9. An example of force on a curved sluice gate with fluid both above and below the gate.

A 1.5m long cylinder, radius 1m, lies as shown in the figure. It holds back oil of relative density 0.8. If the cylinder has a mass of 2250 kg find

a) the reaction at A b) the reaction at B

Horizontal force R_H = projection of vertical plane

Reaction at A = -R_H = pressure at centroid  x area =  0.5rgH x (H Length) = 0.5 rgH² x Length

= 0.5 x 0.8 x 10³ x 9.81 x 2² x 1.5

= 23544 N, to the left

Vertical force = Reaction at B

R_V = Force due to weight of fluid, DCE, above (down)

+ Force due to fluid below BD (upward)

+ Force due to weight of cylinder

Force of fluid above = area of sector DCE x length x rg

Force from below = area of real or imaginary fluid above BD

= area of BDEC x length x rg

Taking downward as positive

R_V = 0.8 x 10³x9.81 x(1x1 - pr²/4) x 1.5

- 0.8 x10³ x9.81 x( 1x1 + pr²/4) ) x 1.5

+ 22509.81

= 2526 -

= 3580 N

Reaction at B = 3580 N, vertically up

The resultant and angle of application are given by: