# How long does it take for this bucket to empty?

## Homework Statement

Ignore turbulence and viscosity.
A cylinder with radius r is filled to depth d. There's a leak in the bottom of the cylinder.
When suspended from a rope, the depth is reduced to d/2 after 10 minutes. From this point, how long should it take to empty completely?

## The Attempt at a Solution

My initial thought would be that it would take 10 minutes, but that must be oversimplifying things. As the bucket empties, the pressure will surely decrease, and so, therefore, will the rate of loss of water.
The equation P=ρgd comes to mind, where P is the pressure, ρ the water density, g the acceleration due to gravity and d the depth.
I am unsure how to relate this to time though.
Would I need to find an equation for d, relating it to time, and take the integral, with limits d to d/2, then compare that to the current situation? If so, I have no idea how to actually do this.

Any help appreciated
Thanks

You need to find an equation for the "rate of loss of water". From that, you can form an equation relating the depth with time, and, using the data specified, solve the problem.

haruspex
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Would I need to find an equation for d, relating it to time, and take the integral, with limits d to d/2,
Yes, but create a new variable for the depth at time t. (d is constant here.)
What relationship can you find between the pressure at the bottom of the bucket and the leak rate?

rude man
Homework Helper
Gold Member
Too bad the OP would not follow up. I would like to compare my approach with the other two posters & am sending them mine to that end.

Oops, looks like voko is out.

TSny, mfb, Chet et al?

I would go Bernoulli -> Torricelli -> diff. eq. for depth. Then I would plug in the 10 minutes - d/2 datum, determine unspecified parameters, and solve. I have not actually done so, so there might be some gotchas I don't know about.