This might be a dumb question,, but its art of my assignment and im stuck at the last part.So here it goes.

I have a mass-spring system, i am supposed to workout its equation using conservation of energy, and then graph the final result and prove that the area enclosed by the curve obtained after graphing, is proportional to the total energy of the oscillator.

so here is what i have so far..

Total energy of the simple harmonic system (considering a mass spring system in 1D)

E = Kinetic energy + Potential Energy = $0.5 v^2 + 0.5 w^2x^2 *(where .w^2= k/m)*$

now i know that this is the equation of an ellipse, now i am not sure how to proceed and show the proportionality between, the area of the curve and the total energy of the system?

This might be a dumb question, but its art of my assignment and im stuck at the last part. So here it goes.

I have a mass-spring system, i am supposed to workout its equation using conservation of energy, and then graph the final result and prove that the area enclosed by the curve obtained after graphing, is proportional to the total energy of the oscillator.

so here is what i have so far..

Total energy of the simple harmonic system (considering a mass spring system in 1D)

E = Kinetic energy + Potential Energy = $0.5 v^2 + 0.5 \omega^2x^2 $ (where $ \omega^2= k/m$)

now i know that this is the equation of an ellipse, now i am not sure how to proceed and show the proportionality between, the area of the curve and the total energy of the system?