Problem $4(25$ points$)$ Drawn from A$.$ Gerhart et$.$ Al$,9-$th Ed$,$ Chapter $2$

Concepts: Hydrostatic force on vertically submerged flat surfaces.

A thin $4-$ft$-$wide, right$-$angle gate with negligible mass is free to pivot

about a frictionless hinge at point $O,$ as shown. The horizontal portion

of the gate covers a $1-$ ft diameter drainpipe that contains air at

atmospheric pressure. Determine the minimum water depth, h$,$ at

which the gate will pivot to allow water to flow into the pipe.