Hydrostatic force on a semicircular wall

[Ryker]

Homework Statement

This isn't really a homework problem, it's a passage in the lecture notes that I don't quite get wholly. So here's the passage:

"To compute the hydrostatic force on a semicircular wall of radius a which forms the end of a swimming pool filled with water, between D(y) = 0 and D(y) = a, we compute
$$F = \rho g \int_0^a yw(y)dy.$$If (x, y) is a point on the semicircular edge of the wall, then x² + y² = a². Hence w(y) = 2x = ..."

The Attempt at a Solution

The part I don't get is the bolded part. Why is width of the strip 2x and not πx? It seems I'm not picturing the semicircular wall correctly, because I can't wrap my head around this. But from the description it seems as if the semicircular wall is a quarter of a sphere with radius a. So the x component would denote the radius of the horizontal strip on that sphere, and since that horizontal strip represents half of the perimeter of the whole circle, its width would be πx.

I don't know, any help here would be greatly appreciated.

edit: Or is this semicircular wall meant in a way that the going from the middle of the pool to this wall, that wall in your view would represent the bottom half of the circle as viewed in, say, a circle in the coordinate system sketched in your notes? Then if y is the down component, and x is the lateral one, 2x would make sense, of course, but is this how you picture the wall from the wording, as well?

 

[HallsofIvy]

Because this this is a semi-circle that is symmetric about the y-axis. The distance from (0, y) to (x, y) on any curve is x. The distance from (-x, y) to (x, y) is 2x. That has nothing to do with the curve being a circle.

 

[Ryker]

So you interpreted the wall as being like what I posted under "edit" then (ie. a vertical wall with the shape of a semicircle), and not the spherical thing I was talking about in the part of the post prior to the edit?

 

[tiny-tim]

Hi Ryker!

"To compute the hydrostatic force on a semicircular wall of radius a which forms the end of a swimming pool filled with water, …"

Yes, "semicircular" must mean planar, so the wall is flat.

 

[Ryker]

Hi Ryker!

Yes, "semicircular" must mean planar, so the wall is flat.

Ah, thanks, it's easy to see 2x in this case, and it seems I shouldn't have gone 3D on the problem