Problem:
A mass of 6 kg is suspended by a rope of length 2 m from a ceiling. A force of 50 N in the horizontal direction is applied at the mid-point of the rope. What is the angle the rope makes with the vertical in equilibrium?

Solution:
To find the angle the rope makes with the vertical in equilibrium, we can use the concept of equilibrium of forces.

Step 1: Identify the forces:
In this problem, we have the following forces:
1. The weight of the mass, which acts vertically downwards.
2. The tension in the rope, which acts upwards and at an angle.
3. The applied force, which acts horizontally.

Step 2: Resolve the forces:
We can resolve the forces into their horizontal and vertical components.

1. Weight of the mass (mg):
The weight of the mass can be resolved into two components:
- Vertical component (mgcosθ): This component acts downwards and is balanced by the tension in the rope.
- Horizontal component (mgsinθ): This component acts horizontally and is balanced by the applied force.

2. Tension in the rope (T):
The tension in the rope can be resolved into two components:
- Vertical component (Tcosθ): This component acts upwards and balances the vertical component of the weight.
- Horizontal component (Tsinθ): This component acts horizontally and balances the horizontal component of the weight.

3. Applied force (F):
The applied force acts horizontally and is balanced by the horizontal component of the tension.

Step 3: Set up the equations:
In equilibrium, the sum of the horizontal and vertical components of the forces must be zero.

For the vertical components:
Tcosθ - mgcosθ = 0

For the horizontal components:
Tsinθ - mgsinθ - F = 0

Step 4: Solve the equations:
1. From the equation Tcosθ - mgcosθ = 0, we can divide both sides by cosθ:
T = mg

2. Substitute T = mg into the equation Tsinθ - mgsinθ - F = 0:
mgsinθ - mgsinθ - F = 0
-F = 0
F = 0

Step 5: Calculate the angle:
From the equation F = 0, we can see that the applied force is zero. This means that there is no horizontal force acting on the system.
Since the only vertical force is the weight of the mass, the rope must be vertical in equilibrium. Therefore, the angle the rope makes with the vertical is 0 degrees.

Conclusion:
The angle the rope makes with the vertical in equilibrium is 0 degrees. Therefore, none of the given options (a) 60 degrees, (b) 40 degrees, (c) 50 degrees, or (d) 30 degrees, are correct.