In fluid dynamics, Bernoulli’s principle states that for an incompressible fluid flowing through a pipe, the sum of the pressure energy, kinetic energy, and potential energy per unit volume remains constant along a streamline. This relationship can be expressed as:

$P + \frac{1}{2} \rho v^2 + \rho g h = \text{constant}$

where:

• $P$ is the pressure,
• $ρ$ is the fluid density,
• $v$ is the flow velocity,
• $g$ is the acceleration due to gravity $(9.8 \, \text{m/s}^2)$,
• $h$ is the height above a reference point.

This principle explains phenomena such as the lift generated by airplane wings and the reduction in pressure in constricted sections of a pipe. For example, when a pipe narrows, the velocity of the fluid increases, and the pressure decreases.

Consider a horizontal pipe with varying cross-sectional areas. At one point, the pipe narrows, causing the fluid velocity to increase from $2 \, \text{m/s}$ to $4 \, \text{m/s}$. The fluid density is $1000 \, \text{kg/m}^3$, and the pressure at the wider section is $3000 \, \text{Pa}.$

Question 1

If the pipe were tilted upward at an angle, increasing the height difference between the wider and narrower sections, how would the pressure in the narrower section change?

A. It would increase.

B. It would decrease.

C. It would remain the same because Bernoulli’s principle does not account for height differences.

D. It would remain the same because the velocity difference dominates the pressure change.