How to Determine Centroid of Gate Hydrostatics

[ScareCrow271828]

Homework Statement

In this image :
http://ezto.mheducation.com/13252703414204806874.tp4?REQUEST=SHOWmedia&media=2.83qs.jpg
Why does the weight of the gate have a centroid at 2R/pi away from the force F?

Homework Equations

The centroid for a half circle in both x and y directions is = 4r/3pi

The Attempt at a Solution

why isn't the centroid location in regards to the x-axis 4r/3pi? How do we get to 2R/pi?

 

[Chestermiller]

Homework Statement

In this image :
http://ezto.mheducation.com/13252703414204806874.tp4?REQUEST=SHOWmedia&media=2.83qs.jpg
Why does the weight of the gate have a centroid at 2R/pi away from the force F?

Homework Equations

The centroid for a half circle in both x and y directions is = 4r/3pi

The Attempt at a Solution

why isn't the centroid location in regards to the x-axis 4r/3pi? How do we get to 2R/pi?

Can you please provide a word for word statement of this problem? Thank you.

 

[Chestermiller]

I solved this problem by two different methods, and got the same result for the force F. Here are some intermediate results for the moments about point B:

counterclockwise moment exerted by normal pressure force from base = $\rho g r^3/2$

counterclockwise moment exerted on right vertical side = $\rho g r^3/6$

clockwise moment exerted by weight of fluid in quarter circle = $\rho g \frac{\pi r^2}{4}\left(r-\frac{4r}{3\pi}\right)$

clockwise moment of force F = $rF$