Density is an expression of how dense or tightly compact a material is. This is expressed in mathematical terms as mass over the unit volume of a material. A very useful representation of density can be related to the different states of matter of a substance. The three known states of matter are gas, liquid, and solid.
When a substance in a gas state is confined within a fixed volume of space, its particles will be spread out into the confined space as seen below. When the same substance is in a liquid form confined within the same fixed volume of space, its particles will be loosely packed. In its solid state, the particles are packed together tightly.
The amount of substance confined in this fixed volume can therefore be expressed in terms of density, where the substance in the gas state is the least dense as it has a lower mass confined in the fixed volume. Similarly, the substance in liquid form will be slightly denser, since it has a larger amount of mass confined in the fixed volume. Lastly, the substance in solid form is the densest, since it has the largest amount of mass confined in the same fixed volume.
The density of a substance in different states of matter, solid, liquid, and gas.
What affects density?
Density is affected by various factors.
High temperature causes a substance to expand, hence raising the temperature causes a decrease in density. Low-temperature results in increased density.
Increasing pressure will reduce the volume in some cases, hence increasing density. The reverse is also true.
Humidity will increase when density is decreased, as it is inversely proportional to density.
What is the formula for density?
Mass density is equal to the mass of a substance over its unit volume as seen in the equation below, where ρ is the density, m is the mass, and V is the volume. Density can be used mathematically to obtain the mass or volume of a substance when density is known or vice versa. The units of density are kg over cubic metres.
\[\rho[kg \space m^3] = \frac{m[kg]}{v[m^3]}\]How can density be used to express other physical quantities?
Density is used in science, in general, to express a physical quantity over unit area or volume. Similar to mass density, other types of densities can also be expressed in a similar manner.
For example, the current density J is the product of the flow of current I, and unit area A, which can mathematically be expressed as shown below. Another example is specific weight, which is an expression of the weight force W over density, ρ.
For specific weight:
\[D [N \cdot kg \cdot m^3] = g[m/s^2] \cdot \rho [kg \space m^3]\]
For current density:
\[J = I[A] \cdot A[m^2]\]
Calculate the density of a fluid with a mass of 1800g and a volume of 235 ml.
Solution:
Convert to SI units,
\(1800 g = 1.8 kg \cdot 235 ml = 2.35 \cdot 10^{-4} m^3\)
\(\rho = \frac{m}{V} = \frac{1.8 kg}{2.35 \cdot 10^{-4}m^3} = 0.766 \cdot 10^4 kg/m^3\)
What is upthrust?
Upthrust is an upward force that is exerted on a body when it is submerged in a fluid due to the pressure difference between the top and bottom of the fluid. Archimedes' principle states that the upthrust on a body submerged in a fluid is equal to the weight of the fluid that is displaced by the body. In mathematical terms, this is expressed as the volume multiplied by the fluid density as seen in the equation below. The upthrust force is described by Fup; this is measured in N, where W is the weight of the object, and V is the volume of the object.
\[\text{Weight of fluid displaced = Upthrust Force} \qquad F_{up} = W[N] = mg= \rho_{fluid} \cdot G[m/s^2]\cdot V_{object}[kg/m^3]\]
How is upthrust related to density?
Upthrust is directly proportional to the density of the fluid. The difference between the density of a body submerged in a fluid and the density of that fluid determines whether the object sinks or floats. The diagram below shows when an object sinks or floats when submerged in fluid.
Upthrust and density relationship.
If the upthrust force is greater than the weight of the body, the object floats.
If the fluid's density is greater than the substance's density, the object floats.
If the density of the substance is greater than the fluid's density, the object sinks.
If the upthrust force is less than the weight of the object, the object sinks.
An object is submerged into a fluid. It has a density four times that of the fluid. Calculate the acceleration of the object when it is sinking.
Solution:
We begin by comparing the forces acting on the object. Based on the info given the object is sinking, hence the weight must be greater than upthrust.
\[\sum F= m \cdot a \text{ sinking: }W > F_{up}\]
Then, we analyse forces acting on the object using Newton's law. We replace weight with the product of mass and gravity, and upthrust force with the product of density, gravity, and volume using the formulas you have learned. We get the following equation (let's call it equation 1).
\[W -F_{up} = m \cdot a m \cdot g - \rho \cdot g \cdot V = m \cdot a \space (1)\]
Then we can use the information given about the density of the object which is four times the density of the fluid. This is written mathematically as shown below
\[\rho_{object} = 4 \cdot \rho_{fluid}\]
Using the relation between density and mass shown below, we can substitute mass with the product of volume and density in equation 1 which was derived earlier.
\[\rho = \frac{m}{V}\]
\[m \cdot g - g \cdot \rho \cdot V = ma \space V \cdot \rho_{obj} \cdot g - \rho_{fluid} \cdot V \cdot g = \rho_{obj} \cdot V \cdot a \space (2)\]
Subsequently, we can substitute each term that contains ρobj with 4ρfluid , using the relation that was obtained earlier. This gives us the following expression.
\[V \cdot (4 \cdot \rho_{fluid}) \cdot g - (\rho_{fluid} \cdot V \cdot g) = (4 \cdot \rho_{fluid}) \cdot V \cdot a\]
We divide both sides by the common terms which are ρfluid and V. Which gives us the expression below.
\[4g - g = 4a \Rightarrow 3g = 4a\]
The last step is to solve for acceleration and substitute g with the acceleration of gravity constant, 9.81 m/s2.
\[a = \frac{3}{4} g = 7.36 m/s^2\]
Density - Key takeaways
Density is a property that can be expressed as the force over area or volume. It describes how dense a material is.
Specific mass density is the mass over volume.
Upthrust is the force exerted on a body by the fluid it is submerged into.
The upthrust determines whether an object will float or sink.
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