Theorems of Perpendicular and Parallel

The parallel axis theorem is one of the  common method which is used to define the moment of inertia of an inflexible body about some axis provided the moment of inertia of the object about the axis through the centre of mass of the object and the perpendicular distance between the axes.
In this article we are going to see about the theorems of parallel and the perpendicular axes.

Parallel axes theorem:

                          Suppose that the perpendicular distance between the axes that lies along the horizontal axis and the centre of the mass that lies at the origin in any Cartesian coordinate plane. Then the moment of inertia which is relative to the z-axis that passes through the centre t of mass is given by,
                                                                                   A_tn = `int` (a² + b²) dn
The moment of inertia that is corresponding to the newer axis, the perpendicular distance d along the horizontal axis from the centre of the mass is given by,  A _≈ = `int` ((a-d) ² + b²) dn
If we solve the above expression we get, A _≈ = `int` (a² + b²) dn + d² `int` dn – 2d `int` a dn
Here the initial term is A_tn and the second term is nd² and the last term is zero because the origin is said to be at the centre of the mass. Then the expression is given by, A _≈ = A_tn+ nd²
Hence the parallel axis theorem has been proved.

Perpendicular axis theorem:

• The perpendicular axis theorem is defined as the one which is used to illustrate the moment of inertia of an inflexible body that lies on a plane about the axis that is right angles to the plane provided the moment of inertia of the object about the two perpendicular axes lies within that plane.

• Let us consider A_x be the moment of inertia of the rigid body about the horizontal axis, A_y be the moment of inertia of the body about the vertical axis and A_z be the moment of inertia of the object about the y axis.

• Then according to the theorem of perpendicular axis

                                                 A_z = A_x + A_y

• The theorem of perpendicular axes can be applied with that of the parallel axis theorem to find the moment of inertia of various objects.

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