Problem 1 - Applied Physics

Question

Water flows through a hose of diameter \(3.90 \mathrm{~cm}\) at a velocity of \(5.00 \mathrm{~m} / \mathrm{s}\). Find the flow rate of the hose in \(\mathrm{L} / \mathrm{min}\).

Step-by-Step Solution

Answer

The flow rate is 358.2 L/min.

Step 1: Calculate the cross-sectional area of the hose

The cross-sectional area of the hose can be calculated using the formula for the area of a circle, \( A = \pi r^2 \), where \( r \) is the radius of the hose. First, convert the diameter from centimeters to meters: \( 3.90 \, \text{cm} = 0.039 \, \text{m} \). Therefore, the radius \( r = 0.039 \, \text{m} / 2 = 0.0195 \, \text{m} \). Now, calculate the area: \[ A = \pi (0.0195)^2 = 1.194 \times 10^{-3} \, \text{m}^2 \]

Step 2: Calculate the volumetric flow rate in cubic meters per second

The volumetric flow rate \( Q \) can be calculated using the formula \( Q = A \times v \), where \( A \) is the cross-sectional area and \( v \) is the velocity. Using the area calculated in Step 1, and given \( v = 5.00 \, \text{m/s} \): \[ Q = 1.194 \times 10^{-3} \, \text{m}^2 \times 5.00 \, \text{m/s} = 5.97 \times 10^{-3} \, \text{m}^3/s \]

Step 3: Convert the flow rate to liters per second

To convert the flow rate from cubic meters per second to liters per second, use the fact that \(1 \, \text{m}^3 = 1000 \, \text{L} \). Multiply the flow rate by 1000: \[ 5.97 \times 10^{-3} \, \text{m}^3/s \times 1000 = 5.97 \, \text{L/s} \]

Step 4: Convert the flow rate to liters per minute

Since 1 minute is 60 seconds, multiply the flow rate in liters per second by 60 to find the flow rate in liters per minute: \[ 5.97 \, \text{L/s} \times 60 = 358.2 \, \text{L/min} \]

Key Concepts

Flow Rate Calculation

Flow rate calculation is essential in determining how efficiently fluid moves through a conduit, like a hose or pipe. In essence, flow rate tells us how much volume of fluid passes through a certain point in a given time.
To calculate the flow rate, you need two key pieces of information: the cross-sectional area of the pipe and the velocity of the fluid. The basic formula for flow rate \( Q \) is given by:

• \( Q = A \times v \)

where \( A \) is the cross-sectional area and \( v \) is the fluid velocity.
This calculation helps in various practical applications, such as irrigation, plumbing, and even in understanding natural water streams. A precise flow rate calculation can prevent overflow in systems and enhance the efficiency of fluid transport systems.

Unit Conversion

Unit conversion plays a significant role in ensuring that calculations are accurate and compatible with different measurement systems. Converting units involves changing a measurement from one unit to another without altering the actual quantity.
For instance, if you have a flow rate in cubic meters per second (\( \, \text{m}^3/ ext{s} \)) and need it in liters per minute, you need to perform these conversions:

• First, convert cubic meters to liters by multiplying by 1000, because \( 1 \, \text{m}^3 = 1000 \, \text{L} \).
• Next, convert seconds to minutes by multiplying by 60, since 1 minute equals 60 seconds.

Accurate unit conversion facilitates communication and comprehension of data across different contexts and ensures that the findings are applicable and understandable globally.

Cross-Sectional Area

The cross-sectional area of a pipe is the area of a cut through the pipe perpendicular to its length. This measurement is crucial for calculating flow rate, as it directly affects the amount of fluid the pipe can handle.
For a pipe with a circular cross-section, the formula to find the area \( A \) is:

• \( A = \pi r^2 \)

where \( \pi \) is approximately 3.14159 and \( r \) is the radius of the circle.
Let's say you have a pipe with a diameter of 3.90 cm. First, you convert the diameter to meters (0.039 m) and then find the radius by halving it (0.0195 m).
Substituting into the formula gives you an area of \( 1.194 \times 10^{-3} \, \text{m}^2 \). This step is vital for further calculations related to flow rate, capacity, and designing pipe systems.

Volumetric Flow Rate

Volumetric flow rate quantifies the volume of fluid that flows through a cross-section per unit time. It is an important metric for understanding how systems manage fluid movement, with implications in engineering, environmental science, and day-to-day applications.
To find this flow rate, the formula \( Q = A \times v \) is used, where \( Q \) represents the volumetric flow rate, \( A \) is the cross-sectional area, and \( v \) is the velocity of the fluid.
In our example with a hose, if the area is \( 1.194 \times 10^{-3} \, \text{m}^2 \) and the velocity is 5.00 m/s, multiplying these gives a flow rate of \( 5.97 \times 10^{-3} \, \text{m}^3/s \). By converting this number into liters and scaling from seconds to minutes, you attain the final flow rate in \( \, \text{L/min} \).
The ability to accurately calculate and manipulate flow rates is key to efficient fluid system design and operation, helping ensure optimized performance and resource use.