How do I set up the pressure integral for a submerged curved surface?

[Alexander_P]

I’m really stuck on how to approach this week’s physics problem about calculating the force on a submerged curved surface. The textbook example uses a flat plate, but our assignment has this irregular aquarium window shape, and I can’t figure out how to set up the integral for the pressure distribution.

[Brandon_P]

Chunk the window into tiny flat pieces, each with area ΔA and a known outward normal. The local hydrostatic value p = ρ g h, where h is the vertical depth from the free surface. The elemental force is dF = p ΔA in the direction of the surface normal, so sum them up: F ≈ ∑ p_i n_i ΔA_i. If you treat it as a continuous surface, turn the sum into the integral F = ∫ p n dA over the surface.

[Zoey_R]

If you want a clean setup, parameterize the surface by r(u,v). Then the differential area vector is dS = (∂r/∂u × ∂r/∂v) du dv, and p = ρ g z(u,v). The resultant force is F = ∫∫ p(u,v) [∂r/∂u × ∂r/∂v] du dv, integrated over the domain D. In components, F_x = ∫∫ p n_x dA, F_y = ∫∫ p n_y dA, F_z = ∫∫ p n_z dA. If you compute with a mesh of panels, this reduces to summing p_i ΔA_i n_i.

[EmilyZW]

Maybe the snag isn’t algebra but the geometry. If parts of the window are near the surface or tilted, h varies nonlinearly and a naive average depth will mislead you. Are you sure the free surface height is defined consistently across the whole shape and that the surface is fully submerged?

[Grace_T]

I tried breaking the window into four triangular patches and used the average depth for each patch, then summed p_i n_i ΔA_i. It looked okay at first but the total swung a bit if I nudged the reference depth. I kept mixing up signs on the vertical component a couple times.

[HannahL]

A quick cross-check: project the curved surface onto a vertical plane and compute the hydrostatic force on that projection. That horizontal component should match the projection result and catch obvious mistakes. It won’t give you the full vector, but it helps.