Velocity in plane polar coordinates

AI Thread Summary
The discussion focuses on understanding the relationship between the change in the radial unit vector (Δer) and the angular change (Δθ) in plane polar coordinates. It highlights that the arc length corresponding to an angular change Δθ is given by δ = |er|Δθ, where |er| is the unit magnitude of the radial vector. The argument is made that since Δer is very small, it closely approximates the arc length, leading to the conclusion that Δer is approximately equal to Δθ. This geometric reasoning clarifies the confusion regarding the magnitude of Δer in relation to Δθ. The explanation emphasizes the importance of understanding the small angle approximation in polar coordinates.
marcom
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Hi,
I have a problem with the following explanation of velocity in plane polar coordinates.
30sik39.jpg

I don't understand why the magnitude of Δer is approximately equal to Δθ.

Thanks
 
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You know from basic geometry that the arc of circumference spanned by ##\Delta \theta## is going to be ##\delta=|\hat{e}_r|\Delta\theta##. However, even if ##\Delta e_r## is actually straight, it is very very small and so it will be close in magnitude to the arc of length. Also since ##\hat{e}_r## has unit magnitude, you find ##\Delta e_r\simeq\delta=\Delta\theta##.
 
Thanks!
 
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