We have two identical beads, each with charge q and mass m, on a horizontal frictionless circular hoop of radius a. One bead is glued to the hoop at a fixed position, while the other bead can move freely along the hoop. Due to electrostatic repulsion, the movable bead will find an equilibrium position and perform small oscillations around it.
At equilibrium, the movable bead positions itself diametrically opposite to the fixed bead, maximizing their separation distance to 2a. When the movable bead is displaced by a small angle theta from equilibrium, it experiences a restoring force that tends to bring it back to the equilibrium position.
The electrostatic force between the two charged beads follows Coulomb's law: F equals k q squared over r squared. When the movable bead is displaced by angle theta, the distance r between the beads changes. To analyze the oscillation, we need to find the tangential component of this force, which provides the restoring force for small oscillations.
Let's derive the angular frequency step by step. First, we find the distance r between the beads using geometry. Then we apply Coulomb's law and find the tangential component of the force. For small angles, we use the approximation sine theta over 2 equals theta over 2. Applying Newton's second law in the tangential direction, we get m a theta double dot equals the tangential force. This gives us the equation of simple harmonic motion, and the square of the angular frequency is k q squared over 8 m a cubed.