A garden hose is attached to a water faucet on one end and a spray nozzle on the other end. The water faucet is turned on, but the nozzle is turned off so that no water flows through the hose. The hose lies horizontally on the ground, and a stream of water sprays vertically out of a small leak to a height of 0.85 m. What is the pressure inside the hose?
? kPa
To determine the pressure inside the hose, we can use the concept of hydrostatic pressure. The pressure at any point in a fluid at rest is given by the equation:
P = ρgh
Where:
P = pressure
ρ = density of the fluid
g = acceleration due to gravity
h = height of the fluid above the point of interest
In this case, the fluid is water, which has a density of approximately 1000 kg/m^3 and the height is given as 0.85 m.
Plugging these values into the equation, we can calculate the pressure:
P = (1000 kg/m^3) * (9.8 m/s^2) * (0.85 m) = 8330 Pa
To convert the pressure from Pascals (Pa) to kilopascals (kPa), we divide by 1000:
8330 Pa / 1000 = 8.33 kPa
Therefore, the pressure inside the hose is approximately 8.33 kPa.
To find the pressure inside the hose, we can make use of Bernoulli's equation, which relates the velocity, height, and pressure of a fluid. In this case, the velocity of the water stream is zero since the nozzle is turned off. The height of the water stream is given as 0.85 m.
Bernoulli's equation can be written as:
P + 1/2 * ρ * v^2 + ρ * g * h = constant
where P is the pressure, ρ is the density of the fluid, v is the velocity of the fluid, g is the acceleration due to gravity, and h is the height.
In this case, since the velocity is zero, the term 1/2 * ρ * v^2 becomes zero. Also, the height h is given as 0.85 m. So the equation simplifies to:
P + ρ * g * h = constant
We can assume that the pressure outside the hose (atmospheric pressure) is the same as the pressure inside the hose when the nozzle is turned off.
Atmospheric pressure is approximately 101.3 kPa.
So the equation becomes:
P + ρ * g * h = 101.3 kPa
To find the pressure inside the hose, we need to know the density of water (ρ) and the acceleration due to gravity (g).
The density of water is approximately 1000 kg/m^3.
The acceleration due to gravity is approximately 9.8 m/s^2.
Substituting these values into the equation, we can solve for P:
P + (1000 kg/m^3) * (9.8 m/s^2) * (0.85 m) = 101.3 kPa
P + 8330 N/m^2 = 101.3 kPa
P = 101.3 kPa - 8330 N/m^2
P = 93 kPa (rounded to nearest whole number)
Therefore, the pressure inside the hose is approximately 93 kPa.
Use the variable-height Bernoulli equation again.
The velocity of water inside the hose and at the top of the stream is nearly zero, so the change in (density)*g*y equals the change in P
P -Po = (density)* g * 0.85 m
Po is atmospheric pressure