Calculation of mass from density. Example through a problem.

Density = mass / volume

Before solving the exercises and practical examples of calculating density, it is important to take into account the units used for mass and volume. The SI unit for density would be: kilogram / meter ³   (kg/m ³ ).

However, it is possible to use other units depending on the mean system used, such as the cegesimal system (cgs), whose unit would be: g/cm 3 ^, it is also possible to use g/mL, and in the pound imperial system /ft ³ (lb/ft ³ ).

For practical purposes, beyond its calculation, knowing the density is useful to know if one body will float in another, and on some occasions to identify what material it is. As the legend tells of the gold crown that was ordered to be made for a king, and that they wanted to identify if it was really made of gold.

It was Archimedes, who tried to calculate the density of the crown, discovering that the volume would be calculated according to the amount of water that it displaced when immersed in a container. In this way, with its mass and having discovered the volume, he could calculate the density knowing that it was pure gold or not, with which the crown had been made.

It also brought with it the discovery known as “Archimedes’ Principle”, to discover the volume of an object according to the amount of volume of water displaced.

Examples of how to calculate density

Example 1: How to calculate density from mass and volume

Problem 1. Determine the density in g/cm ³ of a piece of unknown metal whose mass is 300 mg and has a volume of 0.0155 mL.

m _metal = 300mg

V _metal = 0.0155 mL

d _metal = ? g/ ^cm3

Although you have the necessary data for the formula, the units of mass and volume do not match those requested for density. Before using the equation, the units must be transformed.

The mass must be converted to grams, and the volume to cubic centimeters, which can be done using the following conversion factors:

Now you can use the formula:

Solution: The unknown metal has a density of 19.4 g/cm ³ .

Example 2: How the density of regular solids is calculated

Problem 2 . Determine the density in international system units of a lead sphere that has a mass of 300 g and a diameter of 3.70 cm.

This is a clear case, in which the volume is not known directly, but the shape and dimensions of the object whose density is to be determined are known.

It is also necessary to take into account the units, it is preferable to transform everything to the system of units required in the exercise before starting to carry out the calculations.

So the mass of the sphere from grams to kilograms:

Regarding the diameter:

Now to calculate the volume of a sphere, given the diameter (being unnecessary to calculate the radius):

Now to calculate the density, using the mass and volume in the well-known formula:

Solution: The lead sphere has a density of 11,300 kg/m ³ or 1.13 . 10 ⁴ kg/m ³ .

Example 3: How to calculate the density of irregular solids by liquid displacement

Problem 3. In a graduated cylinder that is half filled with water, where it marks 200 mL, an irregularly shaped object previously weighed is introduced. Once fully submerged, the water level rose to 325 mL. The mass of the object was 246 g. Determine the density of the material.

When it is not possible to calculate the volume of a body because it does not have a regular shape, an alternative way is to use the liquid displacement method. In this case, the increase in the volume of the liquid is due to the introduction of the object that displaced part of the fluid upwards.

In these cases, the volume of the irregular body is easily obtained by subtracting the volumes before and after submerging the body:

Now we can use the density formula:

Solution: The density of the material from which the irregular object is made has a density of 1.97 g/mL.

References

BBC. (nd). Density – Density – GCSE Physics (Single Science) Revision. Retrieved from https://www.bbc.co.uk/bitesize/guides/zbg7hyc/revision/1

Young. (nd). Determining the Density of a Solid and Liquid | Protocol (Translated to Spanish). Retrieved from https://www.jove.com/v/10082/determining-the-density-of-a-solid-and-liquid?language=Spanish

The Science Education Resource Center. (nd). How do I Calculate Density? Retrieved from https://serc.carleton.edu/mathyouneed/density/index.html