Force and moment on submerged surface

[boileroo]

Homework Statement

I am having trouble understanding problems involving the force and bending moment on a submerged surface.

The problem in question is as follows:

variation in density of a muddy lake is given as rho = rho_o + kh, where rho_o = 1000, k = 100 and h = depth of lake.

A vertical plate is used as a dam, and the depth of the water is 10m.

Find the force/meter width and moment/meter width.

My problem lies in part b, calculating the moment

The Attempt at a Solution

F = int(rho*g*y) dy = int((rho_o +kh)*g*y) dy = (rho_o*g*h²)/2 + (k*g*h³)/3 = 815.5x10³ N

It is this next part with the moment I am struggling on. The formula i have been taught to use is:

M = int(r*P) dA

I think that this goes to M = int((rho_o +kh)(y-h) dy but am unsure why, if it is infact it is the case.
Any help would be appreciated!

 

[ideasrule]

To calculate torque, you need to select a reference axis, because you get different answers depending on where you define "r" to be 0. Suppose you choose the bottom of the dam as your reference.

The moment exerted on a small section of the dam, of length dy and width w, would then be y*P*wdy: the distance from the reference axis, times the force exerted on the infinitesimal section. P is just rho_0+k(h-y), and integrating from y=0 to y=h would give you the moment.

 

[boileroo]

thank you, I think that helps!