1. Why Do We Need Theorems of Moment of Inertia?
Finding the moment of inertia (MI) of objects directly from integration can be difficult. The theorems of moment of inertia provide shortcuts that help compute MI about new axes using known values. They are powerful tools that simplify calculations in rotational motion, mechanical design, and engineering.
Two important theorems are used frequently: the Parallel Axis Theorem and the Perpendicular Axis Theorem.
1.1. When Do These Theorems Help?
The theorems help when:
- You know the MI about one axis and need it about another axis.
- The shapes are standard but the required axis is not through their centre.
- You are dealing with composite bodies made of simpler parts.
1.1.1. Real-World Intuition
Consider holding a bat. Rotating it around its centre feels easier than rotating it around the end. The theorems help explain and calculate this difference.
2. Parallel Axis Theorem
The Parallel Axis Theorem allows you to find the moment of inertia about any axis parallel to an axis passing through the centre of mass (COM).
According to the theorem:
\( I = I_{cm} + M d^2 \)
Where:
- \( I \) is the MI about the new axis
- \( I_{cm} \) is the MI about the COM axis
- \( M \) is the mass of the body
- \( d \) is the distance between the two axes
2.1. Understanding the Formula
The extra term \( M d^2 \) accounts for shifting the axis away from the centre. The farther the axis is from the COM, the larger the moment of inertia becomes.
2.2. Example: Uniform Rod
The MI of a rod about its centre is \( I_{cm} = \dfrac{1}{12} M L^2 \). To find the MI about one end, use \( d = \dfrac{L}{2} \):
\( I = \dfrac{1}{12} ML^2 + M \left(\dfrac{L}{2}\right)^2 = \dfrac{1}{3} ML^2 \)
3. Perpendicular Axis Theorem
The Perpendicular Axis Theorem applies only to flat laminar bodies (objects of negligible thickness). It relates the MI about an axis perpendicular to the plane to the MI about two perpendicular axes lying in the plane.
The theorem states:
\( I_z = I_x + I_y \)
Where:
- \( I_z \) is the MI about an axis perpendicular to the plane
- \( I_x \) and \( I_y \) are the MI about two perpendicular axes in the plane
3.1. When Is It Used?
This theorem is extremely useful for shapes like discs, rings, plates, and other 2D structures.
3.2. Example: Thin Circular Disc
For a disc, the MI about a diameter is \( I_x = I_y = \dfrac{1}{4} MR^2 \).
So the MI about the central axis perpendicular to the disc is:
\( I_z = I_x + I_y = \dfrac{1}{4} MR^2 + \dfrac{1}{4} MR^2 = \dfrac{1}{2} MR^2 \)
4. Using Both Theorems in Composite Bodies
Many objects in real life are made by combining simple shapes. To find their total moment of inertia, we calculate the MI of each part about the same axis and then add them.
4.1. Steps to Solve Composite Problems
- Find or calculate the MI about the COM axis for each part.
- Use the Parallel Axis Theorem to shift to the required axis.
- Add the MI of all parts to get the total MI.
4.2. Illustration: L-shaped Block
An L-shaped object can be broken into two rectangular blocks. Using the parallel axis theorem on each block gives the total MI.
5. Importance of These Theorems
The theorems of moment of inertia provide essential shortcuts for rotational dynamics. They eliminate the need for lengthy integration and offer clear insights into how MI changes when axes shift or rotate. Mechanical systems, robotics, machinery, and structural analysis rely heavily on these theorems.
With these tools, you can now compute MI for a wide range of bodies and move forward to understanding angular motion more deeply.