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Reference Frames

A frame of reference is usually a collection of objects, together with the observer, which are at rest relative to each other. Thus, for example, we at rest relative to Earth along with Earth and laboratories built on Earth, etc., constitute a frame of reference. It is with respect to the Earth’s frame of reference that we observe and measure the changes of positions of other objects. Similar frames of reference can be associated with other observers (or objects) who may not be at rest relative to us.

Within a frame of reference, we set up a co-ordinate system, which is used to measure the position of an object. Measurements indicate that the space we inhabit is a three-dimensional space. This means that we need three numbers to specify the position of any (point) object in the space. The choice of numbers is determined by the type of co-ordinate system that we use. The generally used co-ordinate systems are rectangular (Cartesian) co-ordinates (x, y, z), spherical polar co-ordinates (r, θ, Ø), and cylindrical co-ordinates (ρ, Ø, z).

A co-ordinate system may be set up anywhere in the frame of reference, the choice of origin has complete freedom. Given the origin, a Cartesian (rectangular) system is formed by having three mutually perpendicular straight lines, the X, Y and Z axes. There are two independent rectangular systems possible in space, the left-handed and the right-handed which cannot be made to coincide with each other by any means of translation or rotation. Cartesian system, by convention is chosen to be right-handed as shown in fig. The position of a particle P relative to origin O is given by position vector r, characterized by its specific length and direction:

= x i + y j + z k (i)

where, z, j, k denote constant unit vectors in the directions of X, Y, Z axes.

We talk of a point, straight line, perpendicular straight lines, length of line segment, etc. in the Euclidean sense. Our experience tells us that Euclidean geometry is applicable in our ordinary Newtonian space. This is valid if the objects are not very massive, or are not moving with very high speeds of the order of speed of light.