The Tension in the Middle of the Rope in a Gravity-Free Space

When a block is placed on a frictionless surface in a gravity-free space and a heavy string of mass M is connected to it, and a force F is applied to the string, the tension in the middle of the rope can be determined using the principles of Newton's laws of motion and equilibrium.

Principles Involved:
1. Newton's First Law of Motion: An object will remain at rest or move with a constant velocity unless acted upon by an external force.
2. Newton's Second Law of Motion: The acceleration of an object is directly proportional to the net force applied to it and inversely proportional to its mass.
3. Newton's Third Law of Motion: For every action, there is an equal and opposite reaction.

Analysis:
1. Force F: The force F applied to the string will cause the block to accelerate in the direction of the force.
2. Tension in the Rope: The tension in the rope will be transmitted throughout the length of the rope and will be the same at any point on the rope in a gravity-free space.
3. Newton's Second Law: The net force acting on the block is the tension in the rope minus the force F applied. Since the net force is equal to the mass of the block times its acceleration, we can write the equation as: T - F = Ma, where T is the tension in the rope, M is the mass of the block, and a is the acceleration of the block.
4. Newton's Third Law: The force F applied to the string will result in an equal and opposite reaction force F on the block.
5. Equilibrium: In a gravity-free space, the block will remain at rest or move with a constant velocity when the net force acting on it is zero. Therefore, when the block is in equilibrium, the tension in the rope will be equal to the force applied, i.e., T = F.

Tension in the Middle of the Rope:
Since the tension in the rope is the same at any point on the rope, the tension in the middle of the rope will also be equal to the force applied, i.e., T = F.

Conclusion:
In a gravity-free space, when a block is placed on a frictionless surface and a heavy string of mass M is connected to it, with a force F being applied to the string, the tension in the middle of the rope will be equal to the force applied, i.e., T = F. This is because the tension in the rope is transmitted throughout its length, and in equilibrium, the tension in the rope will be equal to the force applied to it.