Question

Io = Ic + mr² prove that
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Answer 1

Step-by-step explanation:

Perpendicular Axis Theorem

This theorem is applicable only to the planar bodies. Bodies which are flat with very less or negligible thickness. This theorem states that the moment of inertia of a planar body about an axis perpendicular to its plane is equal to the sum of its moments of inertia about two perpendicular axes concurrent with the perpendicular axis and lying in the plane of the body.

In the above figure, we can see the perpendicular body. So Z axis is the axis which is perpendicular to the plane of the body and the other two axes lie in the plane of the body. So this theorem states that

IZ = Ix + Iy

That means the moment of inertia about an axis which is perpendicular to its plane is equal to the sum of its moments of inertia about two perpendicular axes.

Let us see an example of this theorem:

Suppose we want to calculate the moment of inertia of a uniform ring about its diameter. Let its centre be MR²/2, where M is the mass and R is the radius. So, by the theorem of perpendicular axes, IZ = Ix + Iy. Since the ring is uniform, all the diameters are equal.

∴ Ix = Iy

∴ IZ = 2 Ix

Iz = MR²/4

So finally the moment of inertia of a disc about any of its diameter is MR²/4.

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