# Simple Harmonic Motion Energy

If you've ever seen an object moving back and forth like a pendulum, you've witnessed Simple Harmonic Motion (SHM). This type of motion happens when an object's acceleration is related to its position. But did you know that SHM also has something called SHM Energy? This is the energy an object has when it moves back and forth in a predictable way.

When an object is in motion, it has kinetic energy - energy because it's moving. On the other hand, when an object is still, it has potential energy - energy because of its position. During SHM, these two types of energy switch back and forth constantly. There are two types of potential energy: gravitational and elastic. For example, a pendulum with a mass on it has gravitational potential energy when it's not in its equilibrium position. Meanwhile, a mass on a spring has elastic potential energy.

So, that's a brief overview of SHM Energy and the types of energy involved in it. Understanding this concept is important for anyone studying physics or just curious about the world around them.

## What is the energy in a simple harmonic system?

In a simple harmonic system, the total energy always stays the same and is made up of kinetic and potential energy. As one form of energy goes up, the other goes down, causing a continuous exchange of energy. When one form reaches its highest value, the other reaches its lowest value, and the system keeps going as the total energy stays constant.

To see this in action, let's take the example of a pendulum that starts swinging from its middle position. As you can see in Figure 1, the pendulum moves from its starting point to the highest point on the right side of the middle point, which is the highest positive point. Then it swings back to the middle point, changing direction, and moves to the highest point on the left side of the middle point, which is the highest negative point. This back-and-forth motion keeps going as the system's energy stays constant.

The energy of a pendulum during oscillation depends on its position. At the beginning of the swing, the pendulum has kinetic energy and its potential energy is at a minimum (0). As the pendulum reaches highest point, it stops momentarily, and its kinetic energy decreases to 0, while its potential energy reaches its maximum. Then, as the pendulum moves in the opposite direction, its kinetic energy starts to increase while its potential energy decreases. Finally, when the pendulum returns to its starting point, after completing one cycle, its kinetic energy reaches its maximum, and its potential energy returns to its minimum value. This cycle of energy conversion continues as long as the pendulum keeps swinging.

**Mechanical energy conservation in simple harmonic motion**

In an ideal oscillator, mechanical energy is conserved during simple harmonic motion. This means that external forces like friction or air resistance are not taken into account. If energy is not being lost due to these external forces, the total mechanical energy in the system remains constant. Mechanical energy is made up of kinetic and potential energy, and it stays the same throughout the motion. As we discussed earlier, energy is continuously exchanging between these two forms during simple harmonic motion. Therefore, we can conclude that when the kinetic energy is at its maximum at equilibrium (when x = 0), the potential energy is 0. And when the potential energy is at its maximum at the two extreme points of motion, the kinetic energy is 0.

## Energy vs time graph in simple harmonic motion

The energy vs time graph shown in Figure 2 demonstrates the conservation of mechanical energy in simple harmonic motion. From this graph, several properties of simple harmonic motion can be derived:

- The kinetic energy is at its maximum when the potential energy is 0, and vice versa.
- Both kinetic and potential energies are periodic functions, represented by sine or cosine waves.
- The two 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, both the kinetic and potential energies go through two complete cycles, as one period reaches the maximum amplitude point twice (negative and positive).

These properties are crucial in understanding the behavior of simple harmonic motion and the conservation of energy in the system.

The average energy in an oscillator performing simple harmonic motion is the total energy of the oscillator in one time period, which is the time it takes for the oscillator to return to its initial equilibrium position after it has reached both of the amplitude points once.

## 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.

Displacement is a vector quantity, and the energy vs time graph in simple harmonic motion has both positive and negative displacement values. The potential energy is at its maximum at the maximum amplitude position (x=±Xmax) and goes to 0 at the equilibrium position (x=0), which is represented by a U-shaped curve. The kinetic energy is at its maximum at the equilibrium position (x=0) and goes to 0 at the amplitude positions (x= represented an upside-down-shaped curve. The total energy is represented by a horizontal straight line above the curves since mechanical energy is conserved in simple harmonic motion. Understanding these characteristics is crucial to analyze the energy vs time graph and gain insight into the behavior of simple harmonic motion.

## Simple harmonic motion energy equation

To calculate the maximum kinetic energy of the object, we can use the equation for maximum kinetic energy in simple harmonic motion, which is:

KE = (1/2) * m * ω^2 * Xmax^2

where m is the mass of the object, ω is the angular frequency, and Xmax is the maximum amplitude of the motion.

Using the given equation for the position of the object, x(t) = 10 sin(2t), we can see that the amplitude, Xmax, is equal to 10 and the angular frequency, ω, is equal to 2.

Substituting these values into the equation for maximum kinetic energy, we get:

KE = (1/2) * (5 kg) * (2)^2 * (10)^2

Simplifying the equation, we get:

KE = 500 J

Therefore, the maximum kinetic energy of the object performing simple harmonic motion is 500 J.

## Energy in Simple Harmonic Motion - Key takeaways

In simple harmonic motion, the energy of the system is constantly being transferred between kinetic and potential energies. At the maximum displacement, all the energy is in the form of potential energy, while at the equilibrium position, all the energy is in the form of kinetic energy. As the oscillator moves between these two positions, the potential energy is converted into kinetic energy and vice versa, resulting in a continuous exchange of energy between the two forms.

It is crucial to note that the total energy of the system remains constant throughout the motion, and mechanical energy is conserved in simple harmonic motion. This means that the sum of kinetic and potential energies at any point is always equal to the total energy of the system. It is also important to note that neither the kinetic energy nor the potential energy can be negative in simple harmonic motion, as both are proportional to the squares of their respective quantities, velocity, and displacement.

Having an understanding of the exchange of energy and the conservation of mechanical energy in simple harmonic motion is vital in interpreting the behavior of oscillatory systems.

## Simple Harmonic Motion Energy

**Why does simple harmonic motion have zero point energy?**

Because the kinetic and potential energies interchange. When one increases, the other decreases. When one reaches a maximum value, the other reaches its minimum value 0.

**What is energy of oscillation?**

Energy of oscillation is the energy that an oscillator possesses when it performs simple harmonic motion.

**What happens to the energy of a simple harmonic oscillator?**

The total energy of a simple harmonic oscillator remains constant.