An open channel has a cross-section in the form of an equilateral triangle with \(2.5 \mathrm{~m}\) sides and a vertical axis of symmetry. Its end is closed by a triangular vertical gate, also with \(2.5 \mathrm{~m}\) sides, supported at each corner. Calculate the horizontal thrust on each support when the channel is brim-full of water.

Hydrostatic pressure is the pressure exerted by a fluid at rest due to the force of gravity. It increases with depth because the weight of the fluid above exerts a force on the layers below. The formula to calculate hydrostatic pressure at a certain depth is given by: \[ P(h) = \rho g h \] where \( \rho \) is the fluid's density, \( g \) is the acceleration due to gravity, and \( h \) is the depth. In open channels, understanding hydrostatic pressure is crucial as it directly influences the force exerted on submerged surfaces like gates and walls. For example, in our exercise, we calculated the hydrostatic pressure at the depth of 2.5 meters, which is the maximum depth for the gate.

An equilateral triangle is a triangle where all three sides are of equal length. Each internal angle in an equilateral triangle is 60 degrees. The properties of equilateral triangles are useful in various engineering and physics applications due to their symmetry and simplicity in calculations. The area (A) of an equilateral triangle can be found using the formula: \[ A = \frac{\sqrt{3}}{4} s^2 \] where \( s \) is the side length. In the given problem, the gate's cross-section is an equilateral triangle with each side measuring 2.5 meters. Using this formula, we determined the area to proceed with further calculations.

Resultant force is the single force that represents the combined effect of multiple forces acting on a body. In fluid mechanics, this often involves integrating pressure over an area to find the total force exerted by a fluid. For a submerged surface, the resultant force can be calculated using the formula: \[ F = P_{avg} \times A \] Here, \( P_{avg} \) is the average pressure, and A is the area over which the pressure acts. In the exercise, we determined the average hydrostatic pressure and multiplied it by the gate's area to find the resultant force acting on the gate.

Fluid mechanics is the study of fluids (liquids and gases) and the forces acting upon them. It's essential for understanding how fluids behave in various situations, such as in open channel flows, pipelines, and around submerged objects. In our exercise, principles of fluid mechanics help us understand how water pressure acts on the triangular gate, allowing us to calculate forces exerted by the fluid. Key concepts in fluid mechanics include viscosity, flow rate, drag, and buoyancy. These principles are instrumental for engineers and scientists in designing efficient systems involving fluids.

Supports and reactions are critical concepts in static systems to keep structures stable. Supports are points or areas where external forces or constraints are applied to hold a structure in place. Reactions are the forces or moments that develop at supports to counteract applied loads and maintain equilibrium. In the problem, the triangular gate is supported at three corners. The horizontal thrust due to hydrostatic pressure is distributed equally among these supports. Each support reacts with a force calculated by dividing the resultant force by the number of supports, ensuring the gate remains stable.