# Nonuniform circular motion

A ball rocks around an arc. In the following illustration, the ball reaches the end of the arc (its velocity magnitude is zero at that particular moment).

Now, I want to know which forces are acting on that ball at that particular moment. We have the tension force $\vec T_2$ acting on the ball, which is the centripetal force. We also have the gravity which consists of two Cartesian components: radial $mg \cos\alpha$ and tangent $mg \sin \alpha$. In the radial axis our net force ($\vec T_2 - mg \cos \alpha = mv^2 / R$) is zero because $v = 0$. However, in the tangent axis our net force is not zero - $mg \sin \alpha$. My question is - how it could be if the ball at that particular moment is not moving because he reaches the edge of his trajectory? If it doesn't move then there should be some opposite force acting on it. My intuition says that that tangent force $mg \sin \alpha$ is forcing the body to slow down in that direction, so it slowly "cancels" the force which caused the body to move initially. But how can I describe it with formulas and/or illustrate it from the point of view of inertial system (i.e., Earth)?

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The instantaneous velocity tells you nothing about the force. $F=ma$. – DilithiumMatrix Mar 13 '13 at 15:00

## 1 Answer

You are right that the force balance is non-zero and that the pendulum-bob is not moving, but this does not mean that the pendulum-bob is not accelerating does it. So, at the moment the 'bob' is still, it is accelerating back to the centre-line of the oscillation with it's maximum absolute velocity.

See here for more information and a nice animation of this phenomenon.

I hope this helps.

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Yes, thanks, I think I got it. It's just the same as in free fall - if i throw a ball vertically, it will reach some max height and will have a zero velocity there, but it is just because he decelerates, and eventually he will start accelerating from there towards the earth. Now when I think of it, my question sounded stupid. Thank you sir. – hsjk04 Mar 13 '13 at 15:13
Exactly. You have got it! – Killercam Mar 13 '13 at 15:13