Explain why the pressure at depth h in a stationary fluid is `rho g h.
To understand why the pressure at a certain depth in a stationary fluid is given by the equation `rho g h`, where `rho` is the density of the fluid, `g` is the acceleration due to gravity, and `h` is the depth, we need to apply the principles of fluid mechanics.
Fluids, such as water or air, exert pressure due to the weight of the fluid above them. The pressure at a given depth in a fluid is determined by the weight of the fluid column above that depth.
Let's break down the equation `rho g h` to understand its components:
- `rho` (rho) represents the density of the fluid. Density is the measure of how much mass is contained within a given volume of a substance. In this case, it represents how "dense" or "compact" the particles of the fluid are. The denser the fluid, the greater the mass there is per unit volume, resulting in a higher pressure.
- `g` represents the acceleration due to gravity. This is the acceleration experienced by objects in free fall near the surface of the Earth. Gravity pulls objects downward, causing them to accelerate. The acceleration due to gravity is constant and has a value of approximately 9.8 m/s^2 on the surface of the Earth.
- `h` represents the depth or vertical distance from the surface of the fluid to the point at which we want to measure the pressure. The depth is measured vertically from the surface of the fluid to the point of interest. As we go deeper into a fluid, the weight of the fluid above that point increases, resulting in higher pressure.
Now, let's combine these components to understand why `rho g h` gives us the pressure at a certain depth in a stationary fluid:
- The density `rho` tells us how much mass is present per unit volume of the fluid. The denser the fluid, the more particles there are in a given volume, and therefore, the greater the weight of the fluid above a specific depth.
- The acceleration due to gravity `g` tells us how much force is pulling objects downward. As we go deeper into a fluid, the weight of the fluid column above that point increases, resulting in more force pushing down on the fluid at that depth.
- The depth `h` tells us how far we are from the surface of the fluid. The greater the depth, the greater the weight of the fluid above that point, and consequently, the greater the pressure.
Combining these factors, we can see that the pressure at a certain depth in a stationary fluid is determined by the weight of the fluid above that depth. Therefore, the equation `rho g h` gives us the pressure at a specific depth, where `rho` represents the density of the fluid, `g` represents the acceleration due to gravity, and `h` represents the depth.