Alexander Zakharov
Institute for Advanced Science in Arad

Technique of Teaching Vector Variables Using Buoyancy Force as an Example

The process of forming the buoyancy force is a basic one in teaching the “Pressure in solids, fluids and gases” section of the course of physics at secondary school. The idea of the “buoyancy force” is closely connected with floating of bodies and aeronautics. This subject matter is taught only once during the school course of physics, at its very beginning, and this is quite rewarding because the basic knowledge of the subject learners receive is not very complicated. The subject is simple, and teachers give a lot of experimental and factual proofs, still the resulting knowledge in most learners is still insufficient. There are different teaching technique approaches to raise the standart of knowledge on this subject.

There are some papers dealing with the analysis of typical mistakes in learners. For instance, N.E. Savchenko quotes such assertions as “if the body is immersed into the fluid it experiences the buoyant force which is equal to the volume of the fluid displaced”, “the body immersed into the fluid forces out the volume of the fluid which is equal to the volume of this body” or “the body forces out the volume of the fluid equal to the weight of the body”.

Several papers are directed to prevent mistakes in training for university entrance exams (A. I. Igoshev, V. I. Igoshev). Several authors offer novel experiment demonstrations (G.A. Gurianov, N.P. Gurianova). Methodical papers on this subject include method qaides edited by A.V. Usova and edited by A.V. Pyoryshkin But the problem is still very acute. Top class pupils cannot correctly formulate the Pascal law, Archimedes’ principle. They do not consider the medium influencing the bodies immersed into it. They cannot find the point of application of the force exerted by the fluid on the body immersed into it. They cannot find the value of the force of the fluid pressure on the vessel’s wall. They cannot explain why the force of pressure on the vessel’s bottom can be bigger or lesser than the weight of the fluid contained in the vessel depending on the vessel’s shape (so called “hydrostatics paradox”).

The necessity of revealing the peculiarities of making ideas on the subject stems mostly not only from purely educational motifs, but also from its didactic value. Namely, correctly formulated essential features of the concepts influence the quality of forming their related ideas in the subsequent section of “Mechanics”, “Molecular physics”, “Thermodynamics”.

We suppose that the basic idea of the subject is the idea of the “vector variable”. The mentioned mistakes necessarily arise if the learners have not become aware of the only knowing four main essential features of the vector variable (module, acting line, direction, point of application) but obtaining stable skills in manipulating them under any vector variables (movement, velocity, acceleration, force, etc.) as well.

Commonly known are the module and the direction as the two main essential features of the vector variable. The point of application and acting line are important features as well, but in analyzing the conditions of a given task, students often forget them or do not pay them attention. They often identify the acting line with the direction. These are not the same. The difference becomes obvious on adding the two forces which are directed at an angle and have different points of application. The stability of the ship when heeled, by a lateral wind can serve as an example of the analysis of the application points and acting lines of the buoyant force and the weight of the ship.

One can easily define the application point of the buoyant force by analyzing three stages of the coming to the surface body: fully immersed, partly immersed and floating at the surface. It becomes obvious that the application point of the buoyant force depends on how the liquid lies around the surface of the body immersed into it. The force of the pressure, the liquid exerts on the body, is applied to the geometrical centre of the mass of the liquid which is forced out by the body (if one supposes that the fluid is homogenous as in common). One can conclude that the point of the buoyant force application is situated in the centre of gravity of the liquid forced out by the body.

To ensure good knowledge of the way of determining the point of application and the lain of the buoyant force action, it is quit appropriate to compare the effect of forces acting on the ship when the sea is calm and when there is a lateral wind. In the former case the acting lines of the weight of the ship and the buoyant force coincide. Their application points are positioned in different places but on the same line. In the second case the two application points are positioned in the plane and each on its “own” line of action.