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We generate the formula from $W=Fs$, but why is it when we form the formula the $F$ becomes the average force applied, making the equation

$$W=(1/2)(0+F)x \, ?$$

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closed as off-topic by Cosmas Zachos, Jon Custer, honeste_vivere, Ryan Unger, Yashas Jul 23 '17 at 2:10

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Because the force applied changes with $x$.

Note that Hookes Law for springs is: $$F = k \Delta x$$

This means that in the $W = F s$ equation, we cannot directly use it because $F$ is not constant throughout the entire $\Delta x$ movement.

Since it varies linearly, we can instead take the average force multiplied by the total distance to find the work done, since we are not doing work with the max force the entire time.

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