In this section we will explore Archimedes principle which relates an upward force acting on an object that is fully or partially submerged within a fluid to the weight of the fluid that the object displaces. We will also build upon our understanding of density, pressure, and hydrostatics to quantify the buoyant force.
Lecture 2 | Hydraulics, Buoyancy
Fluid Statics
Lecture 2 | Hydraulics, Buoyancy
Fluid Statics
Do you ever woder why you don't sink to the bottom of the pool or ocean? Check out this OpenStax concept trailer to find out.
Pre-lecture Study Resources
Watch the pre-lecture videos and read through the OpenStax text before doing the pre-lecture homework or attending class.
Fluid Statics | Hydraulics Pressure and Buoyancy
Hydraulics
Consider the analysis we did in the "pressure at a depth" section. If the pressure $P_0$ above the surface of the water increases by some amount $\Delta P$ so that the new pressure above the surface is now $P^{'}_{0} = P_0 + \Delta P$, then the pressure at all locations within the fluid also increase by the same amount $\Delta P$. Thus the pressure at the bottom of the volume element would be $P_1 + \Delta P = P_0 + \rho_{f} \, g \, d + \Delta P$.
Hydraulic brakes and lifts generally work by applying a force over an area at one location which increases the pressure at that location along with all other locations in the system. Thus hydraulics are an interesting application of Pascal's law. Below is a figure illustrating a simplified hydraulic lift.
Diagram of Hydraulic Lift
Did you ever notice that lifting a partially or fully submerged object in water feels easier to do than lifting the same object when it is outside the water? This effect is a consequence of the pressure gradient that the object experiences when in the water. Recall that pressure gradients cause an associated net force. The net forced caused by a pressure gradient within a fluid is what we refer to as the buoyant force, and it points in the upward direction which is why it feels easier to lift objects in water.
Buoyancy (Archimedes' Principle)
The discovery of how the buoyant force can be related other physical quantities is attributed to Archimedes. Archimedes’ principle can be stated as “when an object is partially or fully submerged in a fluid which is at rest, the pressure gradient within the fluid causes an associated net force in the upwards direction which is equal to the weight of the fluid that the object displaced”. Below is a physical representation to help illustrate Archimedes’ principle which was stated in the descriptive representation above.
Archimedes' principle
Notice that both the purple and green objects have the same dimensions, thus the same volume. However, the purple object is displacing a smaller volume of water than the green object. The volume of water that each object is displacing is shown below the container. Archimedes’ principle tells us that the buoyant force that each object experiences is equal to the weight of the displaced fluid, thus the green object experiences a larger buoyant force.
We can now quantify the buoyant force that any object experiences when either partially or fully submerged in a fluid as shown in the equation below.
However it is often much easier to work with volumes and densities instead of mass, so if we use our definition for mass, m = rho V and substitute this into the equation above, we get a much more handy expression for the magnitude of the buoyant force as seen below.
If you consider an object fully submerged in a fluid and apply Newton’s laws of motion, you can construct the conditions that must be satisfied for an object to sink or float. The conditions for sinking or floating are as follows...
if $\bar{\rho}_{object} > \rho_{fluid}$ , object sinks
if $\bar{\rho}_{object} < \rho_{fluid}$ , object floats
if $\bar{\rho}_{object} = \rho_{fluid}$ , neutrally buoyant
...where the bar over the object density indicates that it is the average density of the object.
Key Equations and Infographics
Buoyancy
Now, take a look at the pre-lecture reading and videos below.
Required Videos
OpenStax Section 11.7 | Archimedes' Principle (Buoyancy)
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