Askiitians Tutor Team
Last Activity: 4 Months ago
To tackle this problem, we need to break it down into manageable parts. We’ll calculate the volume and mass flow rates of water through the hose and then determine the average velocity of water at the nozzle exit. Let’s go step by step.
Calculating Volume Flow Rate
The volume flow rate (\(Q\)) is defined as the volume of fluid that passes through a given surface per unit time. In this case, we know the total volume of the bucket and the time it takes to fill it.
- The volume of the bucket is 10 gallons. To convert this to cubic centimeters (since the diameter of the hose is in centimeters), we use the conversion factor: 1 gallon = 3,785.41 cm³.
- Thus, the volume of the bucket in cubic centimeters is:
Volume of bucket:
\(10 \, \text{gallons} \times 3,785.41 \, \text{cm}^3/\text{gallon} = 37,854.1 \, \text{cm}^3\)
Now, since it takes 50 seconds to fill this bucket, we can calculate the volume flow rate:
Volume flow rate (\(Q\)):
\(Q = \frac{\text{Volume}}{\text{Time}} = \frac{37,854.1 \, \text{cm}^3}{50 \, \text{s}} = 757.082 \, \text{cm}^3/\text{s}\)
Determining Mass Flow Rate
The mass flow rate (\( \dot{m} \)) can be calculated using the volume flow rate and the density of water. The density of water is approximately \(1 \, \text{g/cm}^3\), which means 1 cm³ of water has a mass of about 1 gram.
Mass flow rate (\( \dot{m} \)):
\( \dot{m} = Q \times \text{Density} = 757.082 \, \text{cm}^3/\text{s} \times 1 \, \text{g/cm}^3 = 757.082 \, \text{g/s}\)
Finding Average Velocity at the Nozzle Exit
To find the average velocity of water at the nozzle exit, we can use the formula for flow rate in terms of velocity:
Flow rate formula:
\(Q = A \times v\)
Where \(A\) is the cross-sectional area of the nozzle and \(v\) is the velocity. First, we need to calculate the area of the nozzle exit.
The area \(A\) of a circle is given by:
Area formula:
\(A = \pi \left(\frac{d}{2}\right)^2\)
For the nozzle, the diameter \(d\) is 0.8 cm:
Area of nozzle:
\(A = \pi \left(\frac{0.8 \, \text{cm}}{2}\right)^2 = \pi \left(0.4 \, \text{cm}\right)^2 \approx 0.5027 \, \text{cm}^2\)
Now, we can rearrange the flow rate formula to solve for velocity:
Velocity (\(v\)):
\(v = \frac{Q}{A} = \frac{757.082 \, \text{cm}^3/\text{s}}{0.5027 \, \text{cm}^2} \approx 1507.5 \, \text{cm/s}\)
Summary of Results
To summarize our findings:
- Volume flow rate: 757.082 cm³/s
- Mass flow rate: 757.082 g/s
- Average velocity at the nozzle exit: 1507.5 cm/s
This analysis illustrates how fluid dynamics principles apply in practical scenarios, such as filling a bucket with water using a hose and nozzle system. If you have any further questions or need clarification on any part, feel free to ask!
