Given:
Length of the rope (L) = 5 m
Force applied (F) = 20 N

To find:
The ratio of tension at a distance of 2 m from the force to the tension at a distance of 2 m from the free end

Solution:

Understanding the Problem:
The problem states that a rope is being pulled by a horizontal force at one end. We are required to find the ratio of tension at a distance of 2 m from the force to the tension at a distance of 2 m from the free end. In order to solve this problem, we need to analyze the tension in the rope at different points.

Analysis:
When a rope is pulled by a force at one end, the tension in the rope is not constant throughout its length. The tension is maximum at the end where the force is applied and gradually decreases towards the free end. At any point along the rope, the tension depends on the distance of that point from the force. The tension is higher closer to the applied force and lower closer to the free end.

Calculating the Tensions:
To find the tension at a distance of 2 m from the force, we need to calculate the tension at that point. Let's assume that the tension at a distance of 2 m from the force is T1.

At the end where the force is applied:
The tension at this end is maximum and equal to the applied force. Therefore, the tension at this end is 20 N.

At a distance of 2 m from the force:
To find the tension at this point, we can use the concept of proportionality. The tension in a rope is directly proportional to the distance from the force. Mathematically, we can express this as:

Tension ∝ Distance

T1/20 = 2/5

T1 = (2/5) * 20

T1 = 8 N

At the free end:
The tension at the free end is zero, as there is no force acting on it.

Calculating the Ratio:
The ratio of tension at a distance of 2 m from the force to the tension at a distance of 2 m from the free end can be calculated as:

T1/T2 = 8/0

Since the tension at the free end is zero, the ratio is undefined.

Conclusion:
The ratio of tension at a distance of 2 m from the force to the tension at a distance of 2 m from the free end is undefined.