Moment of Inertia of Cone

Moment of inertia of a cone can be expressed using different formulas depending on the structure of the cone. We have to take into account two main types – hollow and solid cones.

Solid Cone

For a solid cone the moment of inertia is found by using the given formula;

I = 3 / 10 MR2

Hollow Cone

For a hollow cone, we determine the moment of inertia using;

I = MR2 / 2

Students can further check out the full derivation of the formulas from the given links:

Moment of Inertia of Circular Cone Derivation

Here we will look at the derivation as well as the calculation for finding the moment of inertia of a uniform right circular cone about an axis.

Moment of Inertia of Circular Cone

We will divide the cone into a small elemental disc where we consider the cone’s radius to be r at a distance x. of width. We will need to find the mass the disc and it is given as;

dm = M / ⅓ π R2H (πr2dx)

dm = 3 M / π R2H (R2 / H2 X x2) dx

dm = 3Mx2dx / H3

Now if we consider the similarity of the triangle we get;

r / x = R / H

r = R / H X x

We also have to recall the moment of inertia of elemental disc which is;

½ dmr2

Meanwhile,

dI = ½ (3Mx2dx / H3) . R2 / H2 X x2

I = ∫dI = 3/2 MR2 / H5 . H5 / 5 = 3MR2 / 10

Moment of Inertia of Circular Cone 1

So if we consider the z-axis then we get;

IZ = 3 / 10 MR2

For x-axis;

Ix = Iy = 3 / 5 m (r2 / 4 + H2)

We will look at one of the simple problems below.

Solved Example To Find Moment Of Inertia Of A Circular Cone

Calculate the moment of inertia of the right circular cone with regards to the x and y-axis. Given, M = 20, R= 4, Height = 2 m.

Moment Of Inertia Of A Circular Cone

Solution:

We will solve the problem by using the right formulas.

For the z-axis;

Iz = 3 / 10 MR2

Substituting the values;

Iz = 3 / 10 X 20 X 4 X 4

Iz = 96 kg m2

For the x-axis;

Ix = Iy = 3 / 5 m (r2 / 4 + H2)

Ix = Iy = 3 / 5 m ( 42 / 4 + 22)

Ix = 3 / 5 x 20 ( 16 / 4 + 4)

Ix = 3 / 5 X 20 X 32 / 4

Ix = 96 kg m2