How Do You Calculate the Elongation of a Spring Stretched at Both Ends?

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Discussion Overview

The discussion revolves around calculating the elongation of a spring when forces are applied at both ends. Participants explore the implications of applying unequal forces, the conditions for equilibrium, and the effects of acceleration on elongation or compression.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant suggests that the elongation can be calculated as x = (F1 + F2)/k, but others challenge this by stating that unequal forces will cause the spring to accelerate towards the larger force.
  • It is noted that for a spring to stretch, forces must be exerted on both sides, and if tied to an immovable object, the net force must be zero.
  • Another participant questions whether the elongation x is equal to F/k or 2F/k when the forces are equal, leading to further clarification on the nature of vector quantities.
  • A participant introduces the idea that a spring can be compressed or elongated even if it is accelerating, and asks how to calculate elongation or compression in such scenarios.
  • One response proposes taking the average of the tension or compression forces at both ends to compute elongation or compression, providing an example involving a spring hanging from one end.
  • There is a discussion about sign conventions when calculating average tension, particularly when different forces are applied from opposite directions.
  • A method is suggested for deriving elongation by treating the spring as a series of smaller springs connected end-to-end.

Areas of Agreement / Disagreement

Participants express differing views on how to calculate elongation, particularly regarding the addition of forces and the conditions under which the spring is in equilibrium. The discussion remains unresolved with multiple competing perspectives on the topic.

Contextual Notes

Participants highlight the importance of considering the net force and equilibrium conditions, as well as the implications of acceleration on elongation calculations. There are unresolved assumptions regarding the definitions of forces and their effects on the spring's behavior.

bgq
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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 ?
 
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bgq said:
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
 
willem2 said:
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?
 
bgq said:
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.
 
bgq said:
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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