**Herong's Notes on Physics** - Version 3.21, by Dr. Herong Yang

1 Frame of Reference with 2 Coordinate Systems

This section provides an example of a frame of reference associated with 2 coordinate systems. The location of an object can be described as 2 sets of coordinates.

A single frame of reference can be associated with multiple coordinate systems. In this case, the location of an object in the same frame of reference will have different sets of coordinates depending on which coordinate system is used.

For example, we can define a sphere coordinate system and associate it to the same frame of reference presented in the previous section:

- Set the origin of the sphere coordinate system at the center of Earth.
- Set the x-axis along the straight line from the center of Earth to the intersection point of the Equator Plane and the Prime Meridian Plane.
- Set the φ (phi) angle on the Equator plane relative to the x-axis.
- Set the θ (theta) angle on any meridian plane relative to the its intersection with the Equator plane.
- Set the r (radius) distance along the straight line from the center of Earth to the location of the object. Scale the r distance with 1 km per unit.

Now we describe the same location of the space shuttle in this new sphere coordinate system as (47.29, 22.82, 30572), because:

- Its perpendicular projection on the Equator plane has a φ angle of 47.29°.
- Its radius on the meridian plane on a θ angle of 22.82°.
- Its radius is 30,572 km.

Note that the φ angle is also called Longitude, the θ angle is also called Latitude, and the r distance minus Earth radius (6,371km) is also called Altitude. So the location of the space shuttle can also be described:

Longitude = 47.29° Latitude = 22.82° Altitude = 24,201km

*Last update: 2014.*

Table of Contents

►Introducion of Frame of Reference

Frame of Reference with 2 Objects

2-Dimensional Cartesian Coordinate System

3-Dimensional Cartesian Coordinate System

►1 Frame of Reference with 2 Coordinate Systems

Introduction of Special Relativity

Time Dilation in Special Relativity

Length Contraction in Special Relativity

The Relativity of Simultaneity