Parallel Axis & Perpendicular Axis Theorems And Their Applications

Parallel axis theorem and perpendicular axis theorem are used to solve problems on moment of inertia, let us discuss the two theorems,

Parallel axis theorem states that, the moment of inertia of I of a body about any axes is same as the moment of inertia IG about an axis parallel to the body passing through its center of gravity plus Mb2, where M is the mass of that body and b is the distance between the axis and the center of gravity. We can put this statement in an expression as follows,

\( I = I_G~+~MB^2 \)

Parallel axis & Perpendicular axis theorems

Theorem of perpendicular axis , states that for any plane body, example a rectangular sheet the moment of inertia about any of its axes which are perpendicular to the plane is equal to the sum of the moment of inertia about any two perpendicular axes in the plane of the body which intersect the first axis in the plane, this theorem is used when the body is symmetric in shape about two out of the three axes, if moment of inertia about two of the axes are known the moment of inertia about the third axis can be found using the expression,

Parallel axis & Perpendicular axis theorems

\( I_a = I_b~+~I_c \)

Say in an engineering application we have to find the moment of inertia of a body, but the body is irregularly shaped, and the moment of in these cases we can make use of the parallel axis theorem to get the moment of inertia at any point as long as we know the center of gravity of the body, this is a very useful theorem in space physics where calculation of moment of inertia of space-crafts and satellites, making possible for us to reach the outer planets and even the deep space.The theorem of perpendicular axis helps in applications where we don’t have access to one axis of a body and it is vital for us to calculate the moment of inertia of the body in that axis.

Assignment: if the moment of inertia of a body along a perpendicular axis passing through its center of gravity is 50 kg·m2 and the mass of the body is 30 Kg. What is the moment of inertia of the same body along another axis which is 50 cm away from the current axis and parallel to it?

Solution: From parallel axis theorem,

I = \( I_G~+~Mb^2 \)

I = 50 + ( 30 \times \( 0.5^2 \) )

I = 57.5 kg – \( m^2 \)

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Practise This Question

One quarter section is cut from a uniform circular disc of radius R. This section has a mass M. It is made to rotate about a line perpendicular to its plane and passing through the centre of the original disc. Its moment of inertia about the axis of rotation is