The maximum displacement of a pendulum about its equilibrium position is the same for both sides because of the nature of the restoring force acting on the pendulum.
In a simple pendulum, the restoring force is proportional to the displacement from the equilibrium position and acts in the opposite direction. According to Hooke's Law, the restoring force is given by F = -kx, where F is the restoring force, k is the spring constant, and x is the displacement from the equilibrium position.
When the pendulum is displaced to one side, it experiences a force pulling it back towards the equilibrium position. This force causes the pendulum to accelerate and eventually reaches its maximum displacement on that side. However, the force continues to act in the opposite direction, causing the pendulum to decelerate and start moving back towards the equilibrium position.
As the pendulum passes through the equilibrium position, the force changes direction and now acts in the opposite direction, causing the pendulum to decelerate and eventually reach its maximum displacement on the other side. The same process repeats, resulting in a sinusoidal motion.
Due to the symmetry of the pendulum and the restoring force acting in the opposite direction for both sides, the maximum displacement on one side is equal to the maximum displacement on the other side. This symmetry is what gives rise to the sinusoidal nature of the pendulum's motion.