“How often will I need Pythagoras when I’m older though”, “I’m never going to have to find ‘𝑥’ in real life”, “But when will I ever use this”, questions that teachers (and a lot of students) are tired of hearing. Unfortunately, after policing 30 energetic students and trying to at least teach what is needed to pass the exams, teachers have to leave many of these queries unanswered. But that does not mean there is not one! Clearly everything taught in schools has a purpose and application in the world around us even if this can get lost in the lesson. Whether it’s a calculation we will do personally or just present in the technology and systems we use, it would be almost impossible to go a day without those strange equations!

To celebrate these equations and demonstrate their importance in our lives, we wanted to share some interesting applications for a different beautiful equation each week! So to start here are three concepts that simply couldn’t work without understanding Simple Harmonic Motion

𝑇 =2π√𝑚𝑘

1. A Grandfather’s clock

Not long ago when all clock’s were mechanical, you couldn’t see a clock without also hearing the comforting ‘tick tock’ of the time-keeper behind it. This was due to a swinging rod, a pendulum, inside that would keep the time. In a grandfather’s clock, these pendulums are large and can often be seen (which is why we are using it as the example, all mechanical clocks use simple harmonic motion!). As the pendulum swings back and forth, it converts kinetic energy to potential and back to kinetic in a closed system which is why each complete oscillation (‘swing’) takes the same time. It is this property that we define as ‘Simple Harmonic Motion’ and was crucial to keeping time for over 700 years!

2. Bungee Jumping

While hurtling towards the ground after jumping off a cliff with only some elastic tied around your ankles, simple harmonic motion is probably the last thing on your mind... However, this is a very real and practical example of a mass-spring system (like the one described in the above equation)! The bungee jumper is the mass in this example and the elastic round their legs the spring. This is not quite a closed system; air resistance will cause some energy losses and so the time period for each oscillation will slowly decrease. This means they will not bounce up quite as high after each fall but also thankfully that the system will not carry on indefinitely. Even the biggest thrill lovers might not appreciate hanging upside down forever.

3. Hearing

When a noise is made, for instance playing a note on the guitar, the movement of the string will cause the surrounding air to vibrate, these vibrations are transferred through the air until they reach the ear. The ear drum will then be moved, vibrating in simple harmonic motion at the same frequency as the original note! The frequency, and hence time period, of the system is what the brain then interprets as the pitch of that specific sound, and so music and hearing itself couldn’t be understood without simple harmonic motion!