Fluids at Rest — Question 4

Question 4

A hinge feels the moment of a triangular load

A vertical rectangular gate has width $b=0.80\,\mathrm{m}$ and height $H=1.50\,\mathrm{m}$. Its top edge is at the free surface of water, and a hinge lies along that edge. Air at the same pressure is on the dry side. Take $\rho=1000\,\mathrm{kg/m^3}$, $g=9.80\,\mathrm{m/s^2}$.

1. Find the net hydrostatic force and its depth below the hinge.

2. What horizontal force at the bottom edge holds the gate closed? Ignore gate weight and friction.

Question 4 - Solution

A hinge feels the moment of a triangular load

(a) Derivation and calculation

$$dF=\rho ghb\,dh,\quad F=\rho gb\int_0^Hh\,dh=\tfrac12\rho gbH^2.$$$$\boxed{F=\tfrac12(1000)(9.80)(.80)(1.50)^2=8820\,\mathrm N}.$$$$M=\rho gb\int_0^Hh^2\,dh=\tfrac13\rho gbH^3=8820\,\mathrm{N\,m}.$$$$\boxed{h_R=M/F=2H/3=1.00\,\mathrm m}.$$

(b) Result and interpretation

The bottom holding force must oppose the water moment:$$TH=M\Rightarrow\boxed{T=8820/1.50=5880\,\mathrm N}.$$$$F(H/2)=6615\,\mathrm{N\,m}<M.$$A midpoint resultant underestimates the moment because the lower pressure forces are greater.

Physical check

The holding force is $2F/3$ since it acts farther from the hinge than the pressure resultant.