A horizontal pipe has a cross sectional area of 40.0 cm^2 at the wider portions and 10.0 cm^2 at the constriction. Water is flowing in the pipe, and the discharge from the pipe is 6.00*10^-3 m^3/s(6.00 L/s). Find (a) the flow speeds at the wide and the narrow portions; (b) the pressure difference between those portions; (c) the difference in height between the mercury columns and the U-shaped tube.
To find the flow speeds at the wide and narrow portions of the pipe, use the principle of continuity, which states that the product of cross-sectional area (A) and flow speed (v) remains constant along a flow path.
The equation for the principle of continuity is:
A1v1 = A2v2
In this case, A1 and A2 are the cross-sectional areas at the wider and narrower portions of the pipe, respectively.
(a) Find the flow speed at the wide portion of the pipe:
A1 = 40.0 cm^2 = 40.0 * 10^-4 m^2 (convert cm^2 to m^2)
v1 = (A2v2) / A1
v1 = (10.0 * 10^-4 m^2 * 6.00 * 10^-3 m^3/s) / (40.0 * 10^-4 m^2) = 1.5 m/s
Repeat the same steps to find the flow speed at the narrow portion:
v2 = (A1v1) / A2
v2 = (40.0 * 10^-4 m^2 * 1.5 m/s) / (10.0 * 10^-4 m^2) = 6.0 m/s
(b) To find the pressure difference between the wide and narrow portions of the pipe, we can use Bernoulli's equation for incompressible fluids:
P1 + 1/2ρv1^2 + ρgh1 = P2 + 1/2ρv2^2 + ρgh2
Assuming the pipe is horizontal, the height difference (h1 - h2) equals zero.
Furthermore, since the pressure difference (P1 - P2) is what we want to find and h1 = h2, the equation simplifies to:
P1 + 1/2ρv1^2 = P2 + 1/2ρv2^2
Let's assume the pressure at the narrow portion of the pipe (P2) is zero.
P1 + 1/2ρv1^2 = 0 + 1/2ρv2^2
Therefore, the pressure difference P1 is:
P1 = 1/2ρv2^2 - 1/2ρv1^2
(c) To find the difference in height between the mercury columns and the U-shaped tube, we use the equation of hydrostatic pressure:
P = ρgh
Let's assume the density of mercury is ρ_mercury = 13,600 kg/m^3, and the height difference between the mercury columns is h.
Using the equation above, we can write the pressure difference between the two columns of mercury:
P_mercury = ρ_mercury * g * h
Comparing this equation with the pressure difference obtained in part (b), we have:
P1 = P_mercury
Therefore, h = P1 / (ρ_mercury * g)
To complete the calculations, you need to know the value of ρ_mercury and g.
Mercury's density (ρ_mercury) is 13,600 kg/m^3, and the acceleration due to gravity (g) is approximately 9.8 m/s^2. Use these values to calculate the difference in height (h).