Circular Motion

Step 1: Mechanical Energy in a Vertical Circle.
For a particle revolving in a vertical circle, the mechanical energy is conserved. The total energy at any position is the sum of kinetic energy and potential energy. At the top-most position, the tension in the string is zero (just before slackening). The forces acting on the particle are gravitational force and centripetal force.
Step 2: Lowest Position Velocity.
At the lowest position, the total energy is purely kinetic: The potential energy at the lowest position is zero, so the total energy is just the kinetic energy.
Step 3: Mid-Way Position Velocity.
At the mid-way position, the energy is split between kinetic and potential energy. We can derive the velocity by considering the work-energy theorem.
Step 4: Top Position Velocity.
At the top-most position, the velocity is derived using the balance of forces. For circular motion to continue, the centripetal force must be provided by the tension and gravity. The condition for slackening is: Thus:

Centripetal force is the force acting on an object moving along a circular path, directed towards the center of the circle. It is calculated as: where represents the mass, is the velocity, and is the radius of the circular path. \bigskip