The time taken by a pendulum to complete one oscillation depends on the amplitude (length) and not on the mass or force acting on it because of the fundamental principles governing simple harmonic motion.
A simple pendulum consists of a mass (the bob) attached to a string or rod of negligible mass. When the pendulum is displaced from its equilibrium position and released, it undergoes periodic motion known as simple harmonic motion (SHM).
The time period of a pendulum, which is the time taken to complete one full oscillation (back and forth), is primarily determined by its effective length, also known as the amplitude. The effective length of a pendulum is the distance from the point of suspension (the pivot) to the center of mass of the bob.
The equation that governs the time period of a simple pendulum is:
T = 2π * √(L / g)
Where: T = Time period L = Length of the pendulum g = Acceleration due to gravity
From this equation, it is evident that the time period of a pendulum depends only on its length (amplitude) and the acceleration due to gravity. The mass of the bob or the force acting on it does not appear in the equation.
This behavior can be understood by considering the forces acting on the pendulum during its motion. As the pendulum swings, the force of gravity acts as a restoring force, bringing the pendulum back towards its equilibrium position. The mass of the bob affects the magnitude of the force but does not affect the time taken for the pendulum to swing back and forth.
The reason for this independence of mass is due to the conservation of energy in simple harmonic motion. As the pendulum swings, energy is continuously transferred between kinetic energy (motion) and potential energy (height). The period of the pendulum is determined by the interplay between these energy transfers, which is primarily influenced by the length of the pendulum.
In summary, the time taken by a pendulum to complete one oscillation depends on its length (amplitude) and not on the mass or force acting on it because of the principles of simple harmonic motion, where the length of the pendulum primarily determines the period of oscillation.