To solve this problem, we need to use the principle of conservation of mass for an incompressible fluid, which is represented by the continuity equation. The continuity equation states:

$A_1 v_1 = A_2 v_2$

Where:

• $A_1$ and $A_2$ are the cross-sectional areas of the larger and smaller pipes, respectively,
• $v_1$ and $v_2$ are the average velocities in the larger and smaller pipes, respectively.

Given:

• Diameter of the larger pipe, $d_1 = 23 \, \text{cm} = 0.23 \, \text{m}$,
• Velocity in the larger pipe, $v_1 = 0.73 \, \text{m/s}$,
• Velocity in the smaller pipe, $v_2 = 1.66 \, \text{m/s}$.

We are required to find:

1. The diameter of the smaller pipe ($d_2$),
2. The flow rate ($Q$).

Step 1: Find the cross-sectional area of the larger pipe ($A_1$)

The cross-sectional area of a pipe is given by the formula:

$A = \pi \left(\frac{d}{2}\right)^2$

For the larger pipe:

$A_1 = \pi \left(\frac{0.23}{2}\right)^2 = \pi \times (0.115)^2 = \pi \times 0.013225 \, \text{m}^2$

$A_1 = 0.04154 \, \text{m}^2$

Step 2: Use the continuity equation to find $A_2$

From the continuity equation:

$A_1 v_1 = A_2 v_2$

Solving for $A_2$:

$A_2 = \frac{A_1 v_1}{v_2} = \frac{0.04154 \times 0.73}{1.66} = \frac{0.0303242}{1.66} = 0.01827 \, \text{m}^2$

Step 3: Find the diameter of the smaller pipe ($d_2$)

Now, use the area formula to find $d_2$:

$A_2 = \pi \left(\frac{d_2}{2}\right)^2$

Solving for $d_2$:

$\left(\frac{d_2}{2}\right)^2 = \frac{A_2}{\pi} = \frac{0.01827}{\pi} = 0.005815 \, \text{m}^2$

$\frac{d_2}{2} = \sqrt{0.005815} = 0.07625 \, \text{m}$

$d_2 = 2 \times 0.07625 = 0.1525 \, \text{m} = 15.25 \, \text{cm}$

So, the diameter of the smaller pipe is approximately 15.25 cm.

Step 4: Calculate the flow rate $Q$

The flow rate $Q$ is the product of the cross-sectional area and the velocity:

$Q = A_1 v_1 = 0.04154 \times 0.73 = 0.0303 \, \text{m}^3/\text{s}$

Thus, the flow rate is approximately 0.0303 cubic meters per second.

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Final Answers:

• The diameter of the smaller pipe is 15.25 cm.
• The flow rate is 0.0303 m³/s.

Would you like further details or have any questions?

Here are 5 related questions for further practice:

1. How does the velocity change if the diameter of the smaller pipe is reduced further?
2. What would the flow rate be if the larger pipe had a diameter of 30 cm instead of 23 cm?
3. How would the diameter of the smaller pipe change if the velocity increased to 2 m/s?
4. What happens to the flow rate if the velocity in both pipes doubles?
5. How does the flow rate depend on the cross-sectional area in a steady flow?

Tip: The continuity equation is crucial for analyzing fluid dynamics, as it reflects the conservation of mass in fluid flow systems.