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.
For a solid cone the moment of inertia is found by using the given formula;
I = 3MR2 / 10
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 Solid 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.
We will divide the cone into a small elemental disc where we consider the cone’s radius to be r at a distance x from the top and of thickness dx. We will need to find the mass of the disc and it is given as;
Using the value of r in equa (1)
Now, let us find the moment of inertia of the elemental disc
dI = (1/2) dmr2
I = 3MR2 / 10
So if we consider the z-axis then we get;
IZ = 3MR2 / 10
Ix = Iy = 3m (r2 / 4 + h2) / 5
We will look at one of the simple problems below.
Solved Example To Find Moment Of Inertia Of A Solid 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.
We will solve the problem by using the right formulas.
For the z-axis;
Iz = 3 MR2/ 10
Substituting the values;
Iz = 3 x 20 x 4 x 4/ 10
Iz = 96 kg m2
For the x-axis;
Ix = Iy = 3 m [(r2 / 4) + H2]/ 5
Ix = Iy = 3 x 20 x [( 42 / 4) + 22)] / 5
Ix = 3 x 20 x [( 16 / 4) + 4)]/ 5
Ix = 3 x 20 x 8/5
Ix = 96 kg m2
Parallel Axis Theorem