find the force in newtons

Question

The Deligne Dam on the Cayley River is built so that the wall facing the water is shaped like the region above the curve y = 0.6x^2 and below the line y = 100.

(Here, distances are measured in meters.) The water level is 24 meters below the top of the dam.

Find the force (in Newtons) exerted on the dam by water pressure. (Water has a density of 1000 kg/m^3, and the acceleration of gravity is 9.8 m/sec^2 .)

Answer 1

If I understand you correctly, this problem is a little tricky because there is no mention of how long the dam is across the river. Therefore, we can't calculate the total force of the water on the dam. Rather, we can find the force per meter in the dimension across the river, the z-direction, instead.

With that little bit of trickiness out of the way, we can now get on with the problem. :-)

From the laws of physics, we know that the pressure in a fluid is given by

P = ρgh,

where ρ is the density of the fluid (here, given to us as 1000 kg/m³), g is the acceleration due to gravity (also given to us, 9.8 m/s²), and h is the height (or depth) of the fluid. The height can be found by subtracting the height of the dam from the base, y(x), from the surface height. The surface height can be found by subtracting 24m from the max height of 100m to get 76m. The height function is therefore

h(x) = (76 m) - 0.6 x².

The pressure force F_P (per meter in this case!) is related to the pressure P by

F_P / z = P · x

where x is the distance at depth h(x). Unfortunately the height (or depth) of the water is changing constantly. The curvature of the dam means we are going to have to use calculus! Go figure. :-)

Instead of the formula for F_P as given above we must write

F_P / z = ∫ P · dx,

where the bounds of integration are from 0 m to x_max m. We can find x_max by solving where y(x) = 76 m to find x_max ≈ 11.25 m.

Plugging in our formula for P, we have

F_P / z = ∫ ρ g h(x) dx

= ρ g ∫ [(76 m) - 0.6 x²] dx

= ρ g [(76 m) · x_max - 0.6 x_max³]

= (1000 kg / m³ ) ( 9.8 m / s² ) [(76 m) (11.25 m) - 0.6 (11.25 m)³]

≈ 5.588 x 10⁶ N / m

= 5.588 MN / m

I hope this helps.