To determine the pressure at a certain point in a liquid with a free surface, we can use the well-known formula for calculating hydrostatic pressure. The pressure at a specific depth in a liquid is given by:
[tex]\[ p = h \cdot d \cdot g \][/tex]
where:
- [tex]\( p \)[/tex] is the pressure at the point in the liquid,
- [tex]\( h \)[/tex] is the height (or depth) of the liquid column above the point,
- [tex]\( d \)[/tex] is the density of the liquid,
- [tex]\( g \)[/tex] is the acceleration due to gravity.
The steps to derive this formula involve understanding how pressure in a fluid column works. The pressure at a depth in a static fluid is due to the weight of the fluid above that point. This can be derived from the principles of fluid statics:
1. Identify the Depth (Height) [tex]\( h \)[/tex]: Determine the vertical distance from the free surface of the liquid down to the point where you want to calculate the pressure. This height [tex]\( h \)[/tex] is crucial as it represents the fluid column above the point.
2. Determine the Density [tex]\( d \)[/tex]: Determine the density of the liquid. This value ([tex]\( d \)[/tex]) may already be given or might need to be looked up based on the properties of the liquid.
3. Gravitational Acceleration [tex]\( g \)[/tex]: The standard acceleration due to gravity on the Earth's surface is approximately [tex]\( 9.81 \, m/s^2 \)[/tex]. This value is considered a constant in these calculations.
Combine these into the pressure formula:
[tex]\[ p = h \cdot d \cdot g \][/tex]
Finally, compare this derived formula to the given answer options:
- [tex]\( p = g / h d \)[/tex]
- [tex]\( p = h d g \)[/tex]
- [tex]\( p = h = d g \)[/tex]
- [tex]\( p = h / d g \)[/tex]
Based on our formula [tex]\( p = h \cdot d \cdot g \)[/tex], the correct choice is:
[tex]\[ p = h \cdot d \cdot g \][/tex]
So, the correct answer is:
- [tex]\( p = h d g \)[/tex]
Thus, the answer option corresponding to the correct formula is the second option:
[tex]\( p = h d g \)[/tex]
