Theorem of parallel axis
Theorem of Parallel Axis
The theorem of parallel axis, also known as the parallel axis theorem or Steiner's theorem, is a fundamental result in classical mechanics. It relates the moment of inertia of a body about any axis to the moment of inertia of the same body about a parallel axis that passes through the center of mass.
Understanding Moment of Inertia
Before diving into the theorem of parallel axis, it's important to understand the concept of moment of inertia (I). Moment of inertia is a measure of an object's resistance to changes in its rotational motion. It depends on the mass distribution of the object relative to the axis of rotation.
Theorem of Parallel Axis Formula
The theorem can be mathematically expressed as:
$$ I = I_{CM} + Md^2 $$
Where:
- ( I ) is the moment of inertia about the new axis.
- ( I_{CM} ) is the moment of inertia about the center of mass axis, parallel to the new axis.
- ( M ) is the total mass of the body.
- ( d ) is the perpendicular distance between the two parallel axes.
Table of Differences and Important Points
| Aspect | Center of Mass Axis | Parallel Axis |
|---|---|---|
| Moment of Inertia | ( I_{CM} ) | ( I ) |
| Reference Point | Center of Mass | Any point off-center |
| Additional Term | None | ( Md^2 ) |
| Application | Pure rotational motion about the center of mass | Rotational motion about any other axis |
Examples
Example 1: Solid Cylinder
Consider a solid cylinder of mass ( M ) and radius ( R ) rotating about its central axis. The moment of inertia about this axis is given by:
$$ I_{CM} = \frac{1}{2}MR^2 $$
Now, if we want to find the moment of inertia about an axis parallel to the central axis but at a distance ( d ) from it, we use the parallel axis theorem:
$$ I = \frac{1}{2}MR^2 + Md^2 $$
Example 2: Rectangular Plate
Take a rectangular plate with mass ( M ), width ( w ), and height ( h ). The moment of inertia about an axis through the center of mass and perpendicular to the plane of the plate is:
$$ I_{CM} = \frac{1}{12}M(w^2 + h^2) $$
If we want to find the moment of inertia about an axis at the edge of the plate (which is parallel to the center of mass axis and a distance ( d = \frac{h}{2} ) away), we apply the theorem:
$$ I = \frac{1}{12}M(w^2 + h^2) + M\left(\frac{h}{2}\right)^2 $$
Example 3: Rod Rotating About Its End
Consider a uniform rod of length ( L ) and mass ( M ) rotating about an axis through one end, perpendicular to its length. The moment of inertia about the center of mass (which is at ( L/2 ) from either end) is:
$$ I_{CM} = \frac{1}{12}ML^2 $$
Using the parallel axis theorem to find the moment of inertia about the end, we get:
$$ I = \frac{1}{12}ML^2 + M\left(\frac{L}{2}\right)^2 = \frac{1}{3}ML^2 $$
Conclusion
The theorem of parallel axis is a powerful tool in rotational dynamics, allowing us to calculate the moment of inertia for complex systems and different axes of rotation. It simplifies the analysis of rotational motion and is essential for solving problems in physics and engineering where the axis of rotation does not pass through the center of mass.