Question 3.9: Determine the magnitude and direction of the resultant force......
Determine the magnitude and direction of the resultant force due to water on a quadrant shaped cylindrical gate as shown in Fig. Ex. 3.9. Check whether the resultant passes through the centre.
The horizontal force = γ A \overline{h} where A is the projected area.
Considering unit width,
Horizontal force = 9810 × 2 × 4 × 1 = 78480 N, to the right
It acts at the centre of pressure of the projected area
i.e., at = 1.333 m from the bottom (i.e., (1/3) × 4)
Vertical force = the weight to the liquid displaced
= π × 4² × 1 × 9810/4 = 123276 N, upwards.
It acts at 4r/3π = 1.698 m, from the hinge.
Resultant force = (123276² + 78480²)^{0.5} = 146137 N
Angle is determined by \tan θ = 78480/123276 = 0.6366,
∴ θ = 32.48° where θ is the angle with vertical.
To check for the resultant to pass through the centre the sum of moment about O should be zero.
78480 × (4 – 1.3333) – 123276 × 1.698 = 42.
Compared to the values the difference is small and these can be assumed to the equal. Hence the resultant passes through the centre of the cricle.