A windy day

I was at an old house, sitting on a chair and looking to the heavy iron gate through which people have to pass to enter the house. Then I saw the gate trembling slowly, in a back-and-forth motion, and I supposed it was because of the air, since that day was very windy. I thought that the situation could turn into an interesting Brilliant problem, so here it is: consider that the iron gate weights 50 kg, has an elastic constant of 50N/m and is a rectangle with dimensions 1.5x2. The wind collided with the gate with a speed of 1m/s and left (instantly) to the other side of it with a speed of 0.9m/s. Considering that the air collided with the whole gate and that the wind blast lasted 1s, what is the frequency, in Hertz, of the gate's vibration? You may take the density of air to be \(1.22 \text{ kg/m}^{3}\).

Details and Assumptions:

  • The whole volume of air collided instantly with the gate. Do consider that air is always suspended in the atmosphere.

  • The vibration is the back-and-forth motion. Consider that the whole gate moves, because it is bound by chains, which let this happen.

  • Answer in decimals.

  • Consider that there was no energy loss to anything that was not part of the system.

Inspiration: A lazy day
Image credit: National Trust Prints.

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