Final answer:
To calculate the gauge pressure needed at the large end of a tapered pipe for water to emerge at the small end, Bernoulli's equation and the continuity equation are used to relate the pressures, densities, velocities, and heights of the two ends of the pipe, taking into account the elevation between the ends.
Step-by-step explanation:
The question asks us to determine the gauge pressure required for water to emerge from the small end of a tapered pipe with a speed of 12 m/s, given that the small end is 8 meters above the large end. To solve this, we can use Bernoulli's equation, which relates the pressure, the kinetic energy per unit volume, and the potential energy per unit volume of a fluid.
To start, Bernoulli's equation is written as P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂, where P is the pressure, ρ is the density of the fluid, v is the velocity of the fluid, g is the acceleration due to gravity, and h is the height above a reference level. Since the diameters of the two ends of the pipe are given, we know that the area A at the small end is four times less than the area at the large end (A₂ = ¼ A₁) due to A = π(d/2)² where d is the diameter of the pipe. This is important because the velocity and the cross-sectional area at each end of the pipe are related by the continuity equation A₁v₁ = A₂v₂.
Using these equations, we can find that the gauge pressure P₂ - P₁ is needed to make the fluid flow from the large end to the small end with the required velocity, taking into account the difference in height (8m). We must also consider that the speed v₁ will be less than v₂ since the large end has a bigger diameter (v₁ = ¼ v₂), and hence the pressure P₁ at the large end must be higher to account for the loss in height and the increased velocity at the small end.
By rearranging Bernoulli's equation and solving for the gauge pressure (P₂ - P₁), we can then find out the necessary pressure difference to achieve the conditions stated in the problem.