# Solving the Spring Paradox: Uncovering the Error

• LukasMont

#### LukasMont

New user has been reminded to use the Homework Help Template when starting schoolwork threads at the PF
Good evening.

The problem states: Spring paradox. What is wrong with the following argument? Consider a mass m held at rest at y = 0, the end of an unstretched spring hanging vertically. The mass is now attached to the spring, which will be stretched because of the gravitational force mg on the mass. \Vhen the mass has lost gravitational potential energy mgy and the spring has gained the same amount of potential energy so that mgy = ~ Cy²/2, the mass will come to equilibrium. Therefore the position of equilibrium is given b y=2mg/C.

Working out by the principle of forces, where the weight, when equated to the elastic force shows,
Fg= -Fk or Fg + Fk = 0 (in the equilibrium point).
So,
-mg = -Cy, and y = mg/C. (What is different from what would be obtained from the spring paradox).
But i simply don't know how to show that the spring argument is wrong, as well.

Thanks.

A very interesting problem. A couple of observations solve this paradox, so that it is no longer a paradox. When the mass is released, simple harmonic motion occurs where the total energy=kinetic energy plus potential energy is constant. To lead you to the solution to this paradox, what is the kinetic energy the instant the mass is released? Thereby is not the total energy equal to the potential energy? Isn't there another place in the simple harmonic motion where the kinetic energy is zero? What is the potential energy there? Answering these questions should help resolve the paradox. ## \\ ## Meanwhile, along the path of simple harmonic motion, when the kinetic energy is non-zero, is there anywhere that the potential energy is equal to the initial potential energy? ## \\ ## If the spring system is slightly damped by air resistance, where will the equilibrium position be where the mass on the spring finally stops? (Hint: It's the place in the simple harmonic motion where the acceleration is zero).

CWatters
At the point described in the bold text the PE lost might be equal to the KE gained but is the system really in equilibrium?