Chapter 2: The power of vectors
(Picture Source sdsu-physics.org)
” At the heart of all mathematics are numbers. If I were to ask how many marbles you had in a bag, you might answer, “Three.” I would find your answer perfectly satisfactory. The ‘bare’ number 3, a magnitude, is sufficient to provide the information I seek. If I were to ask, “How far is it to your house?” and you answered, “Three,” however, I would look at you quizzically and ask, “Three what?” Evidently, for this question, more information is required. The bare number 3 is no longer sufficient; I require a ‘denominate’ number – a number with a name. Suppose you rejoinder, “Three km.” The number 3 is now named as representing a certain number of km. Such numbers are sometimes called scalars. If I were next to ask “Then how do I get to your house from here?” and you said, “Just walk three km,” again I would look at you quizzically. This time, not even a denominate number is sufficient; it is necessary to specify a distance or magnitude, yes, but in which direction?
“Just walk three km due north.” The denominate number 3 km now has the required additional directional information attached to it. Such numbers are called vectors. Quite often, a vector is represented by components. If you were to tell me that to go from here to your house I must walk three blocks east, two blocks north, and go up three floors, the vector extending from “here” to “your house” would have three spatial components: • Three blocks east, • Two blocks north, • Three floors up. “(NASA/TM—2002-211716)
Vector methods have become standard tools for the physicists.
At first we learn to disentangle physical quantities which requires only magnitude for their complete specifications and having no direction associated (scalar quantities) and those which require both magnitude and direction for their complete specification (vectors). Soon afterwards we realize that vectors are not simply mathematical tools to describe an additional piece of information. The bare fact that some physical quantities (like velocity) are vectors becomes the foundation of certain physical phenomena.
Let me make an example, that of uniform circular motion.
The cat is a classical example of a physical system moving along a circle at constant speed. Such motion is called uniform circular motion. As an object moves in a circle, it is constantly changing its direction. So even though the speed is constant in magnitude the velocity, which is a vector and points in the same direction of the object’s motion (tangent to the circle), is changing over time.
Accelerating objects are those one which are changing their velocity, either their speed (magnitude of the velocity vector) or their direction. Therefore an object moving in a circle is accelerating though it is moving with a constant speed. The acceleration is only due to its change in direction.
By using vector analysis we can also learn that the direction of this acceleration is inwards towards the center of the circle. For this reason it is called centripetal acceleration.
The story could continue : if we there is an acceleration Newton2 (F=ma) tells us that there must be a force acting somewhere pointing in the same direction of the acceleration and Newton3 (Action=Reaction) tells us that the counterpart opposite to such an acting force should also exist (therefore pointing outwards). In the example above I am pretty sure the can is experiencing its own weight as a force pulling it outside the circle (look how stretched are its pawns).
Well the phenomena just described is at the basis of the functioning of some of our daily tools like wash machines and rotating salad spinners. On both cases the water droplets are the physical systems moving along a circle (about 50 cm radius in a wash machine and 12 cm in the spinner). Through rotation they are acquiring a centripetal acceleration. The outward pointing force pushes them on the side of the rotating vessels and through the small openings outside the volume.
Just for curiosity, if you were to calculate which acceleration those droplets experience inside the washing machine (by standard 900 RPM – Rotation per Minute -) you would obtain something like 200 G, i.e. 200 times the gravitational acceleration we experience on earth. You can be sure, that is pretty much and if you don’t believe it, ask any astronauts doing the NASA Training in the “20G centrifuge” (a factor of 10 less that in your washing machine home) how they feel afterwards … and all that only because of vectors.
Associating a direction in addition to the magnitude to a physical quantity as a deeper implication than that of using a mathematical symbol (with a complete set of analysis rules) to describe the direction itself.
For those one who want to learn more about uniform circular motion, there is an extremely didactical short video: