00:01
In this question we have to find out what is the centroid of the given semi -annular, semi -circular annulus.
00:10
So this is our x -axis, this is y -axis, this is the annulus that we have been given.
00:18
Inner radius is r where the outer radius is capital r.
00:26
First of all what we can do, we know that we have to just find out the y coordinate correct it is given by integral y da over a.
00:38
So this simply comes out to be 1 over a double integral of y is rho sine theta rho d rho d theta correct.
00:51
So this will be nothing but how we are getting these things.
00:55
So first of all what we do is consider an element at an angle of theta correct, so this is the element we will consider and this element if we talk about will give da equals to rho d rho d theta where rho is the distance from the origin distance from the origin so y bar comes out to be 1 over a limit for theta.
01:38
It varies from 0 to pi since it is a semicircle.
01:42
And rho the distance is varying from a small r to capital r.
01:47
Here rho times rho will become rho square sine theta d rho d theta.
01:54
Now we have to simplify this as 1 over a integral 0 to pi...