Can you find my mistake (either in numbers or my conceptual understanding)? This

Question

Can you find my mistake (either in numbers or my conceptual understanding)? This is the problem I'm working on:

Is energy conserved regarding a simple harmonic motion problem where a spring with a 1kg mass hangs from a vertical surface? Equilibrium (where the 1kg weight is at rest) is 0.193795m, and k=25.86 N/m. Compare highest to equilibrium position and highest to lowest points. (The 0.193795m was measured experimentally, but doesn't seem to make ssense since it's not half way between the highest and lowest points.)

I keep getting for my answer:

My issues:

-I know energy should be conserved, and at any given point, E = ½mv², + ½kx², + (M *g *y), where x and y are the displacement from the equilibrium position and y+ is up (is this wrong?). However, at equilibrium, I get ½k0²,= 0. On a physics forum, the explination was that spring PE = ½kx², where x is the amount of stretch. The stretch is not zero at the equilibrium point, so ½kx² is not the spring energy. So what IS the spring energy at equilibrium? ½k?

-Also, not sure why my highest and lowest position energy sums are not equal. I tried using the both the actual and absolute value of the y (displacement from equilibrium) value for the lowest point, and still energy didn't conserve. I can't figure out what I'm doing wrong, but I know I'm wrong somewhere!

position velocity time acceleration Highest 0.197225 0.127195833 4.1 0.642897524 Lowest 0.175273 -0.136514 0.05 -0.621674025 equilibrium 0.193795 0.114118958 3.65 -0.344229083

Explanation / Answer

The spring energy at equilbrium is zero. It means some other energy is maximum. So at equilbrium condition, the KE is maximum (so you can also say velocity is maximum).

So calculate the energy at highest and lowest point and then compare.

At the top or upper point, the sting strected is less therefore the potential energy is minimum, but gravitational potential energy or potential energy due to gravity is maximum because the mass is lifted and also KE is zero because mass has stopped at point then go down.

At equilbrium position , the PE is zero and also Velocity become maximum there KE is maximum . pe due to gravity is there because mass is lifted above the ground.

At the lowest point, the PE is maximum because it strected up to maximum and PE due to gravity is minimum because it's is least above the ground and also KE is zero because it stopped at that point.

So in up and down energy keep on changing between PE of the spring, PE due to gravity and KE of the spring but the TE of the system remains constant however this motion will soon to destroy because some losses due to heat takes place which soon to destroy the motion.

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