Determine the Location of the Horizontal Axis $y _ { a }$

Multiple Choice

Determine the location of the horizontal axis $y _ { a }$ for figure (b) at which a vertical gate in a dam is to be hinged so that there is no moment causing rotation under the indicated loading (see figure (a) ) . The model for $y _ { a }$ is $y _ { a } = \bar { y } - \frac { I _ { y } } { h A }$ where $\bar { y }$ is the $y$ -coordinate of the centroid of the gate, $I _ { y }$ is the moment of inertia of the gate about the line $y = \bar { y } , h$ is the depth of the centroid below the surface, and $A$ is the area of the gate.

A) $y _ { a } = - \frac { a ^ { 2 } } { L }$
B) $y _ { a } = - \frac { 4 a ^ { 2 } } { 3 L }$
C) $y _ { a } = - \frac { a ^ { 2 } } { 4 L }$
D) $y _ { a } = - \frac { 5 a ^ { 2 } } { L }$
E) $y _ { a } = - \frac { a ^ { 2 } } { 3 L }$