Overview

The primary purpose of the density simulation is to provide students a realistic environment where they can explore and better understand the concepts governing buoyancy and the relative density of materials. In the density simulation, experiments are performed in a framework consistent with the other virtual laboratories; that is, students are put into a virtual environment where they are free to choose their objects and equipment, build a conceptual experiment of their own design, and then experience the resulting consequences. The focus in the density simulation is to allow students the flexibility to perform fundamental experiments with a wide variety of materials to teach the basic concepts of density and buoyancy that are easier to model in a simulated environment rather than a real laboratory. The important principles, features, and assumptions forming the foundation of the density simulation are listed below.

  1. The simulation contains five 250 mL graduated cylinders modeled after typical graduated cylinders that could be purchased from any scientific supply store. These cylinders are approximately 255 mm tall from the bottom of the cylinder to the 250 mL mark and the diameter averages 35.7 mm. The graduated markings on the virtual cylinders exhibit volume errors typical of uncalibrated, off-the-shelf glassware.

  2. Liquids are dispensed into the cylinders through the liquid dispenser. Of course, the dispenser is fictitious and only serves as a convenient interface to select liquids and fill the cylinders. The volume of liquid dispensed is not the same each time a cylinder is filled. The density and viscosity of the various liquids was obtained from the best sources available. When two liquids are added to the same cylinder, the liquids are assumed to either mix completely (miscible) or not mix at all and thus separate into two layers with the least dense liquid on top. If the least dense liquid is added first followed by a more dense material, a short animation will play showing the liquids switching positions. When the two liquids are miscible, the resulting density and viscosity of the mixture is assumed to be an average of the two.

  3. Liquid transfers between the cylinders and the beaker and from the beaker to any of the cylinders is assumed to be perfect, meaning that no liquid is lost in the transfer. Thus, the density of a liquid can be calculated by measuring the volume of the liquid in the cylinder and its corresponding mass as measured in the beaker on the scale.

  4. The solids located on the wall comprise a wide range of densities that can be used to explore the properties of various materials. All of the solids are spheres with random diameters assigned when they are selected from the wall. It is assumed that the solids do not dissolve in any of the liquids nor do they melt, in the case of ice. Some solids will react with incompatible liquids and will explode. Also keep in mind that it is assumed the solids have no voids (such as in real cork) and thus the density is the true density and not an effectively lower density.

  5. When a solid is dropped into a liquid, the solid is first lowered to the surface of the liquid and then allowed to drop. Consequently, the initial velocity of the solid as it falls through the liquid is zero.

  6. When looking for a set of real materials to cover a wide range of densities and viscosities, it was obvious that there were gaps in the real data. Consequently, a collection of virtual solids and liquids has been included to provide options where there are not real samples available.

  7. The balance has been modeled after a typical top-loading balance found in any instructional laboratory. The Tare button can be used to zero out the balance or account for the mass of the beaker when weighing liquids.

  8. Located below each cylinder is an LCD display used to measure the time required for any solid to drop when it sinks to the bottom of the cylinder. The time required for a solid to drop is a function of the relative density of the two materials and the viscosity of the liquid. The purpose for displaying the time is to provide a relative measure of the viscosity of the selected liquids or to provide sufficient data to estimate the actual viscosity of the liquid. It is assumed that neither the sides of the cylinder nor their proximity to the falling solid affects the rate at which a solid drops.

Simulation Equations

The principle equation governing the calculation of the density is ρ = m/V where ρ is the density of the solid or liquid,m is the mass, and V is the volume. There is not a closed-form equation that can be used to calculate the viscosity of the liquid; however, the rate at which a solid will fall through a given liquid is dependent on the relative densities of the solid and the liquid and the viscosity of the liquid. Given below are the variables and the equations that are used to calculate the position of the solid as a function of time as it falls through the liquid. From these, the viscosity of the liquid can be calculated, but keep in mind that a numerical method will be required to calculate the viscosity using these equations. Also, these equations assume that the initial velocity of the solid is zero before it hits the liquid.


ρs = density of the solid


ρl = density of the liquid


m = mass of the solid


r = radius of the solid = [¾ ∙ m / (ρs ∙ π)] ⅓


η = viscosity of the liquid


g = gravitational constant


t = time required for the solid to fall from the surface of the liquid to the bottom of the cylinder


y = the distance from the surface of the liquid to the bottom of the cylinder


τ = time constant = m / (6π ∙ r ∙ η)


g' = effective gravitational constant = g ∙ [1 - (ρl / ρs)]


vt = terminal velocity = m ∙ g' / (6π ∙ r ∙ η)


y(t) = vt ∙ t + vt ∙ τ ∙ (e-t / τ - 1)