Question

Water flows at 0.25 L/s through a 9.0-m-long garden hose 2.0 cm in diameter that is lying flat on the ground. The temperature of the water is 20 degrees Celsius. What is the guage pressure of the water where it enters the hose?

Express in pascals.

Answer 1

The gauge pressure of the water entering the hose is 785.65 pascals.

Step-by-step explanation:

Given:

Water flow rate = 0.25 L/s

Garden hose length = 9.0 m

Garden hose diameter = 2.0 cm

Water temperature = 20 degrees Celsius

To find the gauge pressure of the water, we can use the equation:

Pressure = (Flow rate * Density * Acceleration due to gravity) / (Cross-sectional area * Constant)

First, we need to convert the flow rate from liters per second to cubic meters per second:

Flow rate = 0.25 L/s = 0.25 * 0.001 m^3/s = 0.00025 m^3/s

Next, we can calculate the cross-sectional area of the hose:

Cross-sectional area = (Pi * (Diameter/2)^2)

Plugging in the values, we get:

Cross-sectional area = (Pi * (0.02 m/2)^2) = 0.000314 m^2

Now, we can calculate the gauge pressure:

Pressure = (0.00025 m^3/s * 1000 kg/m^3 * 9.8 m/s^2) / (0.000314 m^2 * Constant)

Using the given information, we substitute the values:

Pressure = (0.00025 m^3/s * 1000 kg/m^3 * 9.8 m/s^2) / (0.000314 m^2 * 1)

Simplifying the equation, we get:

Pressure = 785.65 Pa

Therefore, the gauge pressure of the water where it enters the hose is 785.65 pascals.

Answer 2

The gauge pressure at which water enters the garden hose can be calculated using the continuity equation to compute the velocity of water and then applying Bernoulli's equation to solve for the pressure at the entrance.

Step-by-step explanation:

To determine the gauge pressure at which water enters the garden hose, we can use the continuity equation and Bernoulli's equation. The continuity equation relates the flow rate (Q) to the velocity (v) and the cross-sectional area (A) of the hose:

Q = v × A

Using the given diameter (d = 2.0 cm), we can find the area (A = π d² / 4). With the flow rate (Q = 0.25 L/s), we can then calculate the velocity (v). Once we have v, Bernoulli's equation allows us to relate the change in pressure to the change in velocity, given the constant height of the hose.

Bernoulli's equation states that P1 + ½ ρ v1² + ρgh1 = P2 + ½ ρ v2² + ρgh2. Assuming the exit velocity (v2) equals the velocity we found using the continuity equation and that the elevation stays constant (hence ρgh1 = ρgh2), we can solve for the pressure at the entrance (P1) which is what is being asked.

Remember that gauge pressure is the pressure relative to the atmospheric pressure, implying P1 is actually P1 - atmospheric pressure.

Gauge pressure = (101325 Pa + 1000 kg/m³ * 0) - 101325 Pa = 0 Pa

The gauge pressure of the water where it enters the hose is 0 Pa. This means the pressure within the hose at that point is equal to the atmospheric pressure.