Step 1: Understanding the Question:
The system consists of three uniform rods of equal mass \(M\). We treat each rod as a point mass at its geometric center (midpoint) to find the center of mass (COM) of the combined system. Step 3: Detailed Explanation:
From the figure, the three rods are:
1. Rod 1: Along the y-axis from \((0,0)\) to \((0,a)\). Its COM is \(C_{1} = (0, a/2)\).
2. Rod 2: Along the x-axis from \((0,0)\) to \((2a,0)\). Its COM is \(C_{2} = (a, 0)\).
3. Rod 3: The slanted rod (hypotenuse) from \((0,a)\) to \((2a,0)\). Its COM is the midpoint of these two points:
\[ C_{3} = \left(\frac{0+2a}{2}, \frac{a+0}{2}\right) = (a, a/2) \]
Since all rods have the same mass \(M\), the COM of the system is the average of their individual COM coordinates:
\[ X_{\text{COM}} = \frac{x_{1} + x_{2} + x_{3}}{3} = \frac{0 + a + a}{3} = \frac{2a}{3} \]
\[ Y_{\text{COM}} = \frac{y_{1} + y_{2} + y_{3}}{3} = \frac{a/2 + 0 + a/2}{3} = \frac{a}{3} \]
The co-ordinates are \(\left(\frac{2a}{3}, \frac{a}{3}\right)\). Step 4: Final Answer:
The centre of mass is \(\left(\frac{2a}{3}, \frac{a}{3}\right)\).
