# I have a question where I need to derive an expression the minimum velocity of a ball as it is swinging in a circle, at the top of the circle. So far I've reached an equation stating that "gravity...

I have a question where I need to derive an expression the minimum velocity of a ball as it is swinging in a circle, at the top of the circle. So far I've reached an equation stating that "gravity + tension = velocity^2/radius". Now my teacher told me that at the minimum, the tension must equal "something". I have no idea what that "something" is, and because of that I keep getting the answer wrong.

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### 2 Answers

To keep the rope stretched, the centrifugal force at the highest point of the movement must be at least equal to the sum of the weight of the ball and the tension in the rope, so that the equation to solve is:

mg + T = mv^2/r

g + T/m = v^2/r

r(g + T/m) = v^2

v =√[ r(g + T/m)]

This is the expression of the minimum value of speed. See, that is a function of constant parameters for a given situation.

Note that, since the centrifugal force must have a minimum value, and it depends on the speed, the speed must also have a minimum value to make it happen

**So, to keep an object rotating in a vertical plane, it must have a minimum tangential velocity:**

**v =√[ r(g + T/m)]**

Adding one final step to above formulation which is correct.

For minimum velocity, T=0, so substitute T=0 in above formula,

minimum velocity = `sqrt(rg)`