Water flows in a pipe of 23-cm diameter with an average velocity of 0.73 m/s. The diameter of the pipe is reduced and the average velocity of the water increases to 1.66 m/s. What is the diameter of the smaller pipe? What is the flow rate?

To solve this problem, we can use the principles of **continuity** for fluid flow, which states that the mass flow rate must remain constant in a closed system. In this case, the equation of continuity can be expressed as:

\[
A_1 v_1 = A_2 v_2
\]

where:
- \( A_1 \) and \( A_2 \) are the cross-sectional areas of the pipes (before and after the diameter reduction),
- \( v_1 \) and \( v_2 \) are the average velocities of the fluid in the pipes (before and after the diameter reduction).

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

\[
A = \pi \left(\frac{d}{2}\right)^2 = \frac{\pi d^2}{4}
\]

Let:
- \( d_1 = 23 \, \text{cm} = 0.23 \, \text{m} \) (diameter of the larger pipe),
- \( v_1 = 0.73 \, \text{m/s} \) (average velocity in the larger pipe),
- \( v_2 = 1.66 \, \text{m/s} \) (average velocity in the smaller pipe),
- \( d_2 \) be the unknown diameter of the smaller pipe.

### Step 1: Find the diameter of the smaller pipe

Using the continuity equation:

\[
A_1 v_1 = A_2 v_2
\]

Substituting the areas \( A_1 = \frac{\pi d_1^2}{4} \) and \( A_2 = \frac{\pi d_2^2}{4} \) into the equation, we get:

\[
\frac{\pi d_1^2}{4} v_1 = \frac{\pi d_2^2}{4} v_2
\]

Canceling out \( \frac{\pi}{4} \) on both sides:

\[
d_1^2 v_1 = d_2^2 v_2
\]

Solving for \( d_2 \):

\[
d_2 = d_1 \sqrt{\frac{v_1}{v_2}}
\]

Substitute the values:

\[
d_2 = 0.23 \, \text{m} \times \sqrt{\frac{0.73 \, \text{m/s}}{1.66 \, \text{m/s}}}
\]

Let’s calculate \( d_2 \).

### Step 2: Calculate the flow rate

The **volumetric flow rate** \( Q \) is given by:

\[
Q = A_1 v_1 = A_2 v_2
\]

Since \( A_1 = \frac{\pi d_1^2}{4} \), we can calculate:

\[
Q = \frac{\pi d_1^2}{4} v_1
\]

Substituting the known values for \( d_1 \) and \( v_1 \):

\[
Q = \frac{\pi (0.23)^2}{4} \times 0.73
\]

Let’s calculate the flow rate as well.### Results:
1. The diameter of the smaller pipe is approximately **0.153 m** (or **15.25 cm**).
2. The flow rate is approximately **30.33 liters per second**.

Do you need any more details or clarifications?

Here are some related questions to expand your understanding:
1. How would the flow rate change if the diameter was reduced further?
2. What is the relationship between flow velocity and pressure in a pipe?
3. How does fluid viscosity affect the flow rate in a pipe?
4. What happens to the flow rate if the liquid is compressible?
5. How would you calculate the flow rate for a non-circular cross-section pipe?

**Tip:** The principle of continuity can also be applied in open channels, though it requires adjustments for non-enclosed flows.

Learn how to calculate the diameter of a smaller pipe and the flow rate of water using the continuity equation in fluid dynamics. This problem involves key concepts such as fluid velocity, cross-sectional areas, and the flow rate formula. Suitable for high school students in Grades 11-12.

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