A liquid column produces calculable pressure
Definition
Liquid pressure
The pressure caused by the weight of liquid above a point.
- The pressure due to a liquid column is P=hρgP=h\rho gP=hρg.
- Here, PPP is pressure in pascals, hhh is the vertical height of liquid above the point in metres, ρ\rhoρ is density in kilograms per cubic metre, and ggg is gravitational field strength in newtons per kilogram.
- The equation gives the pressure caused by the liquid column, not automatically the total pressure including the atmosphere.
The equation follows from weight per area
- For a column with area AAA and height hhh, its volume is V=AhV=AhV=Ah.
- Its mass is m=ρV=ρAhm=\rho V=\rho Ahm=ρV=ρAh, so its weight is F=mg=ρAhgF=mg=\rho AhgF=mg=ρAhg.
- Using P=F/AP=F/AP=F/A gives P=ρAhg/A=hρgP=\rho Ahg/A=h\rho gP=ρAhg/A=hρg.
Pressure differences use changes in depth
- Between two depths in the same liquid, ΔP=Δhρg\Delta P=\Delta h\rho gΔP=Δhρg.
- Use the vertical difference in depth, even if the container is sloping or irregularly shaped.
- At fixed density and gravitational field strength, pressure due to the liquid is directly proportional to depth.
Example
Pressure at one depth
- For water, ρ=1000 kg m−3\rho=1000\,\text{kg m}^{-3}ρ=1000kg m−3 at a depth of 3.0 m3.0\,\text{m}3.0m with g=10 N kg−1g=10\,\text{N kg}^{-1}g=10N kg−1.
- P=hρg=3.0×1000×10=3.0×104 PaP=h\rho g=3.0\times1000\times10=3.0\times10^4\,\text{Pa}P=hρg=3.0×1000×10=3.0×104Pa.
Density changes the pressure gradient
Definition
Density
Density is the mass per unit volume of a substance.
- On a pressure against depth graph, the gradient is ρg\rho gρg.
- A denser liquid gives a steeper graph because pressure rises more rapidly with depth.
Example
Difference between two depths
- In oil of density 800 kg m−3800\,\text{kg m}^{-3}800kg m−3, two points differ in depth by 0.75 m0.75\,\text{m}0.75m.
- ΔP=0.75×800×10=6000 Pa\Delta P=0.75\times800\times10=6000\,\text{Pa}ΔP=0.75×800×10=6000Pa.
Common Mistake
Do not use the container's total liquid depth when the required hhh is the depth of the stated point below the surface.
Self review
- What does each symbol in P=hρgP=h\rho gP=hρg represent?
- Why does the area cancel when the equation is derived?
- How do you calculate the pressure difference between two depths?
- What does the gradient of a pressure-depth graph represent?
- When must atmospheric pressure be added?
