does tension in the string affect its equilibrium?

Question

In my textbook (Sears and Zemansky's University Physics), it is written that the vector sum of the forces on the rope is zero, however the tension is 50 N. Then is tension different than the force? And if not, then why force is zero while tension is not?

A body that has pulling forces applied at its ends, such as the rope in Fig 4.27, is said to be in tension. The tension at any point is the magnitude of force acting at that point (see Fig 4.2c). In Fig 4.27b, the tension at the right end of the rope is the magnitude of $\vec{\mathbf{F}}_{M\ on\ R}$ (or of $\vec{\mathbf{F}}_{R\ on\ B}$). If the rope is in equilibrium and if no forces act except at its ends, the tension is the same at both ends and throughout the rope. Thus if the magnitudes of $\vec{\mathbf{F}}_{B\ on\ R}$ and $\vec{\mathbf{F}}_{M\ on\ R}$ are $50\ \rm N$ each, the tension in the rope is $50\ \rm N$ (not $100\ \rm N$). The total force vector $\vec{\mathbf{F}}_{B\ on\ R}+\vec{\mathbf{F}}_{M\ on\ R}$ acting on the rope in this case is zero!

Answer 1

Assuming the block isn’t accelerating, the sum of the external forces on the rope is zero. But tension is an internal force. You know it exists because if you cut the rope the Mason will go flying. To determine the amount of tension force, you can cut the string removing one end, say the block end, and replace it with the force necessary to maintain equilibrium (Free body diagram). That force has to be equal and opposite to the force exerted by the Mason on the right end. That’s the tension in the string.

Answer 2

Each element of the rope is subjected to two forces which are equal in magnitude and opposite in direction so the net force on each element of the rope is zero.
One of the forces on the left side of the element is due to the left hand part of the rope adjacent to the element pulling left and the other force on the right side of the element is due to the right hand part of the rope adjacent to the element pulling right.

At the end of the rope, the rope is exerting a force on the anchor point holding the rope in position and the anchor point is exerting an equal in magnitude and opposite in direction force on the rope.

All the forces in the rope are called the tension.

Overall if the rope is not accelerating (or is massless) the net force on the rope is zero.

You can think of a rope as transferring a force from one position to another and also changing the direction of a force if there is a bend in the rope eg due to a pulley.

Answer 3

If you pull the ends of a rope with equal and opposite forces (F on the right hand end and –F on the left hand end), the resultant force on the rope is zero.

But the rope will be in a different state from the state it would be in if no forces were being exerted on it. We say that the rope is under tension.

Tension is quantifiable. Consider any cross-section of the wire at any distance along it. The section, R, of rope to the right of the cross-section will be pulling the section, L, of rope to the left of the cross-section with a force F, and the section, L, of rope will be pulling the section, R, of rope with a force –F. We say that the tension in the rope is of size F. Both L and R, will individually be in equilibrium, assuming that the rope is not accelerating; indeed that's how we could deduce the "internal forces" I've just described.

So tension isn't strictly a force, but a rope under tension F will require forces ±F at either end to keep it in equilibrium, and by Newton's third law, will exert forces –F and +F on whatever it's attached to at either end.

Answer 4

Okay, I am a student myself so please re-read it if you don't get at once.

Your textbook says that the Sum of all forces on the rope is zero and yes it is because the rope is in equilibrium.

To understand this first answer these questions,

• Is the rope moving? (Hint: No)
• Is the rope pulling the wall? (Yes)
• Is the rope pulling the man? (Yes)

I hope that you know the answers to these questions. (Hint: Read laws of motion)

Now, the man is applying a force on the rope. If there was no force counter acting this then the rope should move right..? Yes, but it is not moving. That means that there is a force on the rope acting in the opposite direction. That is the force exerted on the rope by the wall and that force is equal to the force applied by the man.

So the net force on the wall adds up to zero. I think this is a good answer for your second question, https://physics.stackexchange.com/a/221169/202990.