# Will there be a component of acceleration perpendicular to the surface along which a particle moves?

A particle is sliding in a cone maintaining contact with it all the time. Analyzing the situation in spherical coordinates I was able to find that in the total acceleration of the particle there will be a component along $$\hat{\theta}$$ ,where $$\theta$$ is the polar angle(here $$\theta=\alpha$$ ),the equation is

$$\ddot{r}=\left(\ddot{r}-r \sin \theta\dot{\phi}^{2}\right) \hat{r}$$+$$(2 \dot{r} \dot{\phi} \sin \theta+r \ddot{\phi}) \hat{\phi}-r \dot{\phi}^{2} \cos \theta \hat{\theta}$$

But looking it intuitively I feel that there should be no component along $$\hat{\theta}$$, because any component along $$\hat{\theta}$$ would pull the particle away from the contact with the cone.

Why is my intuition wrong?

• This is just the centripetal acceleration keeping the particle traveling in a circle. – Triatticus Mar 28 at 20:23