Answer: To find the speed of the water at a specific point in the pipe, we can use the principle of continuity, which states that the mass flow rate of a fluid is constant at any given point in a closed system.
The mass flow rate (m_dot) of water is given by:
m_dot = ρ * A * v
where:
ρ = density of water
A = cross-sectional area of the pipe at the specific point
v = speed of water at the specific point
Since the water is flowing into the pipe at a steady rate of 1.75 m³/s, the mass flow rate remains constant. Therefore, we have:
m_dot = 1.75 m³/s (given)
Next, we need to find the cross-sectional area (A) of the pipe at the specific point where the radius is 0.220 m. The area of a pipe with a circular cross-section is given by:
A = π * r²
where:
r = radius of the pipe at the specific point
Given: r = 0.220 m
A = π * (0.220 m)²
A ≈ 0.153 m²
Now, we can find the density of water (ρ) at room temperature, which is approximately 1000 kg/m³.
Substitute the known values into the mass flow rate equation:
m_dot = ρ * A * v
1.75 m³/s = 1000 kg/m³ * 0.153 m² * v
Now, solve for v:
v = 1.75 m³/s / (1000 kg/m³ * 0.153 m²)
v ≈ 0.011437 s⁻¹
Finally, the speed of the water at the specific point in the pipe is approximately 0.011437 m/s.