Why is tension force in a string non-conservative? I have this doubt for quite a while now. Suppose we have a massless inextensible string with tension as 'T', now, I would like to know that is this tension a non-conservative force? If yes, then what is the reason behind it? I think that the reason for this is that the tension force depends on the path traversed... but I am not so sure about this.
I have seen that tension in the spring is considered a conservative force because it causes a change in potential energy of the system. But if the same logic is applied to a string, then it should also be a conservative force.
 A: Most of the time I would treat the tension force in a string/rope as conservative.  So perhaps that's why you're having such trouble rationalizing why it is non-conservative.
I think the only time where I would consider that force to be non-conservative is if it was being pulled by something "outside" of the problem.  For example, if your system is a book with a string pulling up on it with 100N of tension force, that would be non-conservative because its something outside of your system that's pulling up on the string -- and thus potentially putting energy into the system.  But if my system included the person pulling on the string, then I'd find once again that the string's tension is acting in a conservative manner.
A: The terms "conservative" and "non-conservative" don't apply to the force of a string. This is because these terms are defined for force fields--that is, a situation where the force on an object is a function of its position. A string of constant length does not have his property. Imagine a ball hanging from the string. The tension force is equal to the weight of the ball. Now, I grab the ball and pull down. Because the string does not change length, the tension force has to increase to resist my pulling. Also, because the string does not change length, the ball does not move. The tension force changed without the ball moving, which means the string's tension force does not constitute a force field.
Now, for problems where you are using kinetic and potential energy like a mass swinging from a string, you can use the fact that gravity is a conservative force. This is because the string tension force does no work on the mass, which means that total energy is conserved. Make sure you know why the string does no work when solving a problem.
As for the spring, being able to define a potential energy function is enough to show that the spring force is conservative. The force of the spring is defined by the position of the object attached to the end, so it is a force field. However, the spring force is not unique in being a conservative force field. All one-dimensional force fields are conservative. The potential energy function of a 1D force field $F(x)$ is
$$V(x) = -\int_{x_0}^x F(s) \textrm{d}s$$
where $V(x_0)=0$ by definition. As long as $F(x)$ is defined for all relevant positions, it is conservative.
