Warm-Up: Fluids and Forces
In Lab 1, we saw that proportionality is a powerful tool that can help us understand various relationships. Take for example the following equation:
[latex]m = {\rho}V[/latex]
where ρ is the density of a fluid, m is its mass and V is the volume. Here, mass is linearly proportional to volume and the constant of proportionality is the density. Thus, if we assume that density remains constant, doubling the mass results in doubling the volume and vice versa. We will make use of this equation shortly.
Archimedes’ Principle
When an object is fully immersed in a liquid, it displaces an amount of liquid exactly equal to the volume of the object. This principle is especially useful for determining the volume of oddly shaped objects, which you’ll be doing in this lab. In fact, this is how Archimedes discovered the concept: he was trying to determine the volume of a gold (or not-so-gold?) crown. He knew the mass and the density of pure gold, but without knowing the crown’s volume, he couldn’t determine whether it was made of solid gold or not.
The Buoyant Force
Lastly, we will also be exploring the concept of buoyant force in this lab. The buoyant force is the upward force exerted on an object by a fluid. The relationship between mass and volume is especially important here, since an object’s density ultimately determines whether it floats or sinks. If the average density of the object is less than that of the fluid, it will float; if it is greater, the object will sink. In both cases, a buoyant force acts on the object. However, if the object is floating and only partially submerged, the buoyant force corresponds only to the volume of the object that is submerged. Archimedes’ principle tells us that the buoyant force is equal in magnitude to the weight of the fluid that the object displaces.
This is an idea that we will explore further in Exercises 2 and 3. In the meantime, we can use a useful trick called dimensional analysis to find a possible formula for the buoyant force.
Dimensional Analysis
Fluids: Warm-up
There is a linear relationship between mass and volume. Mass has dimensions of mass ([M]), while volume has dimensions of length cubed ([L3]). Note: Capital letters and square brackets are used to denote dimensions. The density, ρ acts as a constant of proportionality, and must have the appropriate dimensions.
While dimensional analysis can’t always tells us if an equation is correct, it can certainly help us determine if an equation is wrong. Try the following warm-up exercise to exemplify this.
Before moving on!
Make sure you can utilize dimensional analysis as well as understand the buoyant force and Archimedes’ principle.
