Look at the drawing given in the figure which has been drawn with ink of uniform line-thickness. The mass of ink used to draw each of the two inner circles, and each of the two line segments is $$m.$$  The mass of the ink used to draw the outer circle is $$6 \,m.$$
The coordinates of the centres of the different parts are: outer circle $$\left( {0,0} \right),$$  left inner circle $$\left( { - a,a} \right),$$  right inner circle $$\left( {a,a} \right),$$  vertical line $$\left( {0,0} \right)$$  and horizontal line $$\left( {0, - a} \right).$$  The $$y$$-coordinate of the centre of mass of the ink in this drawing is

Solution :
The system is made up of five bodies (three circles and two straight lines) of uniform mass distribution. Therefore we assume the system to be made up of five point masses where the mass of each body is considered at its geometrical centre.

The y-coordinate of the centre of mass is
$$\eqalign{ & {y_{cm}} = \frac{{{m_1}{y_1} + {m_2}{y_2} + {m_3}{y_3} + {m_4}{y_4} + {m_5}{y_5}}}{{{m_1} + {m_2} + {m_3} + {m_4} + {m_5}}} \cr & \therefore {y_{cm}} = \frac{{6m \times 0 + m \times 0 + m \times a + m \times a + m\left( { - a} \right)}}{{6m + m + m + m + m}} \cr & = \frac{{ma}}{{10m}} \cr & = \frac{a}{{10}} \cr} $$