Unizor - Physics4Teens - Mechanics - Kinematics - Space
Channel: Zor Shekhtman on UNIZOR Education
Date: 16/04/2018

Notes to a video lecture on http://www.unizor.com

Unizor - Physics4Teens - Mechanics - Kinematics - Space

Space, as we will be dealing with in this course of classical Physics, is the same three-dimensional space known from Mathematics. To identify a point in this space, we will use a system of coordinates (in most cases, Cartesian or, when we study rotation, cylindrical) and three numerical coordinates within this system (x, y and z for Cartesian system or their analogous in other systems).
Any three real numbers define a point in this space, as long the system of Cartesian coordinates is established.

System of Cartesian coordinates implies a choice of a point that serves as the origin of coordinates, three mutually perpendicular axes and the unit of measurement along each axis. The latter is specified in the International System of Units (SI) as a meter.
Originally a meter was defined as 1/10000000th of a length from the North Pole to equator. Contemporary approach to the definition of a meter is much more precise and states that the speed of light in vacuum equal to 299792458 meters per second. Obviously, it implies a precise definition of a second, but that was a subject of a previous lecture.
So, a meter is, by definition, a distance covered by light in vacuum during 1/299792458th of a second.

All geometric properties of three-dimensional mathematical space are valid in Physics. We can talk about points, straight lines, circles, planes, spheres, cones etc.
Continuity, obviously, is a valid property of this space, which implies that between any two points in space we can draw a straight line and every point on this line is a valid point in space.
It also justifies the usage smaller units of measurement - millimeters (1/1000th of a meter, micrometers (1/1000000th of a meter etc.)

Additionally, it's convenient to consider not just a point defined by three Cartesian coordinates in three-dimensional space, but a vector from the origin of coordinates to this point. This vector is also defined by the same three real numbers - its projections on X-, Y- and Z-axis.
The convenience of this approach lies in the following simple fact.
Assume, an object at time t1 is located at point A{x(t1), y(t1), z(t1)}
and at later time t2 it moved to point B{x(t2), y(t2), z(t2)}.
Then a segment AB it moved has projections on three axes as x(t2)−x(t1), y(t2)−y(t1) and z(t2)−z(t1).
Let's use the vector representation of this movement.
Consider vector a that represents a position of our object at time t1 and vector b that represents a position of our object at time t2. Obviously, vector a has components x(t1), y(t1) and z(t1), while vector b has components x(t2), y(t2) and z(t2).
Then displacement from one position to another can be represented as a simple difference between vectors: d = b − a.
Because of this simplicity, we will prefer in most cases to talk about vector of position instead of a point.

The first property of space we'd like to mention is its uniformity relative to location. In other words, space is homogeneous. Recall the uniformity (or homogeneity) of time, which means that identical experiments conducted at different moments in time produce identical results. Exactly the same meaning is in the uniformity of space: identical experiments conducted at different locations produce the same results.

The second property of space is its uniformity relative to direction or angular symmetry. In other words, space is isotropic. It means that rotating an apparatus that is involved in any experiment will not change the results of this experiment, as long as all other conditions of the experiment remain identical.

CONCLUSION
These important properties of space - being homogeneous and isotropic - are fundamental for an entire subject of Physics. They are intuitively obvious and many physical problems are solved assuming these properties are indeed the real properties of our space.
What's more interesting is that certain conservation laws, like conservation of energy, momentum and angular momentum, that guide the behavior of any physical system, can be derived from these properties. We will address these issues later in the course.