# Swing velocity at highest point

1. Jan 3, 2014

### Slyforce

1. The problem statement, all variables and given/known data
A swing is rotates with the distance 5m to the center of rotation. A full 360 degree rotation is possible.

Find the minimal value of the velocity at the highest point, so that the person operating the swing doesn't fall.

Given data:
r = 5m

2. Relevant equations
Downward accelaration:
x = 0.5 * -9.81 * t^2

3. The attempt at a solution
So that someone never falls from the swing, then the accelaration upward should be 9.81 m/s^2, but at the highest point (x = 0m, y = 5m), there isn't any vertical velocity. Because there is no velocity, the derivative from the vertical velocity at the highest point is 0, meaning that the downward accelaration is greater than the upward accelaration, meaning that the person would fall if not properly secured.

What am I missing here?

2. Jan 3, 2014

### voko

In circular motion, acceleration is always centripetal, that is, directed toward the center.

3. Jan 3, 2014

### Slyforce

Wouldn't that mean that the person in the swing will always fall at the highest point?

4. Jan 3, 2014

### voko

Would not that mean that anything in circular motion will always "fall" into the center?

Does that happen in reality?

5. Jan 3, 2014

### Slyforce

Hmm if the body doesn't move, the centripetal force must be equal to the gravitational force right?

6. Jan 3, 2014

### haruspex

Yes, those two forces will be equal here, but it's not because the body "doesn't move".
Zero velocity does not imply zero acceleration. At the highest point, the vertical velocity will be zero, but as others have posted already, the acceleration will be towards the axis of rotation. (You know the formula for this, right?). The centripetal force is the resultant force necessary to provide that acceleration, I.e. it is the sum of the vertical forces (gravity, tension in the rope).