Sinking bucket - differential equations

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SUMMARY

The discussion centers on modeling the sinking of a bucket with a hole underwater using Ordinary Differential Equations (ODEs). The user seeks guidance on establishing a differential equation to describe the rate at which the bucket sinks as it fills with water. Key variables include the mass of the bucket, the density of water, and the area of the bucket's base. The conversation emphasizes the relationship between the rate of water inflow and the bucket's buoyancy, leading to the formulation of a differential equation.

PREREQUISITES
  • Understanding of Ordinary Differential Equations (ODEs)
  • Knowledge of buoyancy principles and fluid dynamics
  • Familiarity with mathematical modeling techniques
  • Basic calculus, particularly differentiation and integration
NEXT STEPS
  • Study the derivation of differential equations in fluid dynamics
  • Learn about buoyancy and its mathematical implications in physics
  • Explore examples of ODEs related to fluid flow and sinking objects
  • Investigate the use of proportionality in modeling rates of change
USEFUL FOR

Students and educators in mathematics and physics, particularly those focusing on fluid dynamics and differential equations. This discussion is beneficial for anyone looking to enhance their modeling skills in real-world scenarios involving fluid mechanics.

markswabinski
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Homework Statement



Let's have a bucket flowing in water. Now we make a hole underwater. How fast will the bucket sink completely under water?

It is a question from course called Ordinary Differential Equation, so I'm supposed to establish an ODE to solve this problem. I understand how to solve such equation, but I'm really bad at creating models.

The details about water, where the hole is or how the bucket looks like are not specified - can either be general or taken to be insignificant.

The Attempt at a Solution



I've tried by specifying the t0 state: we know that it floats, so buoyancy equals gravity: m0 = rho_w * h * Ab (where m0 is the mass of bucket, rho_w is the density of water, h is how much the bucket has sunk when floating and Ab is the area of base of backet assuming rectangular bucket for simplicity).

Now the hole appears. I have no idea how to employ the rate of flowing water (since differentials are to be used) into the equations to come up with an model to solve.

Any help or the nudge towards right direction (articles / solved examples) highly appreciated.
 
Last edited:
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hi markswabinski! welcome to pf! :smile:

i assume the bucket starts empty

suppose the rate of filling the bucket is v(t)

what do you think v is likely to be proportional to?

write that as a differential equation (don't bother with any constants such as ρ or A, just include them all in one big constant C :wink:) …

show us what you get :smile:​
 

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