Why is work done in a spring positive?

Elias Heath 2022-09-26 Answered
We know that a stretched spring obeys Hooke's law, such that F = k x
We can find the potential energy of stretching/compressing this spring by x, given by :
U x U 0 = 0 x F . d x = 1 2 k x 2
Setting U 0 = 0 as reference, we have U x = 1 2 k x 2
However, this is also sometimes described as the work done by the spring.
Shouldn't the work done W be given by F . d r, such that W = Δ U = 1 2 k x 2 in this case ?
Isn't the work done by the spring negative ?
Also, in this case the potential energy comes to be negative.. In general, can we set any point as reference and set it to be 0 and perform the integral between any two limits, to get either a positive or a negative U ?
For example, in forces of the nature r n , ( n > 1 ) we usually take the reference at r = and integrate from to some point r. In case of forces of the nature r n , we usually take 0 as the reference and integrate from 0 to some r. In general, we are free to choose any reference and any limit, even though some are much more convenient, right ? In theory, we can choose any point, right ?
As long as we have :
U a U b = b a F . d x
we can choose any a and b, and set either of U a or U b to be the reference and equal to 0, right ?
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Answers (1)

pagellera10
Answered 2022-09-27 Author has 7 answers
Setting x=0 as the reference point means you are looking at the work done by the spring from x=0 to the end position x. Since W = Δ U = 1 2 k x 2 , this will always be negative, which makes sense since the spring force always points towards x=0, and thus will point opposite the displacement.
In general
W a b = ( U ( x b ) U ( x a ) ) = 1 2 k ( x a 2 x b 2 )
and this is positive whenever x a 2 > x b 2
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