Parallel Axis Theorem in Physics

The parallel axis theorem is a fundamental concept in physics that relates the moment of inertia of an object about an axis parallel to a different axis passing through its center of mass (COM). It allows us to calculate the moment of inertia of an object about an axis parallel to a known axis passing through its COM.

Understanding the Parallel Axis Theorem

The parallel axis theorem states that the moment of inertia of an object about an axis parallel to another axis passing through its COM is equal to the sum of the moment of inertia about the COM and the product of the object's mass and the square of the perpendicular distance between the two axes.

Mathematical Representation

The parallel axis theorem can be expressed mathematically as follows:

I = I_cm + md²

Where:
- I is the moment of inertia about the parallel axis
- I_cm is the moment of inertia about the COM
- m is the mass of the object
- d is the perpendicular distance between the two axes

Applicability to One of the Two Parallel Axes Passing Through COM

According to the parallel axis theorem, one of the two parallel axes must pass through the COM of the object. This is essential for the theorem to hold true. If one of the axes does not pass through the COM, the theorem cannot be applied.

When one of the axes passes through the COM, the perpendicular distance (d) between the two axes becomes zero. As a result, the term md² becomes zero, and the equation simplifies to:

I = I_cm + 0

This simplification shows that the moment of inertia about the parallel axis is equal to the moment of inertia about the COM. In other words, if one of the axes passes through the COM, the moment of inertia about that axis is the same as the moment of inertia about the COM.

Conclusion

In conclusion, the parallel axis theorem is applicable to an object when one of the two parallel axes passes through its COM. This condition is necessary for the theorem to be valid. The theorem allows us to calculate the moment of inertia about a parallel axis by adding the moment of inertia about the COM and the product of the object's mass and the square of the perpendicular distance between the axes. Understanding and applying the parallel axis theorem is crucial in various physics problems and calculations involving rotational motion.