406-L: Vertical Circles 1: Constant Speed

These problems involve a person swinging a mass vertically. Vertical circles are good practice for understanding the nature of the centripetal force, because in a vertical circle, the centripetal force is not always equal to the same force. For practice, note that these problems are very similar to elevator problems.

Principles

Centripetal Force is the summation of forces in the inward direction on something moving in a circle.
Centripetal force must be identified as one or more otherwise existing forces, such as gravity, tension, normal force, or friction.

The magnitude of centripetal force:

For an object to move in a circle at a constant speed, the magnitude of centripetal force must equal the value:

F_c = \frac{m v^2}{r}

Forces

In a vertical circle, there are only two forces acting on the mass:

Gravity:
- Gravity always has direction down
- Gravity always has a magnitude given by \( F_g = mg \).
Tension:
- Tension always has the direction inwards, towards the center of the circle.
- Tension is a constraint force. The magnitude of tension will adjust to the situation. You must determine the net force acting on the object first, then determine the tension.