Fluid Mechanics: Worked Out Problem, Check Work?

[verd]

Hi,

I have two problems that I'm having difficulty with, and I don't think that my answers are correct.


Problem:
A swimming pool measures 5.0m long x 4.0m wide x 3.0m deep. Compute the force exerted by the water against a) the bottom; b) either end. (Hint: Calculate the force on a thin, horizontal strip at a depth h, and integrate this over the end of the pool.) Do not include the force due to air pressure.

This what I did:
(I worked out the side of the pool, not the bottom yet)
http://www.synthdriven.com/images/deletable/14_help03.jpg


The second problem is just a more difficult version of the first. ...And I know I got it wrong:
A dam has the shape of a rectangular solid. The side facing the lake has area A and height H. The surface of the freshwater lake behind the dam is at the top of the dam. a) Show that the net horizontal force exerted by the water on the dam equals 1/2(rho)gHA, that is, the average gauge pressure across the face of the dam times the area. b) Show that the torque exerted by the water about an axis along the bottom of the dam is (rho)gH^2A/6. c) How do the force and torque depend on the size of the lake?

What I got:
http://www.synthdriven.com/images/deletable/14_help04.jpg

Could someone tell me where I went wrong? What are the mistakes I've made?


Thanks,
HM

 

[andrevdh]

One problem is that the [itex]P_o[/itex] term is for the air pressure on top of the water, which should not be included in the calculation. Also the pressure increases in the water with an increase in depth, working with height is rather confusing - what does it mean anyway - when the height is zero the pressure is zero? I might be wrong, but your formula [itex]P=P_o-\rho g h[/itex] is for the air pressure above the surface?

 

[quicknote]

Hi HM,

Thank you for reaching out for help with these problems. I am happy to assist you in understanding the concepts and finding any mistakes in your solutions.

For the first problem, you have correctly calculated the force on the side of the pool, but the question asks for the force on the bottom and either end. To find the force on the bottom, you can use the same method you used for the side, but instead of integrating over the length of the pool, you would integrate over the width. For the force on either end, you would use the same method, but instead of integrating over the length or width, you would integrate over the depth of the pool.

For the second problem, your solutions are mostly correct. For part a), you have correctly shown that the net horizontal force is equal to the average gauge pressure multiplied by the area. However, you have not included the factor of 1/2, which is necessary for the solution to be complete. For part b), your solution is correct, but you have not explained how you arrived at the answer. You can show this by integrating the force over the height of the dam and then multiplying by the distance from the bottom of the dam to the axis of rotation. For part c), the force and torque will increase with the size of the lake, as the pressure and height of the water will also increase.

I hope this helps you to better understand these concepts and identify any mistakes in your solutions. Keep practicing and don't hesitate to reach out for help in the future. Good luck with your studies!

Best,