Consider a spring with spring constant k attached at one end to a wall. When an external force stretches it by distance l in the positive direction the force does work 1 2 k l 2 and that energy is stored in the spring as elastic potential energy.Let the stretching take place uniformly over a time t for definiteness.If this operation is viewed by an observer moving at velocity v with respect to the lab, the starting and ending position of the stretching operation will appear to be different, say 0 and l − v t, but the average force exerted on the spring would remain 1 2 k l, which seems to make the work done (and energy stored in the spring) W = ( 1 2 k l ) ⋅ ( l − v t ) .How is this (potentially vast) difference reconciled?

Question

Consider a spring with spring constant k attached at one end to a wall. When an external force stretches it by distance l in the positive direction the force does work       1    2    k      l    2   and that energy is stored in the spring as elastic potential energy.Let the stretching take place uniformly over a time t for definiteness.If this operation is viewed by an observer moving at velocity v with respect to the lab, the starting and ending position of the stretching operation will appear to be different, say 0 and   l  −  v  t, but the average force exerted on the spring would remain       1    2    k  l, which seems to make the work done (and energy stored in the spring)  W  =      (          1      2        k    l    )    ⋅  (  l  −  v  t  )    .How is this (potentially vast) difference reconciled?

Cindy Mercer · Accepted Answer

In the moving frame the &quot;fixed&quot; end of the spring moves over a distance −vt, always exerting the same magnitude force as that applied, but in the negative direction so the work                              W          fixed                                    =                  (          −                      1            2                    k          l          )                ⋅        (        −        v        t        )                                          =        +                  1          2                k        l        v        t                .            That makes the net work done to the spring      W    net    =      (          1      2        k    l    (    l    −    v    t    )    )    +      (          1      2        k    l    v    t    )    =      1    2    k      l    2      ,just as before.

How does the math of storing energy in a spring work in a moving frame?

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