Moment of inertia of a quarter circle is usually found or calculated using the given formula;
I = π R4 / 16
In this lesson, we will learn how to derive the formula as well as its application in problems.
Moment Of Inertia Of Quarter Circle Derivation
When we are deriving the moment of inertia expression for a quarter circle, we can partly use the same derivation that is followed for determining the moment of inertia of a circle. The concept is more or less the same.
1. We will first have a look at a full circle formula. It is given as;
I = πr4 / 4
If we want to derive the equation for a quarter circle then we basically have to divide the results obtained for a full circle by two and get the result for a quarter circle. Notably, in a full circle, the moment of inertia relative to the x-axis is the same as the y-axis.
With that concept we get;
Ix = Iy = ¼ πr4 = ¼ πr2) R2
Jo = Ix + Iy = ¼ πr4 + ¼ πr4 = ½ πr4
We will need to determine the area of a circle as well. When we are solving this expression we usually replace M with Area, A.
Jo = ½ (πr2) R2
Jo = ½ (πr2) R2
Now if take a quarter circle, the moment of inertia relative to the x-axis and y-axis will be one quarter the moment inertia of a full circle. However, the part of the circle rotating about an axis will be symmetric and the values will be equal for both the y and x-axis. With that, we will solve the equation below.
Ix = Iy = 1/16 πr4
= 1/16 (πr2) R2
= 1 /16 (A) R2
= ¼ (¼ Ao) R2
The next step involves finding the moment of inertia of a quarter circle. For this, we will simply add the values of both x and y-axis.
M.O.I relative to the origin, Jo = Ix + Iy
= 1 / 16 (A)R2 + 1 / 16 (A)R2
= ⅛ (A)R2
= ⅛ (πr2)R2
= ⅛ πr4