A particle moving with zero radial acceleration in polar coordinates

[Leo Liu]

In the example above, the authors claim that when $r=r_0e^{\beta t}$, the radial acceleration of the particle is 0. I don't quite understand it because they did not assume $\beta=\pm \omega$.
Can anyone please explain it to me? Many thanks.

 

[Herman Trivilino]

In the example above, the authors claim that when $r=r_0e^{\beta t}$, the radial acceleration of the particle is 0.

No, they do not make that claim. They claim that the radial acceleration is zero when $\beta=\pm \omega$.

All you have to do is replace $\beta$ with $\pm \omega$ in the expression for a and you'll see that the coefficient of $\hat{r}$ vanishes.

 

[Leo Liu]

No, they do not make that claim. They claim that the radial acceleration is zero when $\beta=\pm \omega$.

All you have to do is replace $\beta$ with $\pm \omega$ in the expression for a and you'll see that the coefficient of $\hat{r}$ vanishes.

I see, thanks.