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 $6m$. The coordinates of the centres of the different parts are: outer circle $(0, 0)$, left inner circle $(-a, a)$, right inner circle $(a, a)$, vertical line $(0, a)$ and horizontal line $(0, -a)$. The $y$-coordinate of the centre of mass of the ink in this drawing is

Step 1: Identify the mass and coordinates of each component.

The system consists of five components with the following masses and center coordinates:

• Outer circle: mass $6m$ at $(0, 0)$
• Left inner circle: mass $m$ at $(-a, a)$
• Right inner circle: mass $m$ at $(a, a)$
• Vertical line: mass $m$ at $(0, 0)$
• Horizontal line: mass $m$ at $(0, -a)$

Step 2: Calculate the total mass of the system.

Sum the masses of all components.

$$M = 6m + m + m + m + m$$ $$M = 10m$$

Step 3: Calculate the weighted sum of the $y$-coordinates.

Multiply each mass by its corresponding $y$-coordinate and sum them.

$$\sum m_i y_i = 6m(0) + m(a) + m(a) + m(0) + m(-a)$$ $$\sum m_i y_i = ma$$

Step 4: Calculate the $y$-coordinate of the center of mass.

Apply the formula for the center of mass.

$$y_{cm} = \frac{\sum m_i y_i}{M}$$ $$y_{cm} = \frac{ma}{10m}$$ $$y_{cm} = \frac{a}{10}$$

Answer: The $y$-coordinate of the center of mass is $\dfrac{a}{10}$.