Resultant thrust and overturning turnning moments

[speedy46]

Homework Statement

A vertical sluice gate is 5 meters wide and 3 meters deep. calculate the resultant thrust and overturnning moment when the water is level with the top edge of the gate.
Density water = 1000kg/m^3

Homework Equations

Newton's First Law F = ma

Moment = Force x Distance or M = (F)(d)

The Attempt at a Solution

I wondered if theses where the correct equations to use.

 

[tiny-tim]

Hi speedy46!

Newton's First Law F = ma

Moment = Force x Distance or M = (F)(d)
…
I wondered if theses where the correct equations to use.

F = ma is Newton's second law … and it's a law of motion, and there's no motion here, is there?

Yes, moment = force x distance, but you'll also need an equation relating force (which you don't know) to pressure (which you do).

 

[Mene]

I did some research and found that the correct equations to use for this problem are:

Resultant Thrust: F = ρghA
Overturning Moment: M = ρghA(d/2)

Where:
ρ = density of water (1000 kg/m^3)
g = acceleration due to gravity (9.8 m/s^2)
h = height of water (3 m)
A = area of gate (5 m x 3 m = 15 m^2)
d = distance from the bottom of the gate to the center of pressure (d = 3/2 = 1.5 m)

Using these equations, the resultant thrust would be 44100 N and the overturning moment would be 66150 Nm. These values represent the force and moment exerted on the gate by the water at equilibrium. It is important to consider these values when designing structures such as sluice gates to ensure they can withstand the forces and moments exerted by the water.