Hydrostatic Pressure and Force — Question 8

Question 8

A semicircular gate of radius $R$ has its diameter along the water surface and lies below it. Find the force using centroid depth $4R/(3\pi)$.

Question 8 – Solution

Step 1: Use the centroid form of hydrostatic force. For a submerged vertical plane region, $$F=\gamma A\bar y.$$

Step 2: Find the area of the semicircle. $$A=\frac12\pi R^2.$$

Step 3: Use the given centroid depth. Because the diameter lies on the water surface and the semicircle lies below it, $$\bar y=\frac{4R}{3\pi}.$$

Step 4: Substitute and simplify. $$\begin{align*} F&=\gamma\left(\frac12\pi R^2\right) \left(\frac{4R}{3\pi}\right)\\ &=\gamma\left(\frac{4R^3}{6}\right) =\frac23\gamma R^3. \end{align*}$$ $$\boxed{F=\frac23\gamma R^3}$$ The factors of $\pi$ cancel because the semicircle’s area contains $\pi$ while its centroid depth contains $1/\pi$.