Question

Gate ab is semicircular, hinged at b, and held by a horizontal force p at a. What force p is required for equilibrium?

Answer 1

Answer:

final answer

Explanation:

Centroid of semicircle is at $4R/3$ ≈$1.273$ m off bottom,

The water force F is

$\begin{aligned}\mathrm{F}=\gamma \mathrm{h}_{\mathrm{CG}} \mathrm{A} &=(9790)(5.0+1.727)_{2}^{\pi}(3)^{2} \\&=931000 \mathrm{~N}\end{aligned}$

Line of action of F lies below

$\mathrm{y}_{\mathrm{CP}}=-\frac{\mathrm{I}_{\mathrm{xx}} \sin \theta}{\mathrm{h}_{\mathrm{CG}} \mathrm{A}}=-\frac{(0.10976)(3)^{4} \sin 90^{\circ}}{(5+1.727)(\pi / 2)(3)^{2}}=-0.0935 \mathrm{~m}$

Summing moments about B yields the proper support force p

$\sum \mathrm{M}_{\mathrm{B}}=0=(931000)(1.273-0.0935)-3 \mathrm{P}, \quad \text { or: } \quad \mathrm{P}=366000 \mathrm{~N}$