The correct option is **B. The density of the fluid and the depth below the surface**. ### Detailed Explanation To understand why option B is correct, we need to delve into the principles of fluid statics, particularly how pressure behaves in a fluid at rest. 1. **Understanding Pressure in Fluids**: - Pressure is defined as the force exerted per unit area. In a fluid, pressure can vary with depth due to the weight of the fluid above. - The formula for pressure at a certain depth \( h \) in a fluid is given by: \[ P = P_0 + \rho g h \] where: - \( P \) is the pressure at depth \( h \), - \( P_0 \) is the atmospheric pressure at the surface (if the fluid is open to the atmosphere), - \( \rho \) is the density of the fluid, - \( g \) is the acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \)), - \( h \) is the depth below the surface of the fluid. 2. **Why Density and Depth Matter**: - The pressure increases with depth because the weight of the fluid above exerts a force on the fluid below. The deeper you go, the more fluid there is above you, and thus the greater the pressure. - The density of the fluid (\( \rho \)) is crucial because it determines how much weight is exerted by a column of fluid of a given height. For example, a denser fluid (like mercury) will exert more pressure at the same depth compared to a less dense fluid (like water). 3. **Example Calculation**: - Suppose we have a column of water (density \( \rho = 1000 \, \text{kg/m}^3 \)) at a depth of \( h = 5 \, \text{m} \). The pressure at this depth can be calculated as follows: \[ P = P_0 + \rho g h \] Assuming \( P_0 \) (atmospheric pressure) is \( 101325 \, \text{Pa} \): \[ P = 101325 \, \text{Pa} + (1000 \, \text{kg/m}^3)(9.81 \, \text{m/s}^2)(5 \, \text{m}) \] \[ P = 101325 \, \text{Pa} + 49050 \, \text{Pa} = 150375 \, \text{Pa} \] - This shows how pressure increases with depth and density. ### Why Other Options Are Incorrect - **A. The temperature of the fluid**: - While temperature can affect the density of a fluid (and thus indirectly affect pressure), it is not a primary factor in determining pressure at a given depth. The pressure at a specific depth is primarily a function of the fluid's density and the depth itself, not the temperature. - **C. The shape of the container holding the fluid**: - The shape of the container does not affect the pressure at a specific depth in a fluid at rest. Pressure at a given depth is uniform in all directions and is determined solely by the height of the fluid column above that point and the fluid's density. - **D. The presence of air above the fluid**: - While the presence of air does contribute to the total pressure at the surface (atmospheric pressure), it does not affect the pressure at a specific depth in the fluid itself. The pressure due to the fluid is independent of the air above it, as long as the fluid is open to the atmosphere. ### Revision Summary - Pressure in a fluid at rest is primarily determined by the fluid's density and the depth below the surface. - The formula for pressure at depth is \( P = P_0 + \rho g h \). - Temperature, container shape, and air presence do not directly determine pressure at a specific depth. - Understanding the relationship between depth, density, and pressure is crucial for solving fluid statics problems.