Hydrostatic Pressure and Force — Question 9

Question 9

A student uses constant pressure at the bottom of a 3-m-tall vertical rectangular gate. Explain the resulting bias and find the ratio of this estimate to the true force when the top is at the surface.

Question 9 – Solution

Step 1: Write the student’s estimate. The student applies the bottom pressure $p=\gamma(3)$ uniformly over the entire gate of area $A$. Thus $$F_{\mathrm{est}}=3\gamma A.$$

Step 2: Find the true force. For a rectangular gate extending from the surface to depth $3$ m, the centroid is at the average depth $$\bar y=\frac{0+3}{2}=\frac32\text{ m}.$$ Therefore, $$F_{\mathrm{true}}=\gamma A\bar y =\frac32\gamma A.$$

Step 3: Compare the estimate with the true force. $$\frac{F_{\mathrm{est}}}{F_{\mathrm{true}}} =\frac{3\gamma A}{\frac32\gamma A}=2.$$ Thus the estimate is twice the true value: $$\boxed{F_{\mathrm{est}}=2F_{\mathrm{true}}}.$$

Step 4: Express the bias as a percent error. $$\frac{F_{\mathrm{est}}-F_{\mathrm{true}}}{F_{\mathrm{true}}} \times100\% =(2-1)100\%=100\%.$$ $$\boxed{\text{The bottom-pressure estimate is }100\%\text{ too high.}}$$