I just need help with c
A long horizontal hose of diameter 4.2 cm is connected to a faucet. At the other end, there is a nozzle of diameter 2 cm. Water squirts from the nozzle at velocity 24 m/sec. Assume that the water has no viscosity or other form of energy dissipation.
a) What is the velocity of the water in the hose ?
5.44 m/s
b) What is the pressure differential between the water in the hose and water in the nozzle ?
273203.2 Pa
c) How long will it take to fill a tub of volume 110 liters with the hose ?
? sec
Q = flow rate = velocity*area
= 5.44 *( pi /4) (2/100)^2
Q * t = volume
To calculate the time it takes to fill a tub with the hose, we need to determine the flow rate of water through the hose. The flow rate is the volume of water passing through a certain cross-sectional area per unit of time.
First, let's calculate the cross-sectional area of the hose and the nozzle.
The cross-sectional area of a circle is given by the formula A = π * r^2, where A is the area and r is the radius.
For the hose:
The diameter of the hose is given as 4.2 cm, so the radius (r) can be determined by dividing the diameter by 2: r = 4.2 cm / 2 = 2.1 cm = 0.021 m.
Using the formula for the area of a circle, we can find the cross-sectional area of the hose: A_hose = π * (0.021 m)^2.
For the nozzle:
The diameter of the nozzle is given as 2 cm, so the radius (r) can be determined by dividing the diameter by 2: r = 2 cm / 2 = 1 cm = 0.01 m.
Using the formula for the area of a circle, we can find the cross-sectional area of the nozzle: A_nozzle = π * (0.01 m)^2.
Next, we can calculate the flow rate using the equation:
Flow rate = Velocity * Area
In this case, the velocity is given as 24 m/s for the nozzle. We need to find the velocity in the hose for part c). Let's call it V_hose.
For part a), we know that the velocity in the hose, V_hose, is 5.44 m/s.
So we can rewrite the flow rate equation for the hose as:
Flow rate_hose = V_hose * A_hose
Now, we can calculate the flow rate in the hose:
Flow rate_hose = 5.44 m/s * A_hose
For part b), we need to calculate the pressure differential between the water in the hose and water in the nozzle.
Pressure differential can be calculated using Bernoulli's equation, which states that the sum of the pressure, kinetic energy, and potential energy per unit volume is constant along a streamline.
However, since we are assuming there is no energy dissipation or viscosity, the height of the water column is not relevant, and the pressure differential can be calculated simply by considering the change in velocity between the hose and the nozzle.
The pressure differential (ΔP) is given by the equation:
ΔP = 0.5 * ρ * (V_nozzle^2 - V_hose^2)
Where ρ is the density of water, which is approximately 1000 kg/m^3.
Now we can substitute the values into the equation:
ΔP = 0.5 * 1000 kg/m^3 * (24 m/s)^2 - (5.44 m/s)^2
Finally, we reach part c).
To calculate the time it takes to fill a tub of volume 110 liters, we need to know the flow rate in liters per second. We can convert the volume of the tub to cubic meters and then divide it by the flow rate to find the time.
To convert liters to cubic meters, we need to divide the volume by 1000 (since there are 1000 liters in 1 cubic meter):
Volume in cubic meters = 110 liters / 1000 = 0.11 cubic meters
Now we can calculate the time:
Time = Volume in cubic meters / Flow rate_hose
Just substitute the respective values into the equation.