Motion described in polar coordinates
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Change of basis
Equations are omitted for technical reasons - download the original pdf
The unit vectors [Equation goes here - download the original to see it.] and [Equation goes here - download the original to see it.] represent fixed directions, but the vectors [Equation goes here - download the original to see it.] and [Equation goes here - download the original to see it.] change direction as varies. In two dimensions both sets of unit vectors- that is [Equation goes here - download the original to see it.] and [Equation goes here - download the original to see it.] form a basis- that is , any two dimensional vector [Equation goes here - download the original to see it.] can be written in terms of either set [Equation goes here - download the original to see it.] so it would be useful to know how to convert from one basis to the other. [Diagram goes here - download the original to see it.] As the diagram indicates, [Equation goes here - download the original to see it.] This can be written in matrix form [Equation goes here - download the original to see it.] the matrix [Equation goes here - download the original to see it.] represents a rotation through [Equation goes here - download the original to see it.] degrees. Its inverse is a rotation through [Equation goes here - download the original to see it.] degrees. [Equation goes here - download the original to see it.] So [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] Example A particle has velocity [Equation goes here - download the original to see it.] What is its velocity when referred to Cartesian co-ordinates? Solution Let the velocity in [Equation goes here - download the original to see it.] co-ordinates be [Equation goes here - download the original to see it.] then [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] This indicates that the velocity is constant in the [Equation goes here - download the original to see it.] direction.
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Contents of
Motion described in polar coordinates
1 Motion described in polar coordinates. Vectors in polar co-ordinates
2 Change of basis
3 Rate of change of radial direction and radial angle
4 Velocity in polar co-ordinates
5 Acceleration in polar co-ordinates
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