Simple Harmonic Motion Energy

What is the energy in a simple harmonic system?

This can be interpreted as follows:

• As the two forms of energy are interchanging continuously, when one form increases, the other decreases. Hence, when one form of energy reaches its maximum point, the other form reaches its minimum value (zero).
• If the energy is constant, the system is oscillating.

Mechanical energy conservation in simple harmonic motion

In an ideal oscillator, the mechanical energy is conserved in simple harmonic motion. This means that external forces, such as friction, air resistance, etc., are ignored. If energy is not being lost due to external forces, it is conserved in the system. Mechanical energy is the sum of kinetic and potential energy and is constant. Energy is continuously interchanging between the two forms in simple harmonic motion, as mentioned earlier.

Therefore, it can be concluded that:

• KE = KE_max at equilibrium or x = 0
• PE = PE_max at maximum amplitudes
• When KE = KEmax, PE = 0
• When PE = PEmax, KE = 0

Energy vs time graph in simple harmonic motion

The conservation of mechanical energy is illustrated in the energy vs time graph in simple harmonic motion in figure 2, where the following properties can be derived:

• When the potential energy is 0, the kinetic energy is at its maximum point and vice versa.
• Both kinetic and potential energies are represented by periodic functions (sine or cosine).
• The two periodic functions vary in opposite directions.
• Energy is always positive.
• The total energy is represented by a horizontal straight line at the maximum value of both kinetic and potential energy.
• During one period of oscillation, kinetic and potential energy go through two complete cycles, as one period reaches the maximum amplitude point twice (negative and positive).

Energy vs displacement graph in simple harmonic motion

Another graph that can be conducted from the principle of mechanical conservation of energy is the energy vs displacement graph in figure 3, where the total energy is shown as well as the energy at the maximum amplitude points. An interchanging pattern can be seen in the graph.

• As displacement is a vector quantity, the graph has both positive and negative displacement values.
• The potential energy is at its maximum at the maximum amplitude position, where x = ±Xmax, and at 0 at the equilibrium position, where x = 0.
  • This is represented by a U-shaped curve.
• The total energy is represented by a horizontal straight line above the curves, and it is constant.
• The kinetic energy is at its maximum at the equilibrium position, where x = 0, and at 0 at the amplitude position x = ±Xmax.
  • This is represented by an upside-down U-shaped curve.

Simple harmonic motion energy equation

The simple harmonic motion energy equation provides the numerical magnitude of energy of an oscillator. This kinetic energy equation can be derived by starting with the equation of energy in translational motion (see below), where KE is the kinetic energy, m is the mass, and V is the speed.

\[KE = 0.5 \cdot m \cdot v^2\]

By substituting the velocity equation of simple harmonic motion at any given time, we get the following, where X_max is the maximum amplitude, while x is the current position of an object at a given time.

If the position of the object is the equilibrium, the kinetic energy is at its maximum and proportional to the maximum amplitude.

\[v = \sqrt{\pm \omega (X_{max}^2 - x^2)} \qquad v^2 = \omega^2(X_{max}^2 - x^2) \qquad KE = \frac{1}{2} \cdot m \cdot \omega^2 (X^2_{max} - x^2) \qquad KE_{max} = \frac{1}{2} \cdot m \cdot \omega^2 \cdot X_{max}^2\]

Where \(\omega = 2\pi f\)

For the potential energy, we also use the kinetic energy but substitute the spring constant. The potential energy equation is shown below, where PE is the potential energy, k is the spring constant, and ω and t are the angular frequency and time, respectively.

If the position of the object is the equilibrium, the potential energy is at its maximum and proportional to the maximum amplitude.

\[k = m \omega^2 PE = \frac{1}{2} \cdot k \cdot (X_{max}^2 - x^2) \qquad PE_{max} = \frac{1}{2} \cdot k \cdot X_{max}^2\]

The total energy can be determined by the sum of the kinetic and potential energy, which is shown below, where k is the spring constant measured in N/m, and x is the displacement measured in m.

\[E_{total}[J] = KE + PE\]