Resonating Hair

Taking the bus home means you have time to look outside the window. Taking the bus home during rush hour means you have a lot of time to look outside the window. And when you have time, you notice things.
I saw a woman walking on the sidewalk today. Normally this wouldn’t excite me so much, but this woman had the most interesting hair imaginable. It wasn’t curly, or flamboyantly coloured, or dressed up in any special way. What made it marvellous was its length. It was just exactly the right length – just below her shoulders – to resonate along the frequency of her walking.

If that didn’t make 100% sense to you, let me briefly explain. Two interesting things come into effect here: the way we walk, and resonance.
First, is the fact that we do not walk smoothly, because we have a discrete number of feet (2, in most cases). When the left foot steps forward and we push ourselves with our right, we tend a little leftwards. Thus, we actually wobble a bit from left to right as we walk.
Second, let’s consider a child’s swing on the playground. While regular children are fascinated by the windy ride given to them when swinging, physicists (which are usually considered overgrown children) are much more interested in the mathematical description of how they swing. When you push a child on a swing, it’s always better to push him when he at the peak of the climb, and in the direction that he is about to fall at. If you attempt to push faster or slower than that (meaning, at higher or slower frequencies), you will sometimes end up pushing the kid in the opposite direction in which he is moving, which will only slow him down. If you want the swing to go up very high, you have to push it with the same frequency at which it moves – this is the resonance frequency.
We can apply these two principles to hair. If you have long hair, when you walk, it rocks a bit from side to side, just like a child’s swing on the playground. The frequency at which it rocks depends on gravity (which is constant), on its mass (which is virtually unchangeable), and most importantly, on its length.
This particular woman had hair at just the right length, that its swinging frequency was the same as her footsteps. This means that every step she took made her hair rock back and forth higher and higher. It was very neat to see. At first it started low, and we each step you could see how it rose up, and started flailing wildly all about her, until finally she moved her head and the process started all over again.

Of course, it could just be that she rocked her head from side to side, but the phenomenon really looked like the resonance that I saw in physics lab at school. I surely hope I’m not mistaken in my interpretation, because resonating hair is plain awesome.

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For the physics inclined amongst us, lets approximate a bit to see if we have some sort of numeric backup for my hypothesis.
Lets make the horrendous assumption that hair is a simple mathematical pendulum – just a mass on a string. In this case, for small angles its swinging period is:

T = 2 \pi \sqrt{\dfrac{L}{g}}

Which means that its frequency is:

f = \dfrac{1}{T} = \dfrac{1}{2 \pi} \sqrt{\dfrac{g}{L}}

I measured my normal footpace – the period – the time between two left steps – is about one second (from left foot to left foot again), which means that the frequency is one per second. Putting this in the equation above and solving for L will give the length needed in order to match the resonance frequency.

4\pi^2 f^2 = \dfrac{g}{l}
L = \dfrac{g}{4\pi^2 f^2}

Putting the numbers in:

L = \dfrac{9.8}{4 \pi^2 \cdot 1} \approx 0.25[m] .

This does indeed give hair which ends just below the shoulders.
Now, lets think about hair for a minute. A single strand of hair probably won’t resonate at all – it’s way too wobbly, and we know that single strands tend to float around in the air sometime. So if there is any swinging action, it’s because the strands cling together. This is a popular phenomenon with hair – it tends to cluster.
I previously calculated the length for a mass on a massless string – a simple mathematical pendulum. Now lets consider the exact opposite condition – and treat the clustered strands as a single, hard, solid rod. In this case, the period relies on the moment of inertia:

T = 2\pi \sqrt{\dfrac{I}{mLg}}

The moment of inertia for a rod rotating around its far end is:

I = \dfrac{m L^2}{3}

Putting this in, we get that L \approx 0.75[m] .
Which gives a sort of upper bound to the length. The true solution is somewhere in between, because combined strands of hair aren’t completely rigid at all – they are quite flexible, which is a compromise between a massless object on a string and a massive rod rotating around a fixed point. Perhaps a more detailed analysis of the mechanics of swinging hair will be made at a later point.

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