what's the pressure change in water going from a 5.0cm diameter pipe to a 2.0cm diameter pipe if the velocity in the smaller pipe is 4.0m/s
.
Do you mean velocity change?
A1v1 = A2v2
where A = pi r^2
To calculate the pressure change in water when going from a 5.0cm diameter pipe to a 2.0cm diameter pipe, we can utilize the principle of continuity.
The principle of continuity states that the mass flow rate of an incompressible fluid remains constant when it passes through different cross-sectional areas. Mathematically, this can be expressed as:
A1 * v1 = A2 * v2,
where A1 and A2 are the cross-sectional areas of the pipes, and v1 and v2 are the velocities of the water in the pipes.
In this case, we are given that the diameter of the larger pipe is 5.0cm, which means the radius (r1) is 2.5cm or 0.025m. The diameter of the smaller pipe is 2.0cm, so the radius (r2) is 1.0cm or 0.01m.
Using the formulas for the cross-sectional area of a pipe (A = π * r^2), we can calculate the values for A1 and A2:
A1 = π * (0.025m)^2
A2 = π * (0.01m)^2
Next, we are given that the velocity in the smaller pipe (v2) is 4.0m/s.
Now, we can rearrange the equation and solve for v1:
v1 = (A2 * v2) / A1
Substituting the values we have:
v1 = (π * (0.01m)^2 * 4.0m/s) / (π * (0.025m)^2)
Simplifying the equation and canceling out the π term:
v1 = (0.01m)^2 * 4.0m/s / (0.025m)^2
v1 = 0.0004m^2/s / 0.00625m^2
v1 = 0.064m/s
Now that we have both velocities, we can use Bernoulli's equation to calculate the pressure change between the two points.
Bernoulli's equation states that the pressure difference (ΔP) between two points in a fluid flow is given by:
ΔP = 0.5 * ρ * (v2^2 - v1^2)
where ρ is the density of the fluid.
For water at room temperature and atmospheric pressure, the density (ρ) is approximately 1000 kg/m^3.
Substituting the values we have:
ΔP = 0.5 * 1000 kg/m^3 * (4.0m/s)^2 - (0.064m/s)^2
Simplifying the equation:
ΔP = 0.5 * 1000 kg/m^3 * (16.0m^2/s^2 - 0.004096m^2/s^2)
ΔP = 0.5 * 1000 kg/m^3 * 15.995904m^2/s^2
ΔP ≈ 7997.952 Pa
So, the pressure change in water between the 5.0cm diameter pipe and the 2.0cm diameter pipe, when the velocity in the smaller pipe is 4.0m/s, is approximately 7997.952 Pa.