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brainpushups
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- TL;DR Summary
- I get an extra factor of 2 in the expected result
For fun I was trying to use energy considerations to determine the depth to which a solid object will sink in a fluid to reach equilibrium. The first approach that I tried was just to consider the change in potential energy of the block and the fluid as the block is lowered some unknown distance d into the fluid similar to what is shown in the answer to this post. Upon taking the limit as the vessel's cross sectional area approaches infinity I have an extra factor of 2 in the equilibrium depth.
That approach is a rather lengthy process, but I can get the same result when considering the work done on the block as explained below. I'm hoping the source of the error is the same so I'll use this shorter way to explain and hopefully someone can point out what I am overlooking.
Suppose a rectangular block with specific weight ##\gamma_b##,cross section A and length L descends into an ideal fluid of specific weight ##\gamma_f## by a vertical distance d which is the equilibrium depth. The work done on the block is the difference between the work done by gravity and the work done by the buoyant force. For the former, this is just the product of weight and distance which we can write as $$W_g=\gamma_b A L d$$
The work done by the buoyant force must be integrated since it depends on the depth. After integration we obtain $$W_b=-\frac{1}{2} \gamma_f A d^2$$ which would also be the negative change in the potential energy of the fluid. The depth for which the change in the overall potential energy is zero can then be solved for which yields $$\frac{2L\gamma_b}{\gamma_f}$$
Without the factor of 2 we arrive at the expected result. Again, this is the same thing I find when going through the longer process of considering the change in the fluid height in a vessel and taking the limit as the cross sectional area of the vessel approaches infinity. Any insight about what I am missing here?
That approach is a rather lengthy process, but I can get the same result when considering the work done on the block as explained below. I'm hoping the source of the error is the same so I'll use this shorter way to explain and hopefully someone can point out what I am overlooking.
Suppose a rectangular block with specific weight ##\gamma_b##,cross section A and length L descends into an ideal fluid of specific weight ##\gamma_f## by a vertical distance d which is the equilibrium depth. The work done on the block is the difference between the work done by gravity and the work done by the buoyant force. For the former, this is just the product of weight and distance which we can write as $$W_g=\gamma_b A L d$$
The work done by the buoyant force must be integrated since it depends on the depth. After integration we obtain $$W_b=-\frac{1}{2} \gamma_f A d^2$$ which would also be the negative change in the potential energy of the fluid. The depth for which the change in the overall potential energy is zero can then be solved for which yields $$\frac{2L\gamma_b}{\gamma_f}$$
Without the factor of 2 we arrive at the expected result. Again, this is the same thing I find when going through the longer process of considering the change in the fluid height in a vessel and taking the limit as the cross sectional area of the vessel approaches infinity. Any insight about what I am missing here?
