A simple pendulum is one of the most fundamental oscillating systems in physics. It consists of a point mass called a bob, suspended by a massless and inextensible string from a fixed pivot point. The pendulum demonstrates simple harmonic motion when displaced by small angles from its equilibrium position.
To analyze pendulum motion, we must examine the forces acting on the bob and establish geometric relationships. The bob experiences gravitational force mg downward and tension T along the string. The component of weight tangent to the circular path provides the restoring force negative mg sine theta. The arc length s equals L theta, where L is the string length and theta is the angular displacement from vertical.
Using Newton's second law in the tangential direction, we get m times tangential acceleration equals negative mg sine theta. Since tangential acceleration is L times angular acceleration, we obtain the nonlinear differential equation. For small angles, sine theta approximately equals theta, giving us the linearized simple harmonic motion equation with angular frequency omega equals square root of g over L.
The solution to the simple harmonic motion equation is theta equals theta naught cosine omega t plus phi, where omega equals square root of g over L. The period T equals two pi over omega, giving us T equals two pi square root L over g. The frequency is the reciprocal of the period. Notice that the period depends only on the string length and gravitational acceleration, not on the mass or amplitude for small oscillations.
Energy analysis reveals the pendulum's behavior through conservation of mechanical energy. At maximum displacement, all energy is potential, while at equilibrium, energy is purely kinetic. The total mechanical energy remains constant throughout the motion. The phase space diagram shows the relationship between angular position and velocity, forming elliptical trajectories that represent different energy levels.