# Calculate equilibrium position using conservation of energy in vertical spring mass system

I am very confused in the vertical spring mass system right now, below is some of my thinking process:

Say I want to find the equilibrium of a spring mass system, my teacher used the force way to solve it, but I considered to use conservation of energy. For instance, I place a mass of a vertical spring, no initial velocity, no external force, just put it up there, try to find the equilibrum position. However, if I use spring potiential energy equals gravitional potiential energy, the equation is different compare to my teacher's. This made me very confused as my teacher told me conservation of energy will work if no external foce exist, with only conservative force, which do match the current situation. Not sure what happened. Below is my process. Be very thankful if anyone can explain this more detailly!

To use conservation of energy you need an initial and a final state, and set the two energies as the same, here you only have one state, the equilibrium state, so you cannot use conservation of energy to solve it. What you calculated is not the equilibrium position, but something different.

If you assume that the mass starts at rest with the spring not extended, and you set the gravitational potential energy as zero at that point, you have $$E_i=0$$. If then you want to calculate the maximum extension, when the velocity is zero again, then you will have $$E_f=1/2kx^2-mgx$$. From here ($$E_i=E_f$$) you get your result, that is what you actually calculated. But that is not an equilibrium position, the spring will return back and oscillate.

Conservation of energy was a good thought, but it isn't a helpful approach for this problem.

This problem is about finding the equilibrium position. That is the position where the total force is $$0$$. If the velocity is also $$0$$ at that point, the mass will stay there forever. There are two forces on the mass: gravity and the force from the spring. The weight (force of gravity) doesn't change. The spring force is small if the spring is stretched a little, and big if the spring is stretched a lot. It is straightforward to find where the spring is stretched just enough to cancel the weight.

Conservation of energy is useful for a different kind of problem. Suppose you start with the mass in a certain position, and drop it. The mass bounces up and down. Suppose you are asked to find the speed of the weight at a certain height.

One way to solve this is to use forces. But it gets complicated. You know the position and velocity at the start. You know how stretched the spring is, so you know that forces. You can calculate the acceleration. That is enough information to figure out the position and velocity of the mass a short time later.

But the position has changed, so the spring is stretched differently. The forces have changed. You know the new position, velocity, and force, so you can again calculate the position and velocity a short time later.

If you repeat this over and over, you can figure out the position and velocity far into the future. To do it right, you must use infinitely short time intervals. This becomes a calculus problem.

The calculus problem for a spring and mass can be solved. But conservation of energy can be a shortcut for this kind of problem.

First, the forces in this problem are conservative. This means there are no friction-like forces that turn kinetic energy into heat, and bring the system to a stop. Kinetic + potential energy is constant for these kinds of forces.

Second, if you know the position, you know the potential energy. That is the reason conservation of energy is a shortcut. If you know the total energy and potential energy, it is easy to find the kinetic energy. And from that it is easy to find the speed. No need to inch along and figure out the forces every step of the way.

The conservation of energy equation needs to be modified slightly to take into account the Kinetic Energy of the block after the block is released from the "natural length" of the spring. If initially you have placed the block in the equilibrium position you cannot calculate the change in the potential energy of the spring which in this case never moves. Also the work done by gravity in this case would be $$0$$ as the block had never moved. Therefore the Energy conservation equation would become $$0=0$$ which is still valid and does not violate anything. These questions are better solved by considering the forces.Generally energy is used when you want to find the speed or adn other thing when the object is in motion.