The relationship between velocity and radius

Question

I have learned that for an object moving in a circle, there must be an inward force acting upon it in order to cause its inward acceleration. This is sometimes referred to as the centripetal force.

And the law for centripetal force is: $$F = \frac{mv^2}{r}$$

So there is an inverse relationship between the force and radius,and direct proportionality between the force and velocity

And that tells us if the velocity speeds up the force will be stronger and the radius well be smaller.

But as I noticed in the real world when I take a yoyo for example and move it in a circle and I speed up the yoyo, the radius extend..

So why is that happening? What is the relationship between velocity and radius?

Thanks in advance and sorry for my bad English..

Answer 1

And that tells us if the velocity speeds up the force will be stronger and the radius well be smaller.

Only if the (linear) velocity remains the same between the two cases. This can be very hard to arrange if you are swinging a yo-yo on a string.

If may be more helpful in the case of your yo-yo to consider the problem in terms of radius and frequency of rotation $$ F_c = r \omega^2 = \frac{r f^2}{4 \pi^2} \,,$$ where $f$ is the frequency of rotation (in cycle per second, say). The $\omega$ is the "angular frequency" or "angular velocity" and has units of radians per second—it is generally preferred by physicists because radians are the natural unit of angle.

In this form it is clear that if you keep spinning your yo-yo one time each second then a large radius requires a larger force.

---

There is a second thing I'd like to mention in relation to the quote above. You say

if the velocity speeds up

implying that you start under one condition and wonder what happens during a change of condition. The relationship we're talking about here doesn't discuss the evolution of the system under the influence of a tangential acceleration. That requires a little more math.