Problem 83 - Physics

Question

Water flows with a volume flow rate of \(1.50 \mathrm{~m}^{3} / \mathrm{s}\) in a pipe. Find the water speed where the pipe radius is \(0.500 \mathrm{~m}\).

Step-by-Step Solution

Answer

The water flows at approximately 1.91 m/s.

Step 1: Understand the Given Formula

To find the speed of water, we use the relationship between volume flow rate (Q), cross-sectional area (A), and the speed of flow (v). The formula for volume flow rate is: \[ Q = A \cdot v \] where \( Q \) is the volume flow rate, \( A \) is the cross-sectional area, and \( v \) is the speed.

Step 2: Calculate the Cross-Sectional Area

The cross-sectional area \( A \) of the pipe can be found using the formula for the area of a circle: \[ A = \pi r^2 \] Given that the radius \( r \) of the pipe is \( 0.500 \) m, substitute into the formula:\[ A = \pi (0.500)^2 \approx 0.785 \mathrm{~m}^2 \]

Step 3: Solve for the Water Speed

Using the formula from Step 1 and the calculated area from Step 2, solve for the speed of the water flow \( v \):\[ Q = A \cdot v \] \[ 1.50 \mathrm{~m}^3/\mathrm{s} = 0.785 \mathrm{~m}^2 \cdot v \] Re-arranging the formula to solve for \( v \):\[ v = \frac{1.50}{0.785} \approx 1.91 \mathrm{~m/s} \]

Key Concepts

Volume Flow Rate

In fluid dynamics, the volume flow rate is an essential concept that measures how much fluid is moving through a particular point in space over time. It's denoted as \( Q \) and measured in cubic meters per second (m\(^3\)/s). Essentially, it informs us how much of a liquid, such as water, is passing through a location like a pipe every second.

• To calculate it, you multiply the cross-sectional area of the pipe by the speed of the fluid flowing through it.
• This relationship can be expressed as \( Q = A \cdot v \), where \( A \) is the cross-sectional area, and \( v \) is the velocity or speed of the fluid.

Understanding volume flow rate is crucial because it helps determine how fast a liquid can be delivered through systems like pipes or hoses, making it widely used in engineering and physics.

Cross-Sectional Area

The cross-sectional area of a pipe refers to the surface area of the pipe opening that the fluid flows through. In fluid dynamics problems, this is key because it directly influences the flow rate. The larger the cross-sectional area, the more fluid can pass through at a time.

• For a pipe with a circular opening, this area is calculated using the formula for the area of a circle: \( A = \pi r^2 \).
• Here, \( r \) is the radius of the pipe, and \( \pi \) is a constant approximately equal to 3.14159.

For instance, if a pipe has a radius of 0.500 m, its cross-sectional area can be calculated as \( A = \pi (0.500)^2 \), which would be approximately 0.785 m\(^2\). This calculation is vital for determining the possible volume flow rate through the pipe.

Water Speed Calculation

Calculating the speed of water in a pipe requires understanding the relationship between the volume flow rate, the cross-sectional area, and the speed of the water. To find the speed, rearrange the fundamental formula for flow rate: \( Q = A \cdot v \).

• The formula can be reorganized to solve for \( v \): \( v = \frac{Q}{A} \).
• This tells you that the speed equals the volume flow rate divided by the cross-sectional area.

Using the given volume flow rate and calculated cross-sectional area, you can find speed easily. For example, if \( Q = 1.50 \) m\(^3\)/s and \( A = 0.785 \) m\(^2\), then \( v = \frac{1.50}{0.785} \approx 1.91 \) m/s. Thus, this result signifies how quickly the water advances through the pipe.

Pipe Flow

Pipe flow refers to the movement of liquids, often water, through a conduit like a pipe. This is a common scenario in various applications, from household plumbing to large-scale engineering systems. Understanding pipe flow is important for designing systems that efficiently transport fluids.

• Key factors influencing pipe flow include the pipe's diameter (affecting cross-sectional area), the fluid's velocity, and the pressure driving the flow.
• Engineers utilize this knowledge to ensure a sufficient water supply and minimize losses from friction or leaks.

When analyzing pipe flow, it's essential to distinguish it from open channel flow, which involves liquid movement over an open surface, like a river. Pipe flow dynamics are often more predictable and controlled.

Continuity Equation

The continuity equation in fluid dynamics is a key principle that asserts the conservation of mass in a fluid flow system. It signifies that for an incompressible fluid (like water), the mass flow must remain constant from one cross-section of a pipe to another, even if the pipe varies in size.

• The equation can be expressed as \( A_1v_1 = A_2v_2 \), where \( A_1 \) and \( A_2 \) are the cross-sectional areas at two different points in the pipe, and \( v_1 \) and \( v_2 \) are the fluid velocities at these points, respectively.
• This principle helps in understanding how fluid speed changes as it moves through parts of the pipe with varying sizes.

By applying the continuity equation, you can predict changes in velocity and manage flow rates effectively, crucial for designing efficient piping systems.