Final answer:
To find the pressure difference between two segments in the horizontal pipe, Bernoulli's equation is applied, which after substitution and rearrangement shows that the pressure difference is half the product of the density of water and the difference in squares of the velocities in the two segments.
Step-by-step explanation:
To solve the mathematical problem completely regarding the difference in pressure between two segments in a horizontal pipe with water flowing through it, we use Bernoulli's equation because it's an incompressible fluid and the flow is horizontal, meaning the gravitational potential energy term drops out.
Step 1: Write down Bernoulli's equation
Bernoulli's equation is given by:
P + ½ρv^2 = constant,
where P is the pressure, ρ is the density of the fluid, and v is the velocity of fluid flow.
Step 2: Apply Bernoulli's equation to both segments
Since the only change is in the velocity of the fluid, we can set the equation for both points as:
P1 + ½ρv1^2 = P2 + ½ρv2^2,
where the subscript 1 refers to the first segment and the subscript 2 refers to the second.
Step 3: Rearrange the equation to solve for the difference in pressure
By rearranging the equation, we get:
P2 - P1 = ½ρ(v1^2 - v2^2).
Substitute the given values:
P2 - P1 = ½×1185 kg/m^3 ×(5.33 m/s)^2 - (3.17 m/s)^2,
This calculation will then give us the pressure difference between segment 2 and segment 1.