This article is about Coordinate transformation . In this article we will look at coordinate transformation in case of translation, inclination and rotation of S' frame of reference with respect to S frame of reference. This article is mainly for B.Sc. first year and comes under subject Mechanics.
If we define the position of a particle in two different frames of then in both the cases projection of the particle comes out to be different and the relation between the projections of this particle in two different frames of reference is known as their transformation equations.
If there is no relative velocity between the frames of references and they rotate through a certain degree then the transformation does not depend on time. Opposite to this when there is a certain relative velocity between them then transformation equation depends on time.
Transformation equations for frames of reference involving translation
Consider two frames of reference S and S’ whose origins are O and O’ and observers are at the point O and O’ as shown below in figure 1. If position vector of any point P in S is then its position vector in S’ would be .
According to figure
Differentiating it w.r.t. time
If is constant then
Again differentiating it w.r.t. time
So, in translated frame of reference S’ the position of the particle would be different but the acceleration and velocity would be same as measured in S frame of reference. Also the equations 1, 2 and 3 are time independent equations.
Coordinate transformations in reference frames having uniform relative translational motion
In figure 1 if the frame of reference S’ has translational motion with constant velocity w.r.t. the frame of reference S than at any time their axis would be at a constant distance but the position would depend on time. If in the beginning at both the frame of reference has origin O and O’ at the same point. Now at time t position vector of O’ w.r.t. O would be
Differentiating it w.r.t. the time we get
Acceleration of the particle
As is constant.
Hence position and velocity of the particle could change according to equation 4 and 5 when it is moving with uniform relative motion but acceleration remains unchanged.
Equations 4, 5 and 6 are time dependent transformation equations.
Transformation in an inclined frame of reference
Let x, y, z, be the coordinates of a particle in a frame of reference say S as shown in figure 2.
So we have,
Another frame of reference S’ is inclined w.r.t. the frame of reference S such that the origins of both the frame of reference and their z-axis coincides as shown in figure 2.
Now coordinates of point P in S’ frame of reference would be
Since z and z’ coincides we have
Now from right angle we have
Again from right angle we have
This implies that,
Recommended Books and texts
Mechanics: For Students of B.Sc (Pass and Hons.): D.S. Mathur
An Introduction to Mechanics (SIE) by David Kleppner, Robert Kolenkow
Physics for Degree Students B.Sc. First Year
1st Law and Newtonian space and time.
Absolute time and space