Moment of Inertia Proof | Plane Figure Theorem

Perpendicular axis is used to find the moment of inertia for a solid object, that lies about an axis perpendicular to the plane is equal to the sum of moment of inertia of two perpendicular axes lying in the same point within the plane. All the axis passes through the same point in the plane. It is also known as plane figure theorem and is more helpful for objects of regular form like cylinder. This perpendicular axis theorem / plane figure theorem helps you to learn the moment of inertia proof.

Perpendicular Axis Theorem

Statement:

Let us consider a plane lamina lying in the XOY plane. The lamina is made up of a large number of particles. Consider a particle of mass 'm' at P. From P, PN and PN' are drawn perpendicular to X-axis and Y-axis, respectively.
Now PN'= x, PN = y
Moment of inertia about X-axis = my2
MI of the lamina about X-axis
Ix = Σ my2
MI of the lamina about Y-axis
Iy = Σ mx2
MI of the lamina about Z-axis
Iz = Ix + Iy
Where,
Ix = Moment of Inertia X - Axis
Iy = Moment of Inertia Y - Axis
Iz = Moment of Inertia Z - Axis

Proof:

Working in Cartesian co-ordinates, the moment of inertia of the planar body about the z - axis is given by:
Iz = ∫(x2 + y2) dm
Iz = ∫ x2 dm + ∫ y2 dm
Iz = Ix + Iy
On the plane, z=0, so these two terms are the moments of inertia about the x and y axes respectively, giving the perpendicular axis theorem. The converse of this theorem is also derived similarly. Note that ∫ x2 dm = Iy ≠ Ix because in ∫ r2 dm , r measures the distance from the axis of rotation, so for a y-axis rotation, deviation distance from the axis of rotation of a point is equal to its x co-ordinate.

The moment of inertia of any object can be determined dynamically with the Perpendicular Axis Theorem.


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