Moment of Inertia of a Disc About Its Diameter
This simulation demonstrates the calculation of the moment of inertia of a disc about one of its diameters using the perpendicular axes theorem.
Understanding the Problem
We need to calculate the moment of inertia of a disc about one of its diameters. We know the moment of inertia about an axis perpendicular to the disc and through its center (Iz = MR²/2).
The disc is a planar body, so the perpendicular axes theorem is applicable. We consider three axes through the center of the disc (x, y in the plane, z perpendicular).
Key Concepts and Formulas
1. Perpendicular Axes Theorem: For a planar body, Iz = Ix + Iy
Where:
- Iz = moment of inertia about perpendicular axis
- Ix, Iy = moments of inertia about two in-plane axes
2. Symmetry: For a disc, Ix = Iy (same about any diameter)
3. Calculation:
- Iz = MR²/2 (known)
- Iz = Ix + Iy = 2Ix
- Therefore Ix = Iz/2 = MR²/4
Extension Questions
1. Moment of inertia of a ring about any diameter:
Using the same approach, for a ring Iz = MR², so Ix = Iy = MR²/2
2. Applicability to solid cylinder:
The perpendicular axes theorem only applies to planar (2D) bodies. A solid cylinder is a 3D object, so the theorem doesn't apply directly to it.
