Moment Of Inertia Of A Ring

Moment of Inertia of a ring can be determined using various expressions. It includes;

  • When the axis is passing through the centre (z-axis) and is perpendicular to it. For this we use;

I = mR2

  • Along the axis which passes through the diameter. For this we have;

Ix = Iy = mR2 / 2

We will further look at how the equations are derived.

We will derive the moment of inertia of a ring for both instances below.

1. First, we will look at a ring about its axis passing through the centre.

Moment Of Inertia Of Ring

We will assume the mass of the ring to be M and radius be R.

Now we need to cut an elemental ring (dx) at the circumference of the ring. Hence, the mass (dm) of the elemental ring will be;

dm = m / 2πR dx

Next, we calculate I = (dm) R2

Substituting the values, we get;

I = m / 2πR dx R2

Using integration;

I = m / 2π o2πR dx

I = mR / 2π 2πR

I = mR2

2. For the second expression, we will be dealing with the moment of inertia of a ring about an axis passing through its diameter.

Moment Of Inertia Of Ring

Here we will recall the formula;

I = ∫ r′2dm

We need to find dm. We will get;

Dm = M / 2π dθ

Meanwhile, r′ = r cos θ

The next step is to carry out the integration. We will have;

I = o r2 cos2 θ M / 2π dθ

I = Mr2 / 2π o cos2 θ dθ

I = Mr2 / 2π [θ / 2 + sin 2 θ / 4] |o

I = Mr2 / 2π [ ( π + 0) – (0 + 0)]

I = Mr2 / 2