To find the flow velocity and pressure in the bathroom, we can use the principle of continuity and Bernoulli's equation.
a) Flow velocity in the bathroom:
According to the principle of continuity, the flow rate of water must remain constant throughout the pipe. This can be expressed as:
A1 * V1 = A2 * V2
Where A1 and V1 are the cross-sectional area and flow velocity at the inlet pipe, and A2 and V2 are the cross-sectional area and flow velocity at the outlet pipe.
Given that the diameter of the inlet pipe is 2.0 cm and the flow velocity is 4 m/s, we can calculate the area at the inlet pipe:
A1 = π * (d1/2)^2
A1 = π * (0.02 m)^2
A1 = 3.142 * 0.0004
A1 = 0.0001256 m^2
Now we can calculate the cross-sectional area at the outlet pipe, knowing that the diameter is 1.0 cm:
A2 = π * (d2/2)^2
A2 = π * (0.01 m)^2
A2 = 3.142 * 0.0001
A2 = 0.0003142 m^2
Using the principle of continuity, we can find the flow velocity in the bathroom:
0.0001256 * 4 = 0.0003142 * V2
V2 = (0.0001256 * 4) / 0.0003142
V2 = 1.59 m/s
Therefore, the flow velocity in the bathroom is 1.59 m/s.
b) Pressure in the bathroom:
To find the pressure in the bathroom, we can use Bernoulli's equation:
P1 + 1/2 * ρ * V1^2 + ρ * g * h1 = P2 + 1/2 * ρ * V2^2 + ρ * g * h2
Where P1 and P2 are the pressures at the inlet and outlet pipes, respectively, and ρ is the density of water (1000 kg/m^3), g is the acceleration due to gravity (9.81 m/s^2), h1 and h2 are the heights of the pipes, and V1 and V2 are the velocities at the inlet and outlet pipes.
Given that the absolute pressure at the inlet pipe is 4 * 10^5 Pa and the height of the bathroom is 5 m, and using the calculated values of velocities, we can solve for the pressure in the bathroom:
4 * 10^5 + 1/2 * 1000 * 4^2 + 1000 * 9.81 * 0 = P2 + 1/2 * 1000 * 1.59^2 + 1000 * 9.81 * 5
160000 = P2 + 1/2 * 1000 * 2.5281 + 1000 * 9.81 * 5
160000 = P2 + 1264.05 + 49050
P2 = 160000 - 1264.05 - 49050
P2 = 108685.95 Pa
Therefore, the pressure in the bathroom is approximately 108685.95 Pa.
• Water enters a house through a pipe whose inside diameter is 2.0 cm at an absolute pressure of 4 * 10 ^ 5 Pa. The pipe leading to second floor bathroom 5 m above is 1.0 cm in diameter. When the flow velocity at the inlet pipe is 4m * s ^ - 1 find the;
a) flow velocity in the bathroom
b) pressure in the bathroom
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