We know that a stretched spring obeys Hooke's law, such that F = − k xWe 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 &gt; 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 xwe 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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We know that a stretched spring obeys Hooke&#039;s law, such that   F  =  −  k  xWe 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&#039;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&#039;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  &amp;gt;  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  xwe can choose any a and b, and set either of       U    a   or       U    b   to be the reference and equal to 0, right ?

pagellera10 · Accepted Answer

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    &amp;gt;      x    b    2

Why is work done in a spring positive?

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