It’s Simple (HM)

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Have you ever done that trick where you dip your finger into a glass of wine and then proceed to glide said finger around the rim of the wine glass, producing a high-pitched ring? Well I certainly haven’t because I never drink wine, it’s absolutely vile. However, I’m willing to excuse it for today because the effect is kinda interesting, I suppose.

Let’s have a look at why exactly this happens. Whenever you have a system which oscillates back and forth over a fixed point, the motion is known as simple harmonic motion, or SHM. The defining feature of this type of movement is that the restoring force is directly proportional and in the opposite direction to the displacement. A pendulum, for example, if displaced from its ‘rest’ position, will have a restoring force acting in the opposite direction to its displacement, towards its original equilibrium position. As it swings back down, it will overshoot this position due to inertia, and swing up to the same height on the other side (given that there are no external forces like air resistance). Now, a pendulum probably wasn’t the best system to choose because it doesn’t actually completely follow the equations of SHM. Its motion can only be accurately approximated when the angle of swing is very small. A mass oscillating on a spring would be a better example, but perhaps a bit more difficult to imagine.

Any vibrating system has a natural frequency at which it will tend to oscillate in the absence of driving forces. If I let my hypothetical pendulum swing freely, it would eventually be swinging at its natural frequency, determined by its length. Say we applied a bit of force periodically to the pendulum, and this force was given at the same frequency of the natural frequency, the amplitude of the pendulum would increase massively, causing it to swing much higher than would be expected. This phenomenon is known as resonance.

We can demonstrate this idea of resonance using Barton’s pendulums. The set-up consists of a row of pendulums tied to a string, with different length but of the same mass. At one end of the string, a driver pendulum is attached which has a mass considerably greater than the others so that the effect is more noticeable. When the driver pendulum is set into motion, it pushes the string that is attached to, which in turn causes the other pendulums to start oscillating. Most of them will swing with a low amplitude, but the pendulum that has the same length as the driver will swing with a large amplitude. This is because its natural frequency is the same as the driving frequency, and so resonance occurs. If you have the time and effort, try it out for yourself! Get out some string and use blu-tack as your masses. Vary the length of the driver pendulum and see if you can use it to control each other pendulum individually. Alternatively, you know, you could look it up on YouTube.

bartonsexp
Courtesy of one-school.net

Returning to our original alcohol-based conundrum, you may have begun to realise why the effect occurs. The glass of the cup has a specific resonant frequency, dependent on its innate material properties. As you move your finger across the lip, the friction of the glass opposing the movement of your finger will cause the glass to vibrate and oscillate, just as we have seen with the pendulum, in simple harmonic motion. The lubrication of the water will then cause your finger to slip. Each time the glass resists your finger the frictional force acts as a driver, giving the glass oscillation a minute push. If you get this push in such a frequency that it matches the natural frequency of oscillation of the glass, yep you guessed it – it will resonate. The amplified oscillation of the glass vibrates the air molecules surrounding the glass, which propagates to your ears. The resonant frequency of glass is typically within the range of human hearing, which is why we are able to hear the lovely ring.

Your impression may be that resonance barely affects the amplitude of vibration. If so, you’d be sharing the thoughts of engineers designing the Tacoma Narrows Bridge in Washington in the 1920s. They wildly underestimated the impact of this phenomena, and the results were… pretty funky to say the least. I’ll let you judge for yourself. Yes, that is solid concrete and steel. 

Harvey

 

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