etermine the location of the horizontal axis ya 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 ya is ya = y- - Iy- / hA where y- is the y-coordinate of the centroid of the gate, Iy- is the moment of inertia of the gate about the line y=y-, h is the depth of the centroid below the surface, and A is the area of the gate
etermine the location of the horizontal axis ya 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 ya is ya = y- - Iy- / hA where y- is the y-coordinate of the centroid of the gate, Iy- is the moment of inertia of the gate about the line y=y-, h is the depth of the centroid below the surface, and A is the area of the gate
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Determine the location of the horizontal axis ya 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 ya is
ya = y- - Iy- / hA where y- is the y-coordinate of the centroid of the gate, Iy- is the moment of inertia of the gate about the line y=y-, h is the depth of the centroid below the surface, and A is the area of the gate.
Transcribed Image Text: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 \( y_a = \bar{y} - \frac{I_y}{hA} \) 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.
### Diagrams:
**Figure (a):**
- A vertical gate is submerged in water with the water surface at \( y = L \).
- The gate has a trapezoidal shape, submerged at a depth \( h \).
- The line \( y = \bar{y} \) indicates the y-coordinate of the centroid of the gate.
- The line \( y_a = \bar{y} - \frac{I_y}{hA} \) indicates the axis where the gate should be hinged to avoid rotational moments.
**Figure (b):**
- The gate is shown as a rectangle with height \( a \) and width \( b \).
- The top edge of the rectangle is at a depth \( d \) from the water surface (\( y = L \)).
The problem involves finding the precise location for hinging the gate based on principles of moments and buoyancy, ensuring stability under the fluid forces acting on it.
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