Why does gravity dictate min.velocity for centripetal force?

[aeromat]

Homework Statement

You are playing with a yo-yo with a mass of 225 g. The full length
of the string is 1.2 m. You decide to see how slowly you can swing
it in a vertical circle while keeping the string fully extended, even
when the yo-yo is at the top of its swing.
(a) Calculate the minimum speed at which you can swing the yo-yo
while keeping it on a circular path.

Homework Equations

Fg = mg
Fc = v^2/r

The Attempt at a Solution

All I want to know is why is it that the "centripetal force has to be exactly equal to the force of gravity" when the object will be at its smallest possible velocity (aka minimum velocity)?

Why is it that gravity is supplying the centripetal force needed for the object to maintain in circular motion at the least possible velocity?

 

[AC130Nav]

Gravity is not supplying the centripetal force (perhaps that term is causing your problem). The centrifugal force must at least equal gravity or your yo-yo will not make it over the top.

 

[SammyS]

Another way to look at it:

Solve for the force the string needs to exert on the yo-yo to make it swing in in a vertical circle. Then realize that at the top of the circle, if the speed is too small, then the string needs to push up on the yo-yo. But unless you have a stiff string, it can't push.