Explain why the pressure at depth h in a stationary fluid is `rho g h.
To understand why the pressure at depth h in a stationary fluid is given by the equation P = ρgh, where P is the pressure, ρ is the density of the fluid, g is the acceleration due to gravity, and h is the depth, we need to consider the factors that contribute to the pressure in a fluid.
Pressure in a fluid is caused by the weight of the fluid above it and is transmitted equally in all directions. The deeper we go in a fluid, the more fluid is above that point, exerting a greater force on the given point, leading to an increase in pressure.
Now, let's break down the equation P = ρgh step by step:
1. The density of a substance (ρ) is a measure of how much mass is packed into a given volume. In the context of fluids, it represents how closely the molecules or particles are packed together. The denser the fluid, the more mass is present in a given volume.
2. The acceleration due to gravity (g) is a constant that determines the gravitational force exerted by the Earth on objects near its surface. On Earth, the value of g is approximately 9.8 m/s^2. It represents the rate at which objects fall in a vacuum.
3. The depth (h) refers to the vertical distance from the surface of the fluid to the given point under consideration. The deeper the point, the greater the weight of the fluid above it.
Now, combining these factors, we can explain the equation P = ρgh:
The pressure at a given depth h in a stationary fluid is directly proportional to the density of the fluid (ρ), the depth (h), and the acceleration due to gravity (g). This is because the pressure is determined by the weight of the fluid above the given point.
As we go deeper into the fluid, the weight of the fluid above that point increases, leading to a higher pressure. The relationship between pressure, depth, and density is linear, which is why the equation P = ρgh holds true.
In summary, the pressure at depth h in a stationary fluid is given by the equation P = ρgh because the weight of the fluid above that point contributes to the overall pressure. Density, depth, and acceleration due to gravity are the key factors that determine the pressure at a given point in a fluid.