Solving Equation of Motion for Particle on Sphere's Surface

[delve]

Here's a question from a book: A particle of mass m is constrained to move on the surface of a sphere of radius R by an applied force $$\mathbf{F}(\thetha, \phi)$$. Write the equation of motion.

Now here is the answer, but there is something I don't understand about it:

Using spherical coordinates, we can write the force applied to the particle as $$\mathbf{F}=F_r\mbox{e_r}+F_\theta\mbox{e_\theta}+F_\phi\mbox{e_\phi}$$ But since the particle is constrained to move on the surface of a sphere, there must exist a reaction force that acts on the particle.

Why isn't the force just $$\mathbf{F}=F_\theta\e_\theta+F_phi\e_\phi$$, and therefore, no reaction force?

 

[Bob_for_short]

Why isn't the force just $$\mathbf{F}=F_\theta\e_\theta+F_phi\e_\phi$$, and therefore, no reaction force?

That is right. The radial part of the external force is compensated by the reaction force, so in total the force is tangent to the surface. The equation of motion should concern only theta and phi.

But if they speak only of the external force, it is not equal to zero but F_r. It allows calculating the reaction force (and possible damage to the surface).

 

[delve]

great, thank you for your help :)