Why is the energy stored in a spring proportional to 1/2 * distance * force?

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The energy stored in a spring is proportional to 1/2 * distance * force because the force exerted by the spring is not constant as it is compressed. Unlike scenarios with constant force, the spring's force increases linearly from zero to its maximum as it is compressed. To find the energy, one can analyze the compression in small increments, applying varying force at each step. The exact calculation involves calculus, integrating the instantaneous force over the distance compressed. This results in the formula for elastic potential energy in a spring.
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Why is the energy stored in a spring 1/2 * distance * force? Isn't work just force * distance?
 
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GeneralOJB said:
Why is the energy stored in a spring 1/2 * distance * force? Isn't work just force * distance?

The full force is not applied for the full distance.

In the simple case of a constant force over a fixed distance you can just multiply force by distance. But in the case of a spring being compressed, the force is not constant.

You could approximate the answer by analyzing the situation in small steps... You compress the spring the first 1/10th of the distance using 1/10th of the full force, the next 1/10th of the instance using 2/10th of the total force and so on. This approach would give you an answer that is pretty close.

An exact answer can be obtained by using calculus and integrating instantaneous force over incremental distance. That answer turns out to be 1/2 * distance * maximum-force
 
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Thanks, I understand now.
 

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