Which of the following sets of forces acting simultaneously on a particle keep it in equilibrium?

Options

A. `3N,5N,10N`
B. `4N,5N,12N`
C. `2N,6N,5N`
D. `5N,8N,1N`

Explanation

To determine which set of forces acting simultaneously on a particle keeps it in equilibrium, we need to apply the principle of equilibrium. For a particle to be in equilibrium, the vector sum of all forces acting on it must be zero. This can be mathematically expressed as:

\[ F_1 + F_2 + F_3 = 0 \]

Alternatively, for any two forces acting on the particle, their sum must equal the third force, which can be expressed as:

\[ F_1 + F_2 = F_3 \]

or any permutation of the forces.

Let's analyze the given options step by step:

### Step 1: Analyze the Forces
Assume we have three forces \( F_1, F_2, \) and \( F_3 \) with their respective magnitudes. We will check each option to see if the sum of any two forces equals the third force.

### Step 2: Check Each Option
1. **Option 1: Forces 3, 5, and 8**
- Check: \( 3 + 5 = 8 \) (True)
- Check: \( 3 + 8 > 5 \) (True)
- Check: \( 5 + 8 > 3 \) (True)
- Conclusion: This option does not satisfy the equilibrium condition since \( 3 + 5 = 8 \) is equal, but does not indicate equilibrium.

2. **Option 2: Forces 4, 5, and 9**
- Check: \( 4 + 5 = 9 \) (True)
- Check: \( 4 + 9 > 5 \) (True)
- Check: \( 5 + 9 > 4 \) (True)
- Conclusion: This option does not satisfy the equilibrium condition since \( 4 + 5 = 9 \) is equal, but does not indicate equilibrium.

3. **Option 3: Forces 2, 6, and 5**
- Check: \( 2 + 6 = 8 \) (False)
- Check: \( 2 + 5 > 6 \) (True)
- Check: \( 6 + 5 > 2 \) (True)
- Conclusion: This option satisfies the condition since the sum of any two forces is greater than the third force, indicating that they can balance each other out.

### Conclusion
From the analysis, only **Option 3 (2, 6, and 5)** satisfies the conditions for equilibrium.