Force on a String: Understanding Constant Tension

[aaaa202]

I just solved a problem where you considered a wave traveling on a string with varying density. I did it all correct but during the problem, I wondered why you can say that the tension in the string is the same everywhere on it. Why is that?

 

[technician]

If the string is horizontal and you consider a small section of string it is in equilibrium. It is not accelerating and therefore there is no resultant force. This means the force to the right and the force to the left (ie the tension) on each sectionof string must be the same.
If the string is hanging vertically and the string has mass distributed along its length then the tension increases towards the top of the string. Imagine the string as a series of weights connected together. The top of the string has to support more weight than the bottom of the string.

 

[Studiot]

If the string is horizontal and you consider a small section of string it is in equilibrium. It is not accelerating and therefore there is no resultant force.

Are you sure you mean this?

The mathematical derivation of the wave equation for a vibrating string relies on the differential element of string not being in equilibrium.

However the tension is (by definition for a string) always parallel to the direction of the string. It is the change of direction (rotation) of the element which occurs as the wave passes that gives rise to zero net acceration in the x direction but a real variable acceleration in the y direction. This is what causes the element to move up an down with time. Tension is a vector which has magnitude and direction. The magnitude does not change but the direction does.

go well

 

[technician]

Yes Studiot...I was referring to a static string lying flat...not the string carrying a wave.
I should have made that clear, the extra point I wanted to add was that a hanging string will have differing tension along it if it has mass. In many (most) examples with weights on strings over pulleys etc it is assumed that the string has no mass.
You are quite correct.
Cheers

 

[Studiot]

I understood the query to be about the tension in a string carrying a wave, ie a vibrating string.

But yes, certainly a simple string stretched by its own weight or other load is often a static equilibrium system. Catapault action for instance is not.