Uniform Rope Acceleration Problem


This problem involves a uniform rope that is moving with constant acceleration on a smooth horizontal surface. We need to find the ratio of the tension in the rope at its mid-point to the applied force.


To solve this problem, we need to use Newton's second law of motion, which states that the force applied to an object is equal to its mass times its acceleration. We can apply this law to both ends of the rope to find the tension at the mid-point.


We will make a few assumptions to simplify the problem. First, we will assume that the rope has negligible mass compared to the applied force. Second, we will assume that there is no friction between the rope and the surface.


Let F be the applied force, m be the mass of the rope, a be the acceleration of the rope, and T be the tension in the rope at its mid-point.


The force applied to the rope is F, and the tension in the rope is T/2 (since the tension is evenly distributed along the length of the rope). The mass of the rope is m/2 (since we are considering only half of the rope).

Therefore, according to Newton's second law of motion:

F - T/2 = (m/2) * a


The force applied to the rope is F, and the tension in the rope is T/2. The mass of the rope is also m/2.

Therefore, according to Newton's second law of motion:

T/2 - F = (m/2) * a


Adding the two equations, we get:

F - T/2 + T/2 - F = (m/2) * a + (m/2) * a
0 = m * a

This equation tells us that the acceleration of the rope is equal to zero. This is because the tension in the rope is balanced by the applied force.

Therefore, the ratio of the tension in the rope at its mid-point to the applied force is:

T/2 / F = 1/2

This means that the tension at the mid-point of the rope is half of the applied force.

1:2