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Moment Of Inertia Of A Quarter Circle

Moment of inertia of a quarter circle is usually found or calculated using the given formula;

I = Ο€ 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.

Moment Of Inertia Of Quarter Circle

 

1. We will first have a look at a full circle formula. It is given as;

I = Ο€r4 / 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 = ΒΌ Ο€r4

Jo = Ix + Iy = ΒΌ Ο€r4 + ΒΌ Ο€r4 = Β½ Ο€r4

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 = Β½ (Ο€r2) 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 Ο€r4

= 1/16 (Ο€r2) R2

= 1 /16 (A) R2

= ΒΌ (ΒΌ 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

= β…› (A)R2

= β…› (Ο€r2)R2

= β…› Ο€r4

β‡’ Check Other Object’s Moment of Inertia:

Parallel Axis Theorem