Moment Of Inertia Of Sphere

Moment of inertia of sphere is normally expressed as;

I = ⅖ MR2

Here, r and m are the radius and mass of the sphere respectively. Students have to keep in mind that we are talking about the moment of inertia of a solid sphere about its central axis above. Additionally, if we talk about the moment of inertia of the sphere about its axis on the surface it is expressed as;

I = 7/5 MR2

Moment Of Inertia Of Sphere Derivation

Moment Of Inertia Of Sphere

The moment of inertia of a sphere expression is obtained in two ways.

  • First, we take the solid sphere and slice it up into infinitesimally thin solid cylinders.
  • Then we have to sum the moments of exceedingly small thin disks in a given axis from left to right.

We will look and understand the derivation below.

First, we take the moment of inertia of a disc that is thin. It is given as;

I = ½ MR2

In this case, we write it as;

dI (infinitesimally moment of inertia element) = ½ r2dm

Find the dm and dv using;

dm = p dV

p = moment of a thin disk of mass dm

dv = expressing mass dm in terms of density and volume

dV = π r2 dx

Now we replace dV into dm. We get;

dm = p π r2 dx

And finally, we replace dm with dI.

dI = ½ p π r4 dx

The next step involves adding x to the equation. If we look at the diagram we see that r, R and x forms a triangle. Now we will use Pythagoras theorem which gives us;

r2 = R2 – x2

Now if we substitute the values we get;

dI = ½ p π (R2 – x2)2 dx

This leads to:

I = ½ p π -RR (R2 – x2)2 dx

After integration and expanding we get;

I = ½ pπ 8/15 R5

Additionally, we have to find the density as well. For that we use;

p = m / V

p = m / m/v πR3

If we substitute all the values;

I = 8/15 [m / m/v πR3 ] R5

I = ⅖ MR2