According to work - mechanical energy theorem , W = K(final) - K(initial) + U(final) - U(initial) . . . . (1) as we define Potential energy as negative of work done by conservative force and assuming that the only force in this situation is Spring force then , W(spring) = K(final) - K(initial) As work done is calculated by finding component of spring force in direction of displacement. How can we say that U(final) - U(initial) applies for all possible conditions of extension of spring as displacement may not be in direction of force ? Spring force = 0.5kx^{2}
Welcome to PF! First of all, your equation (1) defines the external work done by/on a system. If no energy is added or lost (Wext = 0), Kf + Uf = Ki + Ui. Second, your question is not clear. What do you mean when you say U(final) - U(initial) applies? U(final) - U(initial) is not a mathematical statement. Finally, your statement: Spring force = 0.5kx^{2} is not correct. F = -kx. AM
In case this was a simple slip, the formula [tex]W = \frac{1}{2}k{e^2}[/tex] W = work, e = extension, k = spring constant Refers to the work done in extending a spring = potential energy stored in that spring on extension.