Upthrust and pressure

In a nutshell

Fluids are liquids and gases. Fluids are made of moving particles, and so exert pressure. The pressure depends on the density and depth of the fluid. Fluid pressure is responsible for the upthrust force on objects submerged in a fluid. 

Equations

Variable definitions

​​​​Fluid pressure

Pressure is the force per unit area. The pressure of a fluid is the force exerted by the fluid normal to a surface per unit area.

For a fluid, the pressure depends on the density and depth. More dense liquids have more particles per unit volume colliding with a surface, so they exert more pressure. 

At any given point inside the fluid, the particles above that point weigh down on it. Therefore as the depth increases, the number of particles weighing down on a surface increases. The fluid pressure therefore increases with depth. 

Density is approximately uniform throughout a fluid – it does not increase as depth increases.

The formula for the pressure of a fluid of a certain density at a certain depth is:

​$$pressure=depth\,\times \, density\, \times \, gravitational \,\, acceleration \\ \ \\ p=h\times \rho \times g$$​​

where the variables are defined above. 

Example

Calculate the pressure of liquid hydrogen at a depth of $$2.5\,m$$. Liquid hydrogen has a density of $$71\,kg/m^3$$.

Write down the information provided in the question:

$$h=2.5\,m \\ \ \\ \rho=71\,kg/m^3 \\ \ \\ g=9.8\,m/s^2$$

Write down the relevant formula:

$$p=h\times \rho \times g$$.

Substitute the relevant information into the correct formula:

 $$p=2.5\,\times \, 71\, \times \, 9.8 \\ \ \\ p= \underline{1739.5 \,Pa}$$ 

So the pressure of liquid hydrogen at a depth of $$2.5\,m$$ is $$\underline {1739.5\,Pa}$$.

Upthrust

Definition

The upthrust is the resultant force that a fluid exerts on an object that is (fully or partially) submerged in the fluid. The upthrust is equal to the weight of the fluid displaced by the object. 

When an object is submerged (fully or partially) in a fluid, the fluid exerts pressure on the object from every direction. Because fluid pressure increases with depth, the pressure at the bottom of the object is greater than the pressure at the top. The resultant upwards force is called upthrust. 

An object (fully or partially) submerged in a fluid displaces some of the fluid. An object that is less dense than the fluid will displace its own weight in fluid before it is fully submerged. Therefore, the weight of the object equals the upthrust from the fluid while the object is only partially submerged. The object therefore floats. 

An object that is more dense than the fluid weighs more than the volume of fluid it displaces, so the the weight of the object is greater than the upthrust. The object therefore moves downwards (sinks). 

1 Upthrust 2 Weight

Atmospheric pressure

At higher altitudes, the air pressure decreases. This is because the density of air decreases at higher altitudes, and because at higher altitudes there are fewer air molecules weighing down on a surface.