Understanding the Relationship Between Dripping Rate and Water Clock Design

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SUMMARY

The discussion focuses on the mathematical modeling of water clock design, specifically the relationship between the dripping rate and the height of water. The user proposes a differential equation, dy/dt = -k (πr² pg y(t)), where p represents water density and g is gravitational acceleration. The user notes that the resulting function declines too steeply for practical application and seeks validation of their approach. Adjustments to the constant k are suggested as a potential solution to refine the model.

PREREQUISITES
  • Understanding of differential equations, specifically first-order linear equations.
  • Familiarity with fluid dynamics concepts, particularly pressure and density.
  • Basic knowledge of calculus, including derivatives and exponential functions.
  • Experience with mathematical modeling in physical systems.
NEXT STEPS
  • Explore the derivation and application of first-order linear differential equations.
  • Research fluid dynamics principles related to pressure and flow rates.
  • Investigate the effects of varying the constant k in differential equations.
  • Learn about the design and functionality of water clocks in historical contexts.
USEFUL FOR

Mathematicians, engineers, and hobbyists interested in fluid dynamics and mechanical clock design will benefit from this discussion.

That Neuron
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Okay, this is a really simple question, so to anyone looking for some extraordinarily complex differential equation question turn away now, or be blinded by boredom.

My query is rooted in a question I had about building a water clock... so seemingly relevant to Differentials, I know. Anyways, I realized that the rate of dripping (though probably much more complex than a proportionality) was at simplest proportional to the height, or at least related to it.

Anyway, I was thinking that if the rate of change of the height is proportional to the pressure on the hole at the bottom out of which water drips (or pours) then I could create the differential dy/dt = -k (πr2 pg y(t), where p is equal to the density and g is the acceleration due to gravity, this equation translates to y = y(0) e-kπr2pgt.

But this function seems to decline too steeply for this application, am I doing this right?
 
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That Neuron said:
Okay, this is a really simple question, so to anyone looking for some extraordinarily complex differential equation question turn away now, or be blinded by boredom.

My query is rooted in a question I had about building a water clock... so seemingly relevant to Differentials, I know. Anyways, I realized that the rate of dripping (though probably much more complex than a proportionality) was at simplest proportional to the height, or at least related to it.

Anyway, I was thinking that if the rate of change of the height is proportional to the pressure on the hole at the bottom out of which water drips (or pours) then I could create the differential dy/dt = -k (πr2 pg y(t), where p is equal to the density and g is the acceleration due to gravity, this equation translates to y = y(0) e-kπr2pgt.

But this function seems to decline too steeply for this application, am I doing this right?

Is you're equation dy/dt=-k(...pg)y(t) where "y(t)" is y as a function of t. or the variable y times the variable t?
 
Oh, no y is a function of t!

Not the height!

I actually think that I found the correct function simply by playing around with the constant k. Let me revisit this!
 

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