Question

Find the position of centre of mass of the uniform planner section shown in figure with respect to the origin (O)

Answer 1

The center of mass is a point where all the mass is assumed to be located.

In the above figure , the sheet is divided into 3 segments.

The (x,y) coordinates of the segments can be taken as (a/2,3a/2) for segment 1, for segment 2 it is (a/2,a/2) and for segment 3 are (3a/2,a/2)

Each segment will be having a mass of M/3

Therefore,

$x_{CM}=\frac{1}{M}[\frac{a}{2}\left(\frac{M}{3}\right)+\frac{a}{2}\left(\frac{M}{3}\right)+\frac{3a}{2}\left(\frac{M}{3}\right)]=\frac{5a}{6}$

$y_{CM}=\frac{1}{M}[\frac{3a}{2}\left(\frac{M}{3}\right)+\frac{a}{2}\left(\frac{M}{3}\right)+\frac{a}{2}\left(\frac{M}{3}\right)]=\frac{5a}{6}$