Sound and Oscillation

Mechanical energy of the SHM

We know that the mechanical energy in a spring-mass system is given by the conservation of energy, that is, the total mechanical energy is the sum of the kinetic energy and the potential energy. We represent the kinetic energy by the symbol Ec, the potential energy by the symbol Ep and the mechanical energy by the symbol E. Therefore, the mechanical energy is given by the following equation:

Kinetic energy, which is related to bodies in motion, is represented by the following equation:

And the elastic potential energy, which is related to the position of the body (or object), is given by the following equation:

In the figure below we represent a mass-spring system, where the particle of mass m is attached to a spring whose spring constant is k. This system performs a simple harmonic motion (SHM), of amplitude a, with extremes A and B. In the figure we have any intermediate point C.

1) According to the figure above, we have a body of mass m at one of the extreme points A or B, where the spring elongation can be x = -a or x = +a. At these two points (A or B), the velocity of the body is zero (v = 0), so the kinetic and potential energy are, respectively: 

In such a way, we have that the mechanical energy is the potential energy itself. So:

2) In the figure above we have the body of mass m at the equilibrium point, where the spring elongation is x = 0 and the velocity is maximum, being v = +v or v = -v. At the equilibrium point, the kinetic and potential energies are, respectively:

Therefore, mechanical energy is kinetic energy itself. That way:

3) In the figure above we have the body of mass m at any point C, where the elongation of the spring is x. At this point, the kinetic and potential energy are, respectively:

In this case, we have that the mechanical energy of the system is the sum of the kinetic and potential energy. That way:

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