Motion of particle with acceleration $\hat\theta$

Question

Suppose a particle has acceleration $a=-\hat\theta$ , how can I find its trajectory?( for simplicity lets take $v=0$)

I tried to convert $\hat\theta$ into cartesian ,but the expression turned to be large. Then I tried to attempt it in spherical but it was difficult to integrate with a varying $\hat\theta$. Finally I tried to sketch it ,and it seems that it would spiral upwards.

Could anyone give me a hint how to approach this ? Thank you.

Answer 1

As

$$ \vec a = (\ddot r-r\dot\theta^2)\hat r + (r\ddot \theta+2\dot r\dot\theta)\hat\theta $$

the movement is described by

$$ \cases{\ddot r-r\dot\theta^2=0\\ r\ddot \theta+2\dot r\dot\theta=-1 } $$

Follows a solution plot for $\dot r(0) = \dot\theta(0)=0, r(0) = 1, \theta(0)=0$

Answer 2

The acceleration is always perpendicular to the radius $\hat r$. So if the particle starts from rest, you are moving in a circle, faster and faster. $$\ddot\theta=-1\\\theta=-\frac12t^2$$ If the initial velocity is not zero, then the motion is more complicated.