Moment of Inertia of a Ring About a Tangent to the Circle

The moment of inertia of an object is a measure of its resistance to rotational motion. It is defined as the sum of the products of the mass and the square of the distance from each particle in the object to its axis of rotation. The moment of inertia of a ring of mass M and radius R about a tangent to the circle of the ring can be calculated as follows:

Derivation:

Consider a ring of mass M and radius R. Let us take a tangent to the ring at any point A as shown in the figure below.



Step 1:

Divide the ring into small elemental rings of thickness dx as shown in the figure below.



Step 2:

The moment of inertia of each elemental ring about the tangent at A is given by dI = dm x r^2, where dm is the mass of the elemental ring and r is the distance of the elemental ring from the tangent at A.

Step 3:

The mass of each elemental ring can be calculated as follows:

dm = M x (dx / 2πR)

where dx / 2πR is the fraction of the circumference of the ring that corresponds to the thickness of the elemental ring dx.

Step 4:

The distance of each elemental ring from the tangent at A can be calculated as follows:

r = R - x

where x is the distance of the elemental ring from the point A.

Step 5:

Substituting the values of dm and r in the equation for dI, we get:

dI = M x (dx / 2πR) x (R - x)^2

Step 6:

The total moment of inertia of the ring about the tangent at A can be obtained by integrating dI over the entire ring:

I = ∫dI = ∫(M x (dx / 2πR) x (R - x)^2)

= (M / 2πR) x ∫(R^2 - 2Rx + x^2)dx

= (M / 2πR) x (R^2x - x^2 + x^3/3)|0 to 2πR

= (M / 2πR) x (2πR)^3/3

= (M / 2) x R^2

Hence, the moment of inertia of a ring of mass M and radius R about a tangent to the circle

2
3/4MR