water runs into a house through a 5cm pipe.its pressure at entering the house is 1 times 10^5Pa.the faucent on the second floor 2,5cm above where the water enters the house has an inside diameter of 3cm.when the faucent is in use the flow velocity at the inlet pipe is 1,5m/s ,what is the flow velocity in the flaucent?
To find the flow velocity in the faucet, we can use the principle of continuity. According to the principle of continuity, the mass flow rate of a fluid remains constant along a pipe of varying diameter.
The mass flow rate of the fluid can be calculated using the equation:
m(dot) = ρ * A * V
Where:
m(dot) is the mass flow rate,
ρ is the density of water (approximately 1000 kg/m^3),
A is the cross-sectional area of the pipe, and
V is the flow velocity in the pipe.
We'll start by finding the cross-sectional area and flow velocity in the 5 cm pipe, where the water enters the house.
The cross-sectional area of the 5 cm pipe can be calculated using the formula for the area of a circle:
A = π * r^2
Where r is the radius of the pipe. Since the diameter is given, we can convert it to radius by dividing it by 2.
r = 5 cm / 2 = 2.5 cm = 0.025 m
A = π * (0.025 m)^2 ≈ 0.00196 m^2
Now we can calculate the mass flow rate in the 5 cm pipe:
m(dot) = ρ * A * V
m(dot) = 1000 kg/m^3 * 0.00196 m^2 * 1.5 m/s
m(dot) ≈ 2.94 kg/s
Next, we need to find the cross-sectional area and flow velocity in the 3 cm faucet.
Following the same steps as above, we can calculate the cross-sectional area of the 3 cm faucet:
A = π * r^2
r = 3 cm / 2 = 1.5 cm = 0.015 m
A = π * (0.015 m)^2 ≈ 0.0007069 m^2
Using the principle of continuity, we can now find the flow velocity in the faucet:
m(dot) = ρ * A * V
V = m(dot) / (ρ * A)
V ≈ 2.94 kg/s / (1000 kg/m^3 * 0.0007069 m^2)
V ≈ 4.16 m/s
Therefore, the flow velocity in the faucet when it is in use is approximately 4.16 m/s.