# Why changing the mass of forth object does not change the center of mass in the given problem?

The problem is as follows. Four masses of $$1$$ $$kg$$, $$2$$ $$kg$$, $$3$$ $$kg$$, and $$4$$ $$kg$$ are arranged in square shape. The side length of the square is $$1$$ $$m$$. Find the location of the center of mass of this system.

I have found the solution to it to be ($$1/2$$, $$3/10$$) by representing the masses as points on the cortisone plane like here. However in the calculations, $$4$$ $$kg$$ is always multiplied by zero, so that made me wonder if the center of mass is the same even if the fourth object is $$1000$$ $$kg$$. Why is that?

No, It would change. Note the expression $$\mathbf{r}_\text{cm}=\frac{\sum_i m_i\mathbf{r}_i}{\sum_im_i}=\frac{m_1\mathbf{r}_1+\sum_im_i\mathbf{r}_i}{m_1+\sum_i m_i}$$ If $$\mathbf{r}_1=0$$, $$\mathbf{r}_\text{cm}=\frac{\sum_{i=2}m_i\mathbf{r}_i}{m_1+\sum_{i=2}m_i}$$ Note $$m_1$$ in the denominator.
In the figure, the origin is taken at the position of the 4kg point mass. Now the coordinates of the origin are $$(0,0)$$ i.e. for the 4kg mass, the position is $$x=0$$ and $$y=0$$. As a result, the 4kg is multiplied with zero in the calculation of the centre of mass of the system.