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bigplanet401
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Homework Statement
A body of water of depth D sits behind a vertical dam. The water and dam are in static equilibrium. Calculate the torque on the dam due to the water about an axis at ground level (that is, a depth D below the surface of the water).
Homework Equations
N (torque) = r x F
The Attempt at a Solution
I'm guessing that the force on the dam due to the water acts at a point D/2 above ground since the vertical coordinate of the center of mass of the water would be this high. Then the torque is just DF/2.
But consider the following argument. The force on the dam varies linearly with depth. Wouldn't this mean that I would get the same answer if I divided the water into N "slabs" (like a stack of books) and calculated the contribution to the torque from each slab, each with its own moment arm?
I tried the following: divide the water into N "slabs" of thickness D/N. The height of the ith slab is Di/N. The force on the dam from this slab is F(1 - i/N), where the maximum force (at the bottom of the dam) is F. Then the total torque is
[tex]
\sum_{i=1}^N \; \frac{FDi}{N} \left( 1 - \frac{i}{N} \right)
[/tex]
But if I try to work out the sum and take the limit as N -> infinity, the sum becomes infinite. Is it just my math, or is this not the right argument? Thanks!