Question

Water enters a house from a ground pipe with an inside diameter of 0.02 m at an absolute pressure of 4.0 x 10⁵ Pa. A pipe that is 0.01 m in diameter leads to the second-floor bathroom 5.0m above the ground floor. The flow speed at the ground pipe is 1.5m/s. Find the following:

a) flow speed on the second floor ; and

b) pressure

Answer 1

(a) Let's use the law of continuity:

$$v_1A_1=v_2A_2,$$ $$v_2=\dfrac{v_1A_1}{A_2}=\dfrac{\pi r_1^2}{\pi r_2^2}v_1,$$ $$v_2=\dfrac{\pi\cdot(0.01\ m)^2\cdot1.5\ \dfrac{m}{s}}{\pi\cdot(0.005\ m)^2}=6.0\ \dfrac{m}{s}.$$

(b) We can find the pressure from the Bernoulli's equation:

$$P_1+\rho gh_1+\dfrac{1}{2}\rho v_1^2=P_2+\rho gh_2+\dfrac{1}{2}\rho v_2^2,$$ $$P_2=P_1-\rho g(h_2-h_1)-\dfrac{1}{2}\rho(v_2^2-v_1^{2}),$$

$P_2=4.0\cdot10^5\ Pa-1000\ \dfrac{kg}{m^3}\cdot9.8\ \dfrac{m}{s^2}\cdot(5\ m-0)-\dfrac{1}{2}\cdot1000\ \dfrac{kg}{m^3}\cdot((6\ \dfrac{m}{s})^2-(1.5\ \dfrac{m}{s})^{2})=3.3\cdot10^5\ Pa.$