Question

can someone show how to find the coordinates of center of mass of a disk using double integrals?

Answer 1

Let
\[\sigma=\sigma(x,y), dm= \sigma dS, dS, M=\int_{disk} dm, \bar r\]
be the density of mass, the element of mass of the annulus at distance r from the centre O=(0,0), the element of area of the disk, the total mass of the disk and the position of the point P with respect to O respectively.

Answer 2

Then the centre of mass is given by:
\[\bar r _{CM}=\frac {\int\limits_{disk}\bar r dm}{\int\limits_{disk}dm}=\frac {\int\limits_{disk}\bar r \sigma dS}{M}\]

Answer 3

Since \[\bar r = x \hat x+ y \hat y\]
you have to evaluate these two integrals:
(1)\[\int\limits_{disk} \sigma x dS=\int\limits_0^{2 \pi} \int\limits _0^R \sigma (r \cos \theta) (rd \theta dr)\]
(2)\[\int\limits_{disk} \sigma y dS=\int\limits_0^{2 \pi} \int\limits _0^R \sigma (r \sin \theta) (rd \theta dr)\]
If you know how sigma depends on x,y or on r, theta then you can calculate the coordinates x_CM and y_CM of the centre of mass