### Correct Option: C. The velocity decreases in wider sections and increases in narrower sections. #### Detailed Explanation: The principle of continuity in fluid dynamics states that for an incompressible fluid flowing through a pipe, the mass flow rate must remain constant from one cross-section of the pipe to another. This principle can be mathematically expressed using the equation of continuity: \[ A_1 v_1 = A_2 v_2 \] Where: - \( A_1 \) and \( A_2 \) are the cross-sectional areas of the pipe at two different points. - \( v_1 \) and \( v_2 \) are the fluid velocities at those points. **Step-by-Step Explanation:** 1. **Understanding Cross-Sectional Area:** - The cross-sectional area \( A \) of a pipe is calculated using the formula \( A = \pi r^2 \) for a circular pipe, where \( r \) is the radius. As the diameter of the pipe changes, the area changes accordingly. 2. **Applying the Continuity Equation:** - If the diameter of the pipe decreases (making it narrower), the cross-sectional area \( A \) decreases. According to the continuity equation, if \( A \) decreases, then \( v \) (the velocity) must increase to keep the product \( A \cdot v \) constant. - Conversely, if the diameter of the pipe increases (making it wider), the cross-sectional area \( A \) increases. To maintain the continuity, the velocity \( v \) must decrease. 3. **Conclusion from the Continuity Equation:** - Therefore, when the fluid flows from a wider section of the pipe to a narrower section, the velocity of the fluid increases. When it flows from a narrower section to a wider section, the velocity decreases. #### Why Other Options Are Incorrect: - **Option A: The velocity increases in wider sections and decreases in narrower sections.** - This is incorrect because it contradicts the principle of continuity. In wider sections, the area is larger, leading to a decrease in velocity, not an increase. - **Option B: The velocity remains constant regardless of the diameter changes.** - This option is incorrect because it ignores the relationship between area and velocity. If the area changes, the velocity must also change to maintain the mass flow rate. - **Option D: The velocity increases in both wider and narrower sections.** - This option is incorrect as it suggests that velocity increases regardless of the area change, which is not supported by the principle of continuity. The velocity can only increase in narrower sections, not in wider sections. #### Common Pitfalls: - **Misunderstanding Incompressibility:** The principle of continuity applies to incompressible fluids. For compressible fluids (like gases), the situation can be more complex. - **Forgetting to Consider Area Changes:** Students may overlook the relationship between area and velocity, leading to incorrect conclusions about how fluid behaves in varying diameters. ### Revision Summary: - The principle of continuity states that \( A_1 v_1 = A_2 v_2 \) for incompressible fluids. - Velocity increases in narrower sections and decreases in wider sections of a pipe. - Changes in diameter directly affect fluid velocity to maintain constant mass flow rate. - Always remember the relationship between cross-sectional area and velocity when analyzing fluid flow in pipes.