Why don't we utilize centripetal acceleration for particle accelerators or ion thrusters?

Question

Acceleration of an object in a circle is determined by its radius (r) and speed (v).

Acceleration = (v²)/r

So, acceleration is greater at high speeds and in sharp curves (smaller radius).

Centripetal acceleration is equal to negative of v^2/r but the minus sign here denotes the radial direction from the centre of the circle outwards. If we flip the direction of travel to be inwards then would this make it equal to the postive of v^2/r (i.e. increasing acceleration)?

Can we not increase the speed of particles or thrust from ejecting ions into space by somehow designing these in a way that would force them to move through a spiral that has a decreasing radius? That way the acceleration of the particles or ions would increase as they travel through the spiral (as it decreases in radius).

I have no formal physics training so I would appreciate a simple answer.

Edit: As per the comments and this question* mentioning angular momentum increasing as radius decreases (where L=rmv where L is angular momentum is r radius, m is mass and v is velocity).

You'd have to physically create the spiral that the particles or ions would travel through. Then the particles or ions would have to be pushed into the spiral (by some initial force). What would happen next? Would they increase in velocity as they travel through the spiral until they exit? Could you utilize this increase in velocity? What happens if you attach a spiraling exit pipe to an ion thruster? Why is this effect not utilized in a cyclotron particle accelerator where the radius decreases rather than maintaining a constant?

*Wouldn't decreasing the radius that an object revolves around, increase its velocity to infinity

Answer 1

Centripetal force is a calculation of the net force needed to keep an object moving along a circle of constant radius. It is not a force itself, but the sum of forces applied. If there are radial and tangential forces, centripetal force refers only to the radial component. If the net radial force is inward and larger than $v^2/r$, the object will spiral in to a smaller radius. If the radial force is smaller, then the object will spiral out. If this radial force is zero, then the object will continue in a straight line. Note that the velocity in $v^2/r$ is only the tangential component of velocity.

The radial force cannot affect velocity perpendicular to it, so the "centripetal force" does not directly affect tangential velocity. As the object spirals in, then some speed is added in the radial direction. The object will end up on an essentially elliptical path, oscillating inward and outward. To regain a circular path without spiraling back out, a temporary "backward" tangential force is necessary at the small radius to slow it down so it cannot spiral back out. This is how tangential velocity is decreased to compensate for the decreased radius in $v^2/r$.