Answer
As the ladle is passing through the equilibrium position the spring is not doing any work on the ladle because the spring force is $0$ at the equilibrium position.
Work Step by Step
At the equilibrium position, the force on the ladle by the spring is $0$
We can find the rate at which the spring is doing work on the ladle:
$P = F~v = (0)(v) = 0$
As the ladle is passing through the equilibrium position the spring is not doing any work on the ladle because the spring force is $0$ at the equilibrium position.
