
DYNAMICS  THEORY

Position in Polar Coordinates


For plane motion, many problems are better solved using polar coordinates, r and θ. This requires the development of position, velocity and acceleration equations based on, r and θ. 




Position

Units Vectors for Polar Coordinates


Before velocity and acceleration can be determined in polar coordinates, position needs to be defined. Unlike rectilinear coordinates (x,y,z), polar coordinates move with the point and can change over time. Even though the r coordinate is moving, the position vector r, is measured in the r direction, giving





Velocity



Similar to other coordinate systems, velocity can be determined by taking a time derivative of position,
Since the coordinate system is moving, the time derivative of the unit vector, e_{r}, is not zero. Using a derivation similar to that found in the theory of nt coordinate systems, expressions for the derivatives of the unit radial and unit transverse vectors can be determined as,
The derivative of the e_{θ} unit vector includes a negitve sign due to it changing inward as it moves (inward is a negative rdirection). The chain rule can then be used to express the time derivative of the unit radial vector as
Substituting the above relationships into the velocity equation gives






Acceleration



A time derivative of the velocity will give an expression for the acceleration,
The chain rule can be used to express the time derivative of the unit transverse vector as
By substituting this into the previous equation, and rearranging, gives the acceleration in terms of radial and transverse components,
It is important to note that both r and θ directions have multiple terms. Acceleration in the r and θ direction is not just d^{2}r/dt^{2} and d^{2}θ/dt^{2}, respectively, since the coordinates are moving. 



