# Elongation of Spring if Pulled by Unequal Forces

For example, if a spring is pulled on one end by 5N and 3N on the other end and has a spring constant of k=10 N/M, how much longer will the spring become?

My first thought was that the Tension in the sprint would be 3N and a net force of 2N accelerating the spring, and the spring would elongate by 0.3 meters. However I am unsure if this is right.

Also more generally, if a string on one end is pulled by a force of 5N on one end and 3N on the other, what is the tension of the string? Is the tension in the string constant throughout? And how would it change if the string is not massless?

And how would it change if the string is not massless?

Actually, the spring or string must be massive. If you have unequal forces then there is a net force and so by Newton’s 2nd law there is an acceleration. If the mass is zero then the acceleration is infinite. So therefore the mass cannot be zero for any object with unbalanced forces.

Is the tension in the string constant throughout?

No. Because the spring or string is massive the tension is not constant. Overall you know that it is accelerating, and assuming that it is accelerating smoothly then every part of it is accelerating at the same rate.

You can imagine cutting the string at any point along its length and drawing a free body diagram. If the force is 5 N on one end and 3 N on the other then, if you work it out, you will find that it is 4 N at the center of mass. In fact, the tension varies linearly provided the density is constant.

Because the tension varies the elongation is a little more complicated to calculate. But a reasonable estimate is the elongation at the average tension.

• Ok so if we use average tension though, if the forces are now 0N on one side and 100N on the other side, that would mean the tension is 50N in the center, which is true and so elongation would be 0.5 meters. But at the same time, how does it make sense that the spring would elongate when only one side is being pulled? – Student Jan 12 at 6:56
• Why wouldn’t such a spring elongate. It is exactly equivalent to a massive spring hanging in a gravitational field with the bottom end free. Such a spring will clearly elongate under its own weight. Similarly an accelerating spring will also elongate even if the back side is free. – Dale Jan 12 at 18:06