The circular stream of water from a faucet is observed to taper from a diameter of 21 mm to 10 mm in a distance of 48 cm. determine the flowrate

2.47x10⁻⁴ m³/s

Explanation:

The water goes from point (1), with a diameter of 21 mm, to the point (2), with a diameter of 10 mm. By the continuity equation, the flow rate must be constant in all faucet, and the flow rate in the area (A) multiplied by the velocity (V), so:

A₁V₁ = A₂V₂

The area of a circuference cross section is (π/4)*d², where d is the diameter (in meters) so:

(π/4)*d₁² *V₁ = (π/4)*d₂² *V₂

(π/4)*(0.021)²*V₁ = (π/4)*(0.01)²*V₂

V₁ = (0.01)²*V₂/(0.021)²

V₁ = 0.2268V₂

V₂ = 4.41V₁

By the Bernouli's equation:

p₁ + ρV₁²/2 + γz₁ = p₂ + ρV₂²/2 + γz₂

Where p is the pressure, ρ is the density of the liquid, γ is the specific weight of the liquid (ρ*g, where g is the gravity acceleration), and z is the high of the point. Because both points are subjected to the same surrounding pressure, p₁ = p₂, and the terms are canceled.

ρV₁²/2 + γ*(z₁ - z₂) = ρV₂²/2

z₁ - z₂ = h, which is the distance of the points (48 cm = 0.48 m), so:

(ρ/2)*(V₂² - V₁²) = γh

(V₂² - V₁²) = 2γh/ρ

(V₂² - V₁²) = 2ρgh/ρ

(V₂² - V₁²) = 2gh

With V₂ = 4.41V₁, and g = 9.8 m/s²:

(4.41V₁)² - V₁² = 2*9.8*0.48

18.45V₁² = 9.408

V₁² = 0.51

V₁ = 0.714 m/s

The flow rate (ΔV) is A₁*V₁, so:

ΔV = (π/4)*(0.021)²*0.714

ΔV = 2.47x10⁻⁴ m³/s

mechanics (greek   is that area of science concerned with the behaviour of physical bodies when subjected to forces or displacements, and the subsequent effects of the bodies on their environment. the scientific discipline has its origins in ancient greece with the writings of aristotle and archimedes[1][2][3] (see history of classical mechanics and timeline of classical mechanics). during the early modern period, scientists such as galileo, kepler, and newton laid the foundation for what is now known as classical mechanics. it is a branch of classical physics that deals with particles that are either at rest or are moving with velocities significantly less than the speed of light. it can also be defined as a branch of science which deals with the motion of and forces on objects. the field is yet less widely understood in terms of quantum theory.

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