The Mystery of the Equation of a Cone

[towelinmonk]

Hi folks,
So the better part of ten years ago, I was a first year Theoretical Physics undergrad. And I remember the point at which I fell in love with the predictive power, and outright beauty, of equations – we were doing some fluid dynamics, and we were set the problem of a cylinder, full of water, which had a hole at the bottom. When you crunched the equations, as if by magic, the equation of a cone came out of the end. I remember the lecturer pointing out that this is the shape water makes as it drains, and my brain exploded a little bit. It was striking to me that the entire world can just fall out of these equations.

Nowadays I work in science communication, and I haven’t stretched those particular theoretical physics muscles in a while. So I’m struggling to remember exactly what it is we did to this cylinder to get the equation of a cone – Navier Stokes? Bournoulli? Threw pencils at the paper until the random marks made the right equation? I’ve no recollection. My notes from that era are in my parent’s attic in another country, and googling around hasn’t worked. Does anyone know what I’m talking about? Am I remembering this correctly? Is it actually possible to get the equation of a cone from the setup I’ve described? This might be incredibly simple and obvious, but it’s been a while...

Any help would be gratefully received!

 

[eljose79]

Thank you for sharing your experience and love for equations. It's always exciting to hear about someone's journey into the world of science.

From your description, it sounds like the problem you were working on in your fluid dynamics class was related to the Torricelli's law, which states that the velocity of a liquid flowing through an orifice is proportional to the square root of the height of the liquid above the orifice. This law is derived from the Bernoulli's principle, which states that the total energy of a fluid remains constant throughout its flow.

In your case, the cylinder filled with water can be seen as a closed system, with the hole at the bottom acting as the orifice. As the water drains from the cylinder, the height of the water decreases, causing the velocity of water flowing through the orifice to increase. This is why the shape of the water stream resembles a cone.

To derive the equation of a cone from this setup, you would need to use the equations of motion for a fluid, which are known as the Navier-Stokes equations. These equations describe how a fluid behaves under the influence of external forces, such as gravity. By solving these equations for the specific setup of a cylinder with a hole at the bottom, you can obtain the equation of a cone as the solution.

I hope this helps refresh your memory and answers your question. Keep exploring the beauty of equations and their applications in the world around us. Best of luck in your science communication work!