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I'm reviewing a concept on a system where a mass is hanging from a vertical spring in the presence of gravity. I'm attempting to validate my understanding of conservation of energy when the mass is allowed to slowly extend from its unstretched point to its equilibrium point where the forces cancel out such that there is no kinetic energy at the end point of the movement.

My physics textbook indicates a fundamental equation Wa = -Ws where Wa is the work done by the applied force (gravity in this case) and Ws is work done by the spring force, provided the kinetic energies of the start and end points are zero.

However, I'm confused since this relation seems to be violated. Others on the forum have pointed out that to bring the spring to equilibrium, the work done by gravity is twice as great as that of the spring, as worked out below.

Setting the potential reference level of zero at the unstretched point of the spring (x=0),

Work done by gravity: W = m*g*x

Work done by spring: W = -1/2 * k * x^2

Equating the equations to each other at equilibrium where the net force is zero, we get:

2 * m * g = -k * x

How can this be? Does this violate the work expression I wrote several paragraphs above? Physically, shouldn't the work done by gravity equal to the work done on the spring? I've pored over explanations provided by others on this forum, but I've found them lacking for some reason.

Any insight would be greatly appreciated!

My physics textbook indicates a fundamental equation Wa = -Ws where Wa is the work done by the applied force (gravity in this case) and Ws is work done by the spring force, provided the kinetic energies of the start and end points are zero.

However, I'm confused since this relation seems to be violated. Others on the forum have pointed out that to bring the spring to equilibrium, the work done by gravity is twice as great as that of the spring, as worked out below.

Setting the potential reference level of zero at the unstretched point of the spring (x=0),

Work done by gravity: W = m*g*x

Work done by spring: W = -1/2 * k * x^2

Equating the equations to each other at equilibrium where the net force is zero, we get:

2 * m * g = -k * x

How can this be? Does this violate the work expression I wrote several paragraphs above? Physically, shouldn't the work done by gravity equal to the work done on the spring? I've pored over explanations provided by others on this forum, but I've found them lacking for some reason.

Any insight would be greatly appreciated!

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