State and prove the parallel and perpendicular axis theorem

Dear student,
(a) Perpendicular axis theorem:

The theorem of perpendicular axes states that the moment of inertia of a planar body (lamina) about an axis perpendicular to its plane is equal to the sum of its moments of inertia about two perpendicular axes concurrent with perpendicular axis and lying in the plane of the body.

A physical body with centre O and a point mass m,in the xy plane at (xy) is shown in the following figure.

Moment of inertia about x-axis, Ix = mx2

Moment of inertia about y-axis, Iy = my2

Moment of inertia about z-axis, Iz =mz2



(b)Parallel axis theorem:

The theorem of parallel axes states that the moment of inertia of a body about any axis is equal to the sum of the moment of inertia of the body about a parallel axis passing through its centre of mass and the product of its mass and the square of the distance between the two parallel axes.

Suppose a rigid body is made up of n particles, having masses m1m2m3, … , mn, at perpendicular distances r1r2r3, … , rn respectively from the centre of mass O of the rigid body.

The moment of inertia about axis RS passing through the point O:

Regards,

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