Friction is not considered in any case and I would like to know why the extension in the spring when released slowly is $mg/k$ where as in suddenly released it is $2mg/k$. Can you prove these cases both by work power energy?
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List of two dozen such analyses. When you lower an object slowly, the gravitational field does work on whatever's keeping it from falling freely. This is energy that's then not available to let the object oscillate around the equilibrium point. – Chemomechanics Aug 15 '23 at 04:30