# Spring stretched at both ends

## Main Question or Discussion Point

Hi,

Consider a spring of stiffness k and mass m. If we stretch - simultaneously - both ends of the spring by force F1 and F2. What will be the elongation of the spring? Can I add the forces and say that
x = (F1 + F2)/k ?

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Hi,

Consider a spring of stiffness k and mass m. If we stretch - simultaneously - both ends of the spring by force F1 and F2. What will be the elongation of the spring? Can I add the forces and say that
x = (F1 + F2)/k ?
No, you can't add the forces. If the forces are unequal, the spring itself will start to accelarate in the direction of the largest force. F = ma applies here, where F is the net force.

To stretch a spring, you always have to exert a force on both sides. Even if you tie the spring to an unmoveable object, the object will still exert the same force on one side of the spring, as the force on the other side, because the spring now can't move, so the net force on it has to be 0

No, you can't add the forces. If the forces are unequal, the spring itself will start to accelarate in the direction of the largest force. F = ma applies here, where F is the net force.

To stretch a spring, you always have to exert a force on both sides. Even if you tie the spring to an unmoveable object, the object will still exert the same force on one side of the spring, as the force on the other side, because the spring now can't move, so the net force on it has to be 0
If it is at equilibrium so that F1 = F2 = F. Is elongation x = F/k or x = 2F/k?

AlephZero
Homework Helper
If it is at equilibrium so that F1 = F2 = F.
When it is in equilibrium, F1 = F and F2 = -F.

Forces are vector quantities, they have magnitude and direction.

Is elongation x = F/k or x = 2F/k?
x = F/k. Adding two forces of magnitude "F" that act in different directions, to get "2F", is meaningless.

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The spring may be compressed or elongated even if it is accelerating. My question is how to find the elongation or compression is such a situation.

jbriggs444
Homework Helper
2019 Award
The spring may be compressed or elongated even if it is accelerating. My question is how to find the elongation or compression is such a situation.
Take the average of the tension (or compression) force at the two ends and compute the amount of elongation or compression based on that.

For instance, consider a spring hanging from one end. The tension at the one end is mg. The tension at the other end is zero. So, on average the tension across the length of the spring is mg/2.

Be sure to watch your sign conventions. If a spring is being pushed from the back with a force of 10 Newtons and pulled from the front with a force of 20 newtons, that's 20 minus 10 all divided by 2 for an average tension of 5 Newtons.

You can derive this result by treating the original spring as a bunch of smaller springs connected end-to-end and then adding up the displacements that result from the tension or compression experienced by each of the smaller springs.

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