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Consider the spiral motion of a particle such that the distance from the origin to the particle is

r(t) = e^t

with a constant angular velocity.

Now since the particle rotating with a constant angular velocity, I would think that the net force on the particle would have to be an inward centripetal force in the radial direction.

However, there must also be a net outward force, in the radial direction, to account for the accelerating distance between the origin and the particle.

Obviously this can't be right, but I can't pin down where I'm going wrong.

Cheers!