To calculate the pressure at a certain point in a liquid that has a free surface, we can use the fundamental principles of fluid mechanics. The pressure at a depth within the liquid depends on a few factors: the depth itself, the density of the liquid, and the acceleration due to gravity. The relationship is given by the formula:
[tex]\[ p = h \cdot d \cdot g \][/tex]
Where:
- [tex]\( p \)[/tex] is the pressure at the given depth.
- [tex]\( h \)[/tex] is the height (or depth) of the liquid column above the point where the pressure is being measured.
- [tex]\( d \)[/tex] is the density of the liquid.
- [tex]\( g \)[/tex] is the acceleration due to gravity.
Let's walk through each component:
1. Height (h): This is the vertical distance from the free surface of the liquid to the point where the pressure measurement is being taken.
2. Density (d): This is a measure of how much mass the liquid has per unit volume.
3. Gravity (g): This is the acceleration due to gravity, which is approximately [tex]\( 9.81 \, \text{m/s}^2 \)[/tex] on the surface of the Earth.
By multiplying these three quantities together, we derive the pressure exerted by the liquid at the given depth.
So, taking into consideration the given options:
1. Option 1: p=g/hd
- This indicates that pressure equals gravity divided by the product of height and density. This is incorrect, as it doesn't align with our formula.
2. Option 2: p=h/dg
- This indicates that pressure equals height divided by the product of density and gravity. This is incorrect.
3. Option 3: p=h=dg
- This is an incorrect representation as it includes an incorrect equality with an additional ‘=’ sign.
4. Option 4: phdg
- This is not mathematically plausible as it represents multiplication of all variables without a definitive formula.
Thus, the correct formula among the options provided is:
[tex]\[ p = h \cdot d \cdot g \][/tex]
Therefore, the correct answer is Option 3: [tex]\( p = h \cdot d \cdot g \)[/tex].
