Find an algebraic formula for the net vertical force F on the submerged semicircular projecting structure CDE in Fig. E2.9. The structure has uniform width b into the paper. The liquid has specific weight γ.

Question

Find an algebraic formula for the net vertical force F on the submerged semicircular projecting structure CDE in Fig. E2.9. The structure has uniform width b into the paper. The liquid has specific weight [latex]\gamma[/latex].

Accepted Answer

The net force is the difference between the upward force $F_L$ on the lower surface DE and the downward force $F_U$ on the upper surface CD, as shown in Fig. E2.9. The force $F_U$ equals $\gamma$ times the volume ABDC above surface CD. The force $F_L$ equals $\gamma$ times the volume ABDEC above surface DE. The latter is clearly larger. The difference is $\gamma$ times the volume of the structure itself. Thus the net upward fluid force on the semicylinder is

$F = \gamma_{fluid}$ (volume CDE) = $\gamma_{fluid} \frac{\pi}{2}R^2b$

This is the principle upon which the laws of buoyancy, Sec. 2.8, are founded. Note that the result is independent of the depth of the structure and depends upon the specific weight of the fluid, not the material within the structure.

[latex]F = \gamma_{fluid}[/latex] (volume CDE) = [latex]\gamma_{fluid} \frac{\pi}{2}R^2b[/latex]

This is the principle upon which the laws of buoyancy, Sec. 2.8, are founded. Note that the result is independent of the depth of the structure and depends upon the specific weight of the fluid, not the material within the structure.

Question 2.9: Find an algebraic formula for the net vertical force F on th...

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