of inertia about the Y-axis (in the plane).

Important Points

• The theorem is applicable only to flat, planar bodies.
• The axes must be mutually perpendicular, and the X and Y axes must lie in the plane of the body.
• The Z-axis must be perpendicular to the plane of the body.

Differences and Important Points

Aspect	Description
Applicability	Only applies to planar bodies.
Axes Requirements	X and Y axes must lie in the plane, and Z-axis must be perpendicular to it.
Moment of Inertia Relation	$I_z$ is the sum of $I_x$ and $I_y$.

Formulas

For a planar body with a known mass distribution, the moments of inertia about the X and Y axes can be calculated using the following integrals:

$$ I_x = \int y^2 \, dm $$ $$ I_y = \int x^2 \, dm $$

where $dm$ is the mass element of the body, and $x$ and $y$ are the perpendicular distances from the mass element to the Y and X axes, respectively.

Examples

Example 1: Rectangular Plate

Consider a rectangular plate of mass $M$, width $w$, and height $h$, lying in the XY plane. The moments of inertia about the X and Y axes passing through the center of mass and parallel to the sides of the rectangle are given by:

$$ I_x = \frac{1}{12} M h^2 $$ $$ I_y = \frac{1}{12} M w^2 $$

Using the Theorem of Perpendicular Axes, the moment of inertia about the Z-axis passing through the center of mass and perpendicular to the plate is:

$$ I_z = I_x + I_y = \frac{1}{12} M (h^2 + w^2) $$

Example 2: Circular Disc

For a circular disc of mass $M$ and radius $R$, lying in the XY plane, the moments of inertia about the X and Y axes passing through the center of mass and parallel to the diameter are:

$$ I_x = I_y = \frac{1}{4} M R^2 $$

Using the theorem, the moment of inertia about the Z-axis (perpendicular to the disc and passing through the center) is:

$$ I_z = I_x + I_y = \frac{1}{2} M R^2 $$

In conclusion, the Theorem of Perpendicular Axes is a powerful tool for calculating the moment of inertia of planar bodies about an axis perpendicular to their plane. It simplifies the process by allowing the use of moments of inertia about in-plane axes to find the moment of inertia about the perpendicular axis. Understanding this theorem is crucial for solving problems related to rotational motion in physics exams.