mechanical waves in simple harmonic motion

## Question:

In a vertical mass/spring system that produces a sinusoidal curve on a graph, which areas on the graph would represent the greatest kinetic energy or least kinetic energy as well as the greatest and least potential energy? If mechanical energy is conserved, how should a graph look if the sum of kinetic and potential energy vary with time?

## Answer:

Kinetic energy is 1/2*m*v^2 and potential energy is 1/2*k*y^2. Kinetic energy then will be max when the absolute value of velocity is max which is where the slope of the graph is max, which occurs when the graph is crossing the y axis. Potential energy due to the spring will be max when the absolute value of y is max, which occurs at the plus and minus peaks of the graph. Potential energy due to gravity is of no concern because the neutral point of the spring is taken to be where the weight rests with the force of gravity nulled out by the partially stretched spring. Gravity just shifts the point about which the oscillation takes place but plays no role in the motion.

If the system were non-conservative, then either energy is being transferred out of the system or into it. If energy is being transferred out as for example by friction, the amplitude of each swing will be a bit less than the previous swing and eventually the weight will come to a stop at its equilibrium position. If energy is being added to the system the amplitude of the oscillations will increase over time until either the energy losses match the energy input or the system comes apart due to high amplitude.