# At their own pace

Illustration: Hugo Salais

In 1831, a group of 74 soldiers marching back to their barracks caused the collapse of the Broughton Suspension Bridge near Manchester. The requirement to stop marching when crossing bridges did not prevent the collapse of the Angers bridge in France in 1850, which killed 226 people. The collapse of a bridge due to the cadence of the footsteps of those crossing it is merely anecdotal for physics students. Perhaps not so much for engineering students: on 20 June 2000, five days after its opening, London’s Millennium Bridge was closed to the public because of the alarming vibrations caused by pedestrians crossing it.

An everyday scenario: the car is stationary, the engine is idling, and we hear the glove compartment vibrating. We accelerate a little, increasing the engine speed, and now part of the rear-view mirror is vibrating. If the engine transmits the vibration to the whole car, why does not everything vibrate? Every object has its own natural frequencies at which it vibrates. If we apply a force to the object at the same frequency, we make it vibrate at maximum amplitude (resonance). If we force it to vibrate at frequencies different from its own, it does not even flinch. I remember a concert where a ceiling panel vibrated wildly every time the pianist played a certain key. Of all the objects present, only this one had a frequency (of resonance) that unfortunately coincided with one of the tones emitted by the piano.

Let’s remember our childhood experiences with the swing. The frustration of not knowing how to move in the right rhythm and repeating «can you swing me?», that is, can you push me in the rhythm that will make the swing move with great amplitude and amuse me? If the rhythm is not right, the swing will stop, just like when rocking a cradle. And the fact is that the frequencies of each object are what they are. The frequency at which the branches of a tree vibrate is lower than that of the leaves. The frequency of a hummingbird’s wings (50 Hz or beats per second) is lower than that of a bumblebee (200 Hz). The lowest frequency of the last string of a guitar (82 Hz) is lower than that of the first string (330 Hz). And as the legend attributed to Pythagoras says, each blacksmith’s hammer, when struck, produces a sound with a different frequency, partly due to its dimensions.

Many technological applications (lasers, NMR, atomic force microscopy or AFM) and various phenomena are based on the concept of resonance. The frequency at which a hammer works is close to the frequency of the hand holding it, which vibrates at maximum amplitude (resonance) and is an occupational hazard. The deepest sounds that reach our ears resonate at the thicker end of the basilar membrane, in the cochlea, and the high sounds at the thinner end; with the information from this place, transmitted by the acoustic nerve, the brain generates the sensation of sound.

An important point to remember is that in resonance, even if the force applied to the object is relatively small, it will vibrate with great amplitude if it is applied at the right frequency. Hence the shattering of a crystal glass when exposed to a sound of the same frequency as that produced by the glass when struck, or the oscillation and collapse of a bridge caused by the footsteps of those crossing it. There are those who claim to communicate with spirits using pendulums that oscillate on their own, although it is their hand that imperceptibly moves them in resonance.

In 2011, a 39-storey shopping centre in Seoul was evacuated because of the enormous vibrations apparently caused by seventeen practitioners of tae-bo (a martial art between taekwondo and boxing). Why did nobody tell them to change their pace?

#### TRY IT:

Hang two pendulums of different lengths from a stick (a pen, pencil, etc.), using two clips and thread. For example, one can be 18 cm long and the other 30 cm long.

A) Holding the stick, move the long side of the stick a little bit and pay attention to its oscillation rhythm. Now swing the wand with your hand trying to reproduce and maintain this same rhythm: only the long end oscillates, with great amplitude; the short end will barely move.

B) We repeat the same with the short pendulum, which has a higher frequency (a shorter swing). Now the long one does not move, and the short one oscillates with great amplitude. The two swing from the same stick, but depending on the frequency of our hand movement, we cause the oscillation of one stick or the other.

© Mètode 2022 - 113. Assembled life - Volume 2 (2022)
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Head of the Department of Applied Physics (UV, Spain).