How to Determine Centroid of Gate Hydrostatics

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SUMMARY

The centroid of a gate's hydrostatics is determined to be at a distance of 2R/π from the force F, contrasting with the centroid of a half-circle, which is located at 4r/3π. This discrepancy arises from the specific geometry and distribution of forces acting on the gate. The calculations involve evaluating moments about point B, including contributions from normal pressure forces and the weight of the fluid in the quarter circle. The established equations for these moments are critical for understanding the centroid's location in hydrostatic applications.

PREREQUISITES
  • Understanding of hydrostatics and centroid calculations
  • Familiarity with fluid mechanics principles
  • Knowledge of moment calculations in physics
  • Ability to interpret geometric shapes in engineering contexts
NEXT STEPS
  • Study the derivation of centroids for various geometric shapes in fluid mechanics
  • Learn about the application of hydrostatic pressure equations in engineering
  • Explore the concept of moments and their role in static equilibrium
  • Investigate the implications of centroid location on structural design and stability
USEFUL FOR

Engineering students, fluid mechanics practitioners, and professionals involved in structural analysis and design will benefit from this discussion, particularly those focusing on hydrostatic forces and centroid calculations.

ScareCrow271828
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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?
 
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ScareCrow271828 said:

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.
 
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##
 

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