Simple Harmonic Motion: Explaining the Physics Behind the Motion of an Ideal Pendulum

When you look at a simple pendulum, have you ever wondered why the object at the end of the string (often known as a bob) swings back and forth? Why does the bob move back and forth, and seemingly move without stopping in this regular pattern? This motion is known as simple harmonic motion. This article will explain what this principle is, express this principle in a mathematical equation, and identify several other applications of this principle commonly seen in everyday life. 

What is Simple Harmonic Motion?

According to Britannica (2024), simple harmonic motion (SHM) refers to the repetitive movement back and forth through an equilibrium (or central) position, so that the maximum displacement on one side of this position is equal to the maximum displacement on the other side: 

Image of a simple pendulum (Britannica, 2024)

From the image above, the centre bob that you see is its equilibrium position, whereas the bob at the most left and right of the image is when the bob is at its maximum displacement, away from its equilibrium position. 

Process of the Motion of the Ideal Pendulum

Before we proceed, note that in this ideal pendulum, we will ignore all resistive forces that will prevent the pendulum from moving in constant motion, and any other potential factors capable of causing the object to stop moving. We must also assume that the angle at which the bob is released from is small enough so that through approximations, the pendulum will be in SHM. 

The following explanation would be using the principle of conservation of energy to explain the motion of the pendulum. Basically, this principle states that energy can only be converted from one form to another, and cannot be created nor destroyed.

Now, using the above grounds, we can finally go into the explanation: 

1. Initially, the bob is held at a certain position (referencing the image, let us take this position as the right most bob). At this position, the bob possesses energy known as gravitational potential energy. 

2. When the bob is released, the energy that the bob has is converted into kinetic energy as the bob swings downwards due to gravity (and is only prevented from going downwards further due to the tension from the string that the bob is attached to). 

3. When the bob reaches its equilibrium position, the bob does not stop, but rather continues moving and swings upwards. This is because at this position, the bob will still have kinetic energy and hence will still move. As energy cannot be destroyed, the bob will continue to move. The tension of the string on the bob causes the bob to swing upwards. 

4. Referencing the image above, as the bob heads towards the left most position, the gravity acting on the bob causes the bob to slow down, and kinetic energy is converted back into gravitational potential energy.  This left most position is where the bob is at its maximum displacement. 

5. This causes the motion of the bob to repeat from step 1, only this time the bob is falling from the left most position instead of the right most position. This cycle then repeats itself, causing the pendulum to move back and forth. 

Mathematically…

The relationship in SHM states that the force acting on the object when the object is at maximum displacement (also known as the restoring force) is proportional to the displacement of the object away from the equilibrium position. Generally, this concept is represented by the relationship that the acceleration of the object is directly proportional and oppositely directed to its displacement from equilibrium position, as given by: 

where: 

• a: acceleration of object 

• w: angular frequency of object 

• x: displacement of object 

If you wish to delve deeper into the mathematics of simple harmonic motion, you may watch this video (Physix Daily, 2021) to find out how the above equation can be derived, and even how to derive the equations of the object’s velocity and position! (Note that this video uses the concept of forces acting on the object to derive the various equations seen in the video as opposed to using energy to explain the motion of a pendulum undergoing simple harmonic motion)

Works cited 

Britannica, T. Editors of Encyclopaedia (2024). pendulum. [online] Encyclopedia Britannica. Available at: https://www.britannica.com/technology/pendulum [Accessed 30 Dec. 2024].

Britannica, T. Editors of Encyclopaedia (2024). simple harmonic motion. [online] Encyclopedia Britannica. Available at: https://www.britannica.com/science/simple-harmonic-motion [Accessed 30 Dec. 2024].

GeeksforGeeks. (2024). Simple Harmonic Motion. [online] Available at: https://www.geeksforgeeks.org/simple-harmonic-motion/. [Accessed 30 Dec. 2024].

Physix Daily. (2021). Introduction to Simple Harmonic Motion (SHM) | General Equation and Derivation. [online] www.youtube.com. Available at: https://www.youtube.com/watch?v=8bTvHbrJ8Cg  [Accessed 31 Dec. 2024].