Calculate moment of inertia of uniform disc about diameter using perpendicular axis theorem

Question

Using the perpendicular axis theorem, find the moment of inertia of a uniform disc of mass $M$ and radius $R$ about a diameter.

(JEE Main 2023, similar pattern)

Solution — Step by Step

For a uniform disc, the MOI about an axis through the centre and perpendicular to the plane is:

I_z = \frac{MR^2}{2}

This axis (call it $z$ ) passes through the centre and is normal to the disc.

I_z = I_x + I_y

Valid for planar (flat) bodies only. $x$ , $y$ are any two perpendicular axes in the plane, and $z$ is perpendicular to the plane — all passing through the same point.

Take $x$ and $y$ as two perpendicular diameters. By the theorem:

I_z = I_x + I_y

A disc has circular symmetry — every diameter is equivalent. So $I_x = I_y$ .

\frac{MR^2}{2} = I_x + I_x = 2I_x

\boxed{I_{\text{diameter}} = I_x = \frac{MR^2}{4}}

Why This Works

The perpendicular axis theorem relates three MOIs at the same point. For a flat body, the moment about the normal axis equals the sum of moments about any two perpendicular in-plane axes. This works because $r^2 = x^2 + y^2$ for every mass element, and the moment integrals add up.

The symmetry argument ( $I_x = I_y$ ) is crucial. It only works because a disc looks the same from every diameter. For a rectangle, the two in-plane axes would give different MOIs, and you’d need to know one to find the other.

Alternative Method — Direct integration

Using $I = \int r^2\,dm$ where $r$ is the perpendicular distance from the diameter:

For a disc element at distance $y$ from the $x$ -axis (a diameter), with surface mass density $\sigma = M/(\pi R^2)$ :

I_x = \int_{-R}^{R} y^2 \cdot 2\sqrt{R^2 - y^2} \cdot \sigma\,dy = \frac{MR^2}{4}

(This integral requires the substitution $y = R\sin\phi$ .)

The perpendicular axis theorem avoids this entire calculation by leveraging what we already know about $I_z$ .

This result ( $MR^2/4$ about a diameter) is one you should memorise. Combine it with the parallel axis theorem to find MOI about any axis parallel to a diameter: $I = MR^2/4 + Md^2$ , where $d$ is the distance from the centre.

Common Mistake

Students sometimes try to apply the perpendicular axis theorem to a sphere or solid cylinder. The theorem works ONLY for planar (flat) bodies — disc, ring, rectangular plate, etc. A sphere is 3D; using $I_z = I_x + I_y$ for a sphere gives an incorrect answer. For 3D bodies, use the parallel axis theorem or direct integration instead.

Question

Solution — Step by Step

Why This Works

Alternative Method — Direct integration

Common Mistake

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