Problem 33 - Calculus of a Single Variable

Question

Explain why fluid pressure on a surface is calculated using horizontal representative rectangles instead of vertical representative rectangles.

Step-by-Step Solution

Answer

The use of horizontal rectangles for calculating fluid pressure represents the fact that pressure in a static fluid varies with depth (vertically) due to gravity, but remains constant along a horizontal plane. This is consistent with Pascal's law which states pressure acts equally in all directions, and it doesn't change horizontally for a fluid at rest.

Step 1: Pressure in a fluid at rest

Start by remembering the principle that in fluid physics, pressure in a fluid at rest acts equally in all directions. This concept is known as Pascal's Law.

Step 2: How pressure acts on a surface

Next, understand how pressure acts on a surface. Even though pressure acts in all directions, the effect of pressure on the surface of an object in the fluid is determined by the force acting perpendicularly to the surface.

Step 3: Vertical versus Horizontal Rectangles

Now, if vertical representative rectangles were to be used, it would imply that pressure changes as you move horizontally, which is not true. In a static fluid, the pressure change occurs vertically due to gravity, not horizontally. Hence, horizontal representative rectangles are used where the pressure remains constant along the horizontal dimension, but changes along the vertical plane.

Step 4: Explanation through a tangible example

Imagine a deep swimming pool. If you dive deep, you will feel more pressure acting on you, but it doesn't matter where you are in the horizontal plane. Whether you are in the middle of the pool or at the corner, the pressure will be the same, indicating that pressure variations are vertical, not horizontal.