Hydrostatic Pressure and Force — Question 10

Question 10 – Solution

Step 1: State the force model. Assume the hydrostatic force resisting the downward motion has magnitude $$F(y)=p(y)A=\gamma yA,$$ where $y$ is the hatch depth. The area $A$ is constant.

Step 2: Write a differential amount of work. Moving the hatch through a small downward distance $dy$ requires $$dW=F(y)\,dy=\gamma Ay\,dy.$$

Step 3: Integrate over the change in depth. The hatch moves from $y=d$ to $y=d+h$, so $$\begin{align*} W&=\int_d^{d+h}\gamma Ay\,dy\\ &=\gamma A\left[\frac{y^2}{2}\right]_d^{d+h}\\ &=\frac{\gamma A}{2}\left((d+h)^2-d^2\right). \end{align*}$$

Step 4: Simplify. $$\begin{align*} W&=\frac{\gamma A}{2}\left(d^2+2dh+h^2-d^2\right)\\ &=\gamma A\left(dh+\frac{h^2}{2}\right). \end{align*}$$ $$\boxed{W=\gamma A\left(dh+\frac{h^2}{2}\right)}$$ This is the external work under the stated resisting-force assumption.