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Let's assume a particle starts on an equilateral triangle of side length "A" with some constant speed u. The particle goes on a logarithmic spiral around the centroid. Find the distance covered by the particle when it completes the first circle.

I am a High school student and while solving a physics question, I thought about it, after searching on the net, I found out that its actually a logarithmic spiral, but couldn't find the right method to derive the distance traveled. I think it could be solved using integration but am not able to apply that to the question.

Jyrki Lahtonen
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Harsh Sharma
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1 Answers1

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In polar coordinates, the logarithmic spiral is described by $r=r_0e^{a\theta}$.

The arc length $ds$ in polar coordinates is given by $$ds=\sqrt{r^2+\left(\frac{dr}{d\theta}\right)^2}d\theta=r_0e^{a\theta}\sqrt{1+a^2}d\theta$$

The length of the path is $$\int_0^{2\pi}ds=r_0\sqrt{1+a^2}\int_0^{2\pi}e^{a\theta}d\theta=r_0\sqrt{1+a^2}\frac{e^{2\pi a}-1}{a}$$

Adam
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Andrei
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