why do the shorter pieces of metal make a small, high pitched sound, while the longer metals make a deeper sound?


talking about xylophones

In: 4

Think of a ruler. If you fix it at one end and apply a force at the top, it will move the top pretty far. It has a low stiffness due to being so long and slender. Now cut the ruler to only 2 inches and apply that same force to the top now. It only bends a little bit because it is much shorter. This short piece has a much higher stiffness because it is much shorter.

When we talk about vibrations, we are really talking about what the natural frequency of the piece of material. The frequency will increase with stiffness and will decrease if you add mass. Since we know shorter objects are stiffer, this makes the pitch higher for the short object.

I can get into more detail if you would like, but I wanted to keep it relatively simple for ELI5.

Imagine a ruler that you lay over the edge of a table. The longer it sticks out, the deeper the sound it makes when made vibrate. The more it sticks out, the further it needs to travel in one swing to reach the point where the material stretches so much that it makes it travel back. Therefore the frequency with which it oscillates (how many rounds per time unit does it swing back and forth) is lower. Lower frequency equals lower pitch.

With the xylophpne its similar, only that it rests on two eges and the distance between the define how much material has to move.

Everyone else is talking about rules, so let’s shake it up and talk about yo-yos.

Imagine two yo-yos. One has a short string, and the other a long string. They both get dropped at the same time, and begin falling at the same speed. But the one on the short string soon reaches the maximum length and starts coming back up, while the long string one keeps going down. Maybe at the same time that the short one reaches the top and starts to fall again, the long one reaches the bottom and starts to rise. The two fall at the same speed because gravity is the same for both, but they rise and fall at different *frequencies*, with the short one doing it twice as often, because they have different distances to go.

It’s pretty much the same with metal bars. Each little segment of the bar can wiggle a certain amount. Adding these together gives how far up and down each bar can move. A longer bar can therefore move further up and down so takes more time. This means it completes a wobble less often, the sound has a lower frequency or pitch.

For very small displacements, you can think of the metal oscillators of a xylophone as spring-like things.

The main rule about these objects is that when you apply a force, the entire object stretches or contracts proportionally to that force. That means the longer your string is, softer it is for a set amount of travel. The softness/stiffness is known as the spring constant in springs.

For these small displacements your pitch is determined entirely by the stiffness of the spring and the material’s density.

For further reading you can check out:
Springs and simple harmonic motion,
Strings as an interesting oscillator behaviour,

If you’ve got calculus and/or linear algebra background:
1-dimensional harmonic oscillator equation is a good starting point.
Stress-strain tensors is a good thing to look at too.

P.S before anyone points out: What separates these from linear oscillators is that becaue they have a shear component, the strain will not be linearly dependent on the displacement, so it isn’t really a linear oscillator except under small oscillation limit.