In the question number 25, if the mass placed at vertex A is doubled, then the force acting on the mass 2m placed at the centroid O is

Options

A. zero
B. `(2Gm^(2))/(l^(2))`
C. `(5Gm^(2))/(l^(2))`
D. `(6Gm^(2))/(l^(2))`

Explanation

To solve the problem, we need to analyze the forces acting on the mass \(2m\) placed at the centroid \(O\) of a triangle when the mass at vertex \(A\) is doubled.

### Step-by-Step Solution:

1. **Identify the Initial Setup**:
- Let the mass at vertex \(A\) be \(m\).
- The masses at vertices \(B\) and \(C\) remain unchanged.
- The mass at the centroid \(O\) is \(2m\).

2. **Determine the Forces Acting on Mass at Centroid**:
- The gravitational force acting on the mass at vertex \(A\) (now \(2m\)) is given by:
\[ F_{OA} = \frac{G \cdot 2m \cdot m}{d_{OA}^2} \]
- The distances \(d_{OA}\), \(d_{OB}\), and \(d_{OC}\) are the distances from the centroid to the vertices \(A\), \(B\), and \(C\) respectively. For an equilateral triangle, these distances can be calculated as:
\[ d_{OA} = \frac{L}{\sqrt{3}}, \quad d_{OB} = d_{OC} = \frac{L}{\sqrt{3}} \]
- Thus, the force due to mass at vertex \(A\) becomes:
\[ F_{OA} = \frac{G \cdot 2m \cdot m}{\left(\frac{L}{\sqrt{3}}\right)^2} = \frac{2Gm^2 \cdot 3}{L^2} = \frac{6Gm^2}{L^2} \]

3. **Calculate Forces from Masses at Vertices \(B\) and \(C\)**:
- The gravitational force due to mass at vertex \(B\) and \(C\) (both are \(m\)) is:
\[ F_{OB} = F_{OC} = \frac{G \cdot m \cdot 2m}{\left(\frac{L}{\sqrt{3}}\right)^2} = \frac{G \cdot 2m^2 \cdot 3}{L^2} = \frac{6Gm^2}{L^2} \]

4. **Determine the Resultant Forces**:
- The forces \(F_{OB}\) and \(F_{OC}\) act in opposite directions to \(F_{OA}\).
- The net force acting on mass \(2m\) at centroid \(O\) is:
\[ F_{net} = F_{OA} - (F_{OB} + F_{OC}) = \frac{6Gm^2}{L^2} - \left(\frac{6Gm^2}{L^2} + \frac{6Gm^2}{L^2}\right) \]
- Simplifying this gives:
\[ F_{net} = \frac{6Gm^2}{L^2} - \frac{12Gm^2}{L^2} = -\frac{6Gm^2}{L^2} \]

5. **Final Result**:
- The negative sign indicates that the net force is directed towards the centroid \(O\). Therefore, the force acting on the mass \(2m\) placed at the centroid \(O\) is:
\[ F_{net} = \frac{6Gm^2}{L^2} \]