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

2N,6N,5N

To keep a particle in equilibrium, the following conditions must be met:

1. The vector sum of all the forces acting on the particle must be zero.
2. The vector sum of all the torques (or moments) acting on the particle about any chosen axis of rotation must be zero.

So, to determine which sets of forces keep a particle in equilibrium, we need to consider both force equilibrium and torque equilibrium.

Force equilibrium: The vector sum of all forces acting on the particle must be zero. This means that the forces must balance each other out in all directions.

Torque equilibrium: The vector sum of all torques acting on the particle about any chosen axis of rotation must be zero. This condition ensures that the particle is not rotating or spinning.

Therefore, any set of forces that satisfy both force equilibrium and torque equilibrium will keep the particle in equilibrium.

To determine the set of forces that keep a particle in equilibrium, we need to consider the conditions for equilibrium, which are:

1. The net force acting on the particle is zero.
2. The net torque acting on the particle is zero (if the particle is in rotational equilibrium).

Now, let's analyze each set of forces to see which ones satisfy these conditions:

1. Set 1: Only one force is acting on the particle.
Since there is only one force, the net force cannot be zero in this case, so the particle is not in equilibrium.

2. Set 2: Two forces are acting on the particle, both along the same line of action.
If the two forces are equal in magnitude and opposite in direction, then the net force will be zero and the particle will be in equilibrium. However, if the forces have different magnitudes or are not opposite in direction, the net force will not be zero, and the particle will not be in equilibrium.

3. Set 3: Two forces are acting on the particle, but they are not collinear.
In this case, the net force will not be zero because the forces do not cancel each other out along a single line of action. Therefore, the particle will not be in equilibrium.

4. Set 4: Three forces are acting on the particle, all along the same line of action.
Similar to Set 2, if the three forces are equal in magnitude and opposite in direction, the net force will be zero and the particle will be in equilibrium. However, if the forces have different magnitudes or are not opposite in direction, the net force will not be zero, and the particle will not be in equilibrium.

5. Set 5: Three forces are acting on the particle, not all along the same line of action.
In this case, even if the forces have different magnitudes and directions, the particle can still be in equilibrium if the vectors representing the forces form a closed triangle (i.e., the vector sum of the forces is zero).

Therefore, the sets of forces that keep a particle in equilibrium are:
- Set 2, if the two forces are equal in magnitude and opposite in direction.
- Set 4, if the three forces are equal in magnitude and opposite in direction along the same line of action.
- Set 5, if the three forces form a closed triangle.