 # Moment Of Inertia Of Hollow Cone

Moment of inertia of hollow cone can be determined using the following expression;

 I = MR2 / 2

Here students will learn and understand how the formula is derived as well as its application.

## Moment Of Inertia Of Hollow Cone Formula Derivation

For deriving the moment of inertia formula of a hollow cone we can basically follow some general guidelines which are;

• Determining the mass density in terms of per unit area.
• Defining parameters and finding the mass of small parts.
• Finding the moment of inertia of the small parts corresponding to a ring.
• Integration 1. There is a hollow cone with radius R, height H and mass M.

Now, we will take the element disc at a slant height l and | radius r having thickness dl and mass dm.

Using the similarity of the triangle we get;

r / x = R / √ R2 + H2

r = R / R / √ R2 + H2 X x

The mass of the elemental disc mass is given by;

dm = M / π R√ R2 + H2 X 2 π rdx

dm = M / π R√ R2 + H2 X 2 X R / R√ R2 + H2 X xdx

dm = 2Mxdx / R2 + H2

2. Calculating the moment of inertia of the elemental disk. It will be given as;

dI = r2 dm

dI = 2Mxdx / R2 + H2 X R2 / R2 + H2X x2

3. Finding the moment of inertia through integration.

I = ∫ 2Mxdx / R2 + H2 X R2 / R2 + H2X x2dx

I = 2MR2 / (R2 + H2)2 ∫ x3dx

Using the limits of x where x usually varies from o to √ R2 + H2

We now get;

I = 2MR2 / (R2 + H2)2 X (R2 + H2)2 / 4

I = ½ MR2

Therefore, I = MR2 / 2