Final answer:
The task is to calculate the height to lift a dam gate to achieve a specific flow rate, using Bernoulli's equation and the equation of continuity, considering fluid dynamics principles.
Step-by-step explanation:
The question is asking about determining the height (h) to lift a gate in a dam to achieve a certain water flow rate. To solve this, we will use the principles of fluid dynamics, specifically Bernoulli's equation and the equation of continuity.
Firstly, Bernoulli's equation relates the pressure, velocity, and height of a fluid at two points along its flow path. According to the principle, we have:
P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂,
where P is the pressure, ρ (rho) is the density of water, v is the velocity, g is acceleration due to gravity, and h is the height of the water.
Since the water is at rest behind the dam, the velocity term (v₁) at the starting point (point 1) can be ignored. The pressure at both points is the atmospheric pressure and therefore cancels out. Then we are left with ρgh₁ = ½ρv₂² + ρgh₂.
Next, the equation of continuity which relates the flow rate (Q) to the cross-sectional area (A) and the velocity (v) of the fluid flow, Q = Av. This allows us to solve for the velocity and hence, use it in Bernoulli's equation to find the height.
Using these principles and the given information on the dam width and desired flow rate, we can calculate the necessary height to lift the gate. The given flow rate being 1.0×10⁻² m³/s and gate width 0.80 m helps us determine the required opening height for the desired flow rate.