Energy in Simple Harmonic Motion

Energy in Simple Harmonic Motion Definition

Just like kinetic and potential energy, there is also energy that comes from simple harmonic motion. Additionally, I need to explain an essential distinction right off the bat. I will use the words "simple harmonic motion" and "simple harmonic oscillator" interchangeably throughout this article. However, they are not interchangeable. A simple harmonic oscillator is an object that undergoes simple harmonic motion. The oscillator is the object, and the motion refers to actual oscillatory motion. But to understand both concepts, we must first define restoring forces.

Restoring Forces

The universe likes to be in equilibrium; it wants to be balanced. The universe has laws to correct the imbalance whenever something disrupts that balance. One of these laws involves restoring forces. When an object is in equilibrium, everything is great. To quote the Lego Movie, "Everything is awesome!" But once that object moves from equilibrium, the balance has been destroyed, and the universe gets angry.

A restoring force is a force that acts on that object to put it back to its equilibrium position: to put it back into balance. That is the reason the restoring force always acts opposite to an object's displacement; it is trying to put the object back in its rightful place. That "rightful place" is the equilibrium position.

For energy in simple harmonic motion, the equilibrium position is where there is no energy when the object is stationary because there are no net forces on the object or system.

Fig. 2 - Point B is the equilibrium: the place where the displacement \(x=0\) and the net forces acting on the object \(F_\text{net} = 0\).

Energy in Simple Harmonic Motion and its Derivation

Before we get into the energy of simple harmonic oscillators, we need to talk about simple harmonic motion, a type of periodic motion.

Simple harmonic motion results when the magnitude of a restoring force exerted on an object is proportional to that object’s displacement from its equilibrium position.

The derived equation for simple harmonic motion is given by:

$$ma_x = -k\Delta x$$

where \(m\) is the mass, \(a_x\) is the acceleration in the horizontal direction, \(k\) is the harmonic oscillator constant, and \(\Delta x\) is the displacement.

Energy in Simple Harmonic Motion Formula

The mechanical energy of simple harmonic motion is the sum of its potential and kinetic energies:

$$E_\text{tot} = U + K$$

where \(E_\text{tot}\) is the total mechanical energy, \(U\) is the potential energy, and \(K\) is the kinetic energy.

Fig. 3 - This image shows how potential and kinetic energy have an inverse relationship, that is, as one decreases the other increases. But, due to conservation of energy, their combination cannot go over the total energy of the system.

As you can see from the equation and graph above, potential energy and kinetic energy have quite a rocky relationship. The more potential energy there is, the less kinetic energy there has to be to keep the total energy balanced. Simple harmonic motion is nothing more than a balancing act.

Conservation of energy governs this balance. For energy to be conserved, the total energy of an isolated system with simple harmonic motion must be constant. Therefore, to keep the balance, the potential energy is at a maximum when the kinetic energy is at a minimum. Furthermore, the kinetic energy is at a maximum when the potential energy is at a minimum. As one goes up, the other must go down, or the energy will not remain constant: quite the rocky relationship.

We can think of harmonic motion in terms of potential energy. Potential energy is energy inherent in an object based on its position relative to another object. Therefore, if my two objects were a ball and the earth, the higher I move the ball from the earth's surface, the more potential energy it gains. This is similar to energy stored in simple harmonic oscillators. As the object is displaced from its equilibrium more, the maximum potential energy it can experience is increased, so its total energy is ameliorated.

$$E_\text{total} = \frac{1}{2}\\k A^2$$

is the equation we use to describe the energy for a spring object system with \(E_\text{total}\) as the total energy of the system, \(k\) as the spring constant, and \(A\) as the amplitude.

Fig. 4 - A spring is a perfect example of a simple harmonic oscillator that exhibits simple harmonic motion.

Energy in Simple Harmonic Motion Example

Now armed with all this knowledge on simple harmonic motion, let's do some examples.

Energy in Simple Harmonic Motion Derivation

First, we'll derive the equation for the simple harmonic motion of a spring.

Example 2

Now, we'll use that equation to find the energy of a spring-object system.

Finally, we've reached the bottom of a long staircase; here is the essential knowledge for the energy of simple harmonic oscillators.