When the amplitude of a system undergoing simple harmonic motion (SHM) is large, the departures from ideal SHM can become substantial due to a few reasons:
Nonlinear Effects: Simple harmonic motion assumes a linear relationship between the restoring force and the displacement of the system. However, in real-world situations, the restoring force may not strictly follow a linear relationship at large amplitudes. Nonlinear effects can arise due to factors like stretching, compression, or other material properties of the system. These nonlinearities can cause deviations from the idealized sinusoidal motion of SHM.
Higher-Order Terms: The equation governing SHM, such as for a mass-spring system or a pendulum, is based on approximations using small-angle approximations or Hooke's law. These approximations assume that the motion remains within a small range around the equilibrium position or small angular displacements. When the amplitude becomes large, these higher-order terms that were neglected in the simplifications become more significant, leading to departures from SHM.
Energy Considerations: In simple harmonic motion, the total mechanical energy of the system should ideally remain constant. However, when the amplitude is large, energy considerations become more crucial. The system may lose energy to factors like air resistance, friction, or other dissipative forces, leading to a gradual decrease in amplitude and deviations from the idealized behavior of SHM.
System-Specific Factors: The departures from SHM can also depend on the specific characteristics of the system under consideration. For example, in a simple pendulum, the assumption of small angles breaks down at large amplitudes, resulting in deviations from SHM.
In summary, large amplitudes in simple harmonic motion can lead to substantial departures from ideal SHM due to nonlinear effects, neglected higher-order terms, energy considerations, and system-specific factors. These factors introduce complexities that cause deviations from the simple, idealized sinusoidal motion.