Why Does a Smaller Radius Increase Centripetal Force?

[cclixn]

Homework Statement

My textbook states that, "Traveling in a circular path with a smaller radius of curvature requires a greater centripetal force".

But my question is, why, and how is that true? I would have assumed at first that if the radius was getting shorter, then the centripetal force would be decreasing, and not increasing.

Homework Equations

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The Attempt at a Solution

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I would have assumed at first that if the radius was getting shorter, then the centripetal force would be decreasing, and not increasing.

So why exactly, and how does having a shorter radius result in greater centripetal force?

 

[Orodruin]

Force is proportional to acceleration. Assuming the same speed, a smaller circular orbit must have a larger acceleration. As the orbit grows, it becomes straighter and requires less acceleration.

Edit: If you keep the angular frequency fixed instead of velocity, the force would indeed increase with radius. If the quote is all your textbook gives, then it has failed to specify the underlying assumption.