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The moment of inertia (also known as the second moment) is a physical quantity which quantifies the rotational inertia of an object. The moment of inertia of ellipse can be considered as the rotational analogue of mass in the linear motion. The moment of inertia of a body is always interpreted about a rotation axis. This is a geometrical property of an area that speaks of how its points are distributed in context to an arbitrary axis. The unit of the dimension of moment of inertia of area is length to 4th power, L4. This must not be confused with the mass moment of inertia. If the piece is thin, nonetheless, the mass moment of inertia is equivalent to the area density times the area moment of inertia.

It is not very usual that we find the derivation for the moment of inertia of an ellipse or elliptical object. This makes it even more intriguing to know how to derive the equation for calculating the moment of inertia of an elliptical disc. Remember that there are more than one ways of calculating the moment of inertia of an ellipse. Here we are describing the more generalized method that can also be used for a variety of different shapes. Let’s first check the steps involved in the derivation of moment of inertia of an ellipse.

The steps are as follows:

Identifying the parametric equation of the shape.

Using the equation of parametric as a transformation.

Determining the Jacobian.

Computing the Moment of Inertia.

Let us illustrate the process of derivation via the example of an ellipse.

We will begin with the parametric equation of an ellipse which is given by,

(a cos t) (b sin t)

To obtain the equations of transformation, we are required to modify the parametric equation as,

= (r cos θ, λr sin θ) (r cosθ, λ r sinθ)

In which,

λ = ba λ =ba.

= b / a

Remember that ‘a’ and ‘b’ are the semi-major and semi-minor axis of an ellipse respectively. Here we take ‘a’ as the semi-major axis and ‘b’ as the semi-minor. However, it's only a matter of choice and you can choose however you want. In this case, the positioning of the elliptical plate needs to be rotated to again have the semi-major axis on the x-axis.

Now, the transformation equation shall be,

x= r cosθ

y=λ r sinθ

Next, we require is the Jacobian to obtain the area element here. Jacobian will be given as:

J = \[\begin{vmatrix} ∂x/∂r & ∂x/∂θ \\ ∂y/∂r & ∂y/∂θ \end{vmatrix}\]

J = \[\begin{vmatrix} cosθ & − rsinθ \\ λ sinθ & rλ cosθ \end{vmatrix}\]

=rλ

With this, we are done with the setup of transformation. Now, all we need is the area and then the moment of inertia.

Calculating Area Moment of Inertia

One thing to mention is that r will be integrated from 0 to the largest value it will acquire, which is the semi-major axis i.e. ‘a’.

A=0∫a0∫2πλrdθA

A = λ a2π

=πab

Calculating Moment of Inertia

The Formula for moment of inertia, in this case, will be given by:

I=ρ ∫(x2 +y2)dA

=ρ0∫a0∫2πλ r3(cos2θ + λ2sin2θ)drdθ

=ρλ a4/40∫2π(cos2θ + λ2sin2θ)dθ

Taking into account the symmetry we can express this as,

I=ρλ a4 ½ 20∫2π(cos2θ + λ2sin2θ)dθ

=ρλ a4 ½ [B(1/2 , 3/2) + λ2b[3/2,1/2]

Where ‘B’ denotes the beta function.

Using the beta gamma relation we get,

I=ρ λab / 4(a2+b2)

=M(a2 + b2) / 4

This brings us to the answer with complete derivation

For symmetrical solid like an elliptical cylinder rotated about the symmetry axis or an elliptical or circular disk about any of the axis of symmetry, Routh’s rule implies that the second moment (moment of inertia) “I” of a body of mass M about an axis is mathematically given by:

I = M Sum of the Squares of the Perpendicular Axis/3, 4 or 5

Where,

M = The mass of the body

3 = A rectangular body

4 = An elliptical body

5 = An ellipsoidal body.

Since it is quite simple to express this rule, so let’s workout for one of the body types taking the example of an elliptical disc.

As you might know, area moment of inertia is common in engineering disciplines such as mechanical and civil. However, the mass moment of inertia is much more common in physics. Therefore, working out the moment of inertia shall provide us with an intuitive idea of the working of this object.

We shall work out the analytical treatment of the elliptical body to resolve any doubts you might have with respect to the moment of inertia of an ellipse. Once you are clear with finding the moment of inertia of an ellipse, calculating the moment of inertia for other common object types shall be easy.

[Image will be uploaded soon]

In the above diagram, we have two axes as ‘a’ and ‘b’. Thus, in the denominator, we have:

a2+b2

In the denominator, we are having 4 for an elliptical disc. Therefore by the Routh’s rule we have the moment of inertia as:

Iz – M [a2+b2] / 4

FAQ (Frequently Asked Questions)

Q1. What is the Physical importance of Mass Moment of Inertia?

Ans: Moment of inertia denoted by, I, is involved in explaining the circular motion of an object. MOI is analogous to mass in linear motion.

By Newton's second law F=ma (mass is constant)

For the circular motion to take place, torque τ replaces force F

So,

τ = I α,

where α = angular acceleration.

This suggests that if the angular velocity increases rapidly, then there will be a greater angular acceleration and thus a larger torque is needed.

Also, if ‘I’ increases, a larger torque is required to generate the same angular acceleration.

Also, the rotational kinetic energy = 1/2 I × ω^{2}

Now, I depend on the geometry of the mass distribution. More the object mass is distributed from the centre of rotation, the larger I is. This describes why a skater revolves quicker when they pull their arms. In doing so, they are decreasing their MOI, so, to preserve rotational kinetic energy, their angular velocity increases!