Moment of the Hydrostatic Force on a body

[squire636]

This problem is for a Fluid Dynamics course, but it is mostly math and thus I figured it should be posted here. If it should be moved elsewhere, please let me know and I will do so!

Homework Statement

We saw in class the hydrostatic force acting on a body at rest in an incompressible fluid. Calculate, using a similar procedure, the moment of the hydrostatic force on the
body.

Here is what we did in class: http://imgur.com/wE3O4

Homework Equations

The Attempt at a Solution

Rather than calculating the hydrostatic force, we need to calculate the moment of this force. By definition, we need:

∫_S X x (-pn) dS where X is a position vector, p is the pressure, and n is a normal vector.

Similar to what was done in 6.25 in the image above, I did the following:

= -ρ ∫_S (X x n) (g . X) dS

and then tried to apply the divergence theorem

= -ρ ∫_v (∇ x X) ∇(g . X) dV
= -ρgV ∫_v (∇ x X) dV

I know this isn't right, but it's the best I've gotten after about an hour of work. I know that I need to apply the divergence theorem to get from a surface integral to a volume integral, but the cross product is really throwing me off. I'm also pretty sure that I broke some math rules while trying to mimic step 6.25 in the image above. I'm really just grasping at straws here and trying to do something that makes any sort of sense at all.

Any help will be very appreciated, as I've been stuck for a while. Thanks!

 

[mighty2000]

Thank you for sharing your problem with us. It is always helpful to approach a problem from different perspectives, so I believe it was a good idea to post it here.

I can see that you have made some progress towards finding the moment of the hydrostatic force acting on a body in an incompressible fluid. However, there are a few things that need to be clarified in order to reach the correct solution.

Firstly, the equation you have written for the moment of the hydrostatic force is not entirely correct. It should be:

∫S (X x r) (-pn) dS

where r is the position vector from the origin to the point of application of the force, and p is the pressure at that point.

Secondly, in order to use the divergence theorem, you need to have a volume integral, not a surface integral. So, you will need to convert the surface integral into a volume integral. This can be done by using the fact that the cross product of two vectors can be written as the determinant of a 3x3 matrix, and then applying the divergence theorem.

Finally, you have correctly identified that the cross product is causing some difficulties in your calculation. To simplify things, you can use the fact that the cross product of two vectors is orthogonal to both of the vectors, and therefore, it is also orthogonal to the plane containing the two vectors. This can help you simplify the integration and make it easier to solve.

I hope this helps you make some progress towards finding the correct solution. If you have any further questions or need clarification on any of the steps, please feel free to ask. Keep up the good work!