Arc length in polar coordinates

[Gianmarco]

I know that an arc length in polar coordinates can be computed by integrating $$\int ds$$ using the formula $ds=\sqrt{\rho^2 + \frac{dr}{d\theta}^2}d\theta$. But, seeing that $s=\rho\theta$ and $ds = \rho d\theta$, why is it wrong to calculate arc lengths with this expression for $ds$?

 

[andrewkirk]

The formula ds=ρdθ is not correct for arc length. It assigns a zero length to any radial line, such as the line from (in Cartesian coordinates) (0,0) to (0,1).

 

[Gianmarco]

The formula ds=ρdθ is not correct for arc length. It assigns a zero length to any radial line, such as the line from (in Cartesian coordinates) (0,0) to (0,1).

Right. Thanks

 

[mfig]

The case of motion along a circular arc is a limiting case. Take your expression

$ds=\sqrt{\rho^2 + \frac{dr}{d\theta}^2}d\theta$

and ask what happens if we confine motion to a circular arc. In that specific case, $\frac{dr}{d\theta} = 0$ and you see the familiar formula for the length of arc along a circular arc emerges, $ds = \rho d\theta$.