Final answer:
The gauge pressure in the constricted segment of a pipe is calculated using Bernoulli's equation and the principle of conservation of flow rate, converted to atmospheres.
Step-by-step explanation:
To calculate the gauge pressure in the constricted segment of the pipe, we can use Bernoulli's equation for incompressible flow, assuming negligible fluid viscosity and level horizontal flow. The equation is:
P_1 + \frac{1}{2}\rho v_1^2 = P_2 + \frac{1}{2}\rho v_2^2,
where P is the pressure, \rho is the density of the fluid, and v is the fluid velocity.
To find the velocity in the constricted segment (v_2), we use the continuity equation which states that the flow rate must be conserved:
A_1 v_1 = A_2 v_2,
where A is the cross-sectional area. Thus:
v_2 = v_1 \frac{A_1}{A_2}.
We then solve for P_2 using the rearranged Bernoulli's equation:
P_2 = P_1 + \frac{1}{2}\rho v_1^2 - \frac{1}{2}\rho v_2^2.
Finally, we convert the pressure in pascals to atmospheres using the conversion factor (1 atm = 1.013 \u00d7 10^5 Pa). We'll calculate using the given velocities, diameters, and pressure values:
The cross-sectional areas A_1 and A_2 can be calculated using:
A = \pi \frac{d^2}{4}.
Therefore, the gauge pressure at the constricted segment in atmospheres can be determined.