Calculate the torque of a hinge in a dam

[tinsel89]

Homework Statement

The upper edge of a gate in a dam runs along the water surface. The gate is 2.00 m high and 4.00m wide and is hinged along a horizontal line through its center.
Calculate the torque about the hinge arising from the force due to the water. (Hint: Calculate the torque on a thin, horizontal strip at a depth h and integrate this over the gate.)

Homework Equations

The Attempt at a Solution

I wasn't sure how to calculate torque without force, but I figured maybe I could use atmospheric pressure as follows:
F=100000*8(area of gate)=800000N
but then I don't know how the integration comes to play, I tried just multiplying the force by the distance between the hinge and the surface of water to find the torque:
I assumed it is the distance r from the surface of the water to the center of the gate which is half the height of the gate
d=2*0.5=1m
torque=rF=1*800000
dont know how to properly integrate the toque over the gate and if the distance I have used is correct.

 

[JesseC]

From your diagram, one edge of the gate is at the water surface. The other edge is submerged at a depth of 2m. Do you think the pressure exerted on the gate by the water will be the same at the top as at a depth of 2m? How does pressure change with depth in a column of water?

You are right to consider a distance from the edge of the gate to the centre (axis of rotation). But don't forget that if one edge lies a distance z from the axis, the other lies a distance -z. Meaning, you should realize that the torques will cancel to some extent but not totally because the pressure is changing with depth.