In the context of a pendulum or any oscillatory system, there are a few reasons why it is desirable for the amplitude of the oscillation to be small:
Linearity: Many oscillatory systems exhibit linear behavior over a small range of amplitudes. This means that the motion of the system follows simple, predictable mathematical relationships. However, as the amplitude increases, the system may become nonlinear, and the relationships between variables may become more complex. By keeping the amplitude small, we can approximate the system's behavior as linear, which simplifies the analysis and makes it easier to study and understand.
Periodic Motion: For small amplitudes, the motion of a simple pendulum or a harmonic oscillator can be approximated as simple harmonic motion (SHM). SHM is characterized by a constant period (the time taken to complete one oscillation) and a sinusoidal shape. This simplifies the mathematical description and analysis of the system. However, for larger amplitudes, the motion deviates from perfect SHM, and the period can change with amplitude, making the system more complex to model.
Energy Conservation: In an ideal oscillatory system without external forces or energy losses, the total mechanical energy (kinetic + potential) of the system remains constant. However, as the amplitude increases, the maximum potential energy and the maximum kinetic energy also increase. This means that the energy swings between higher values, making the system more prone to dissipative forces like air resistance or friction, which can lead to energy losses. By keeping the amplitude small, the energy swings are smaller, minimizing the impact of dissipative forces and helping to maintain a more accurate and stable oscillation.
It's important to note that these considerations are often specific to idealized systems and simplifying assumptions. In practical applications, the amplitude of oscillation may be dictated by the specific requirements or constraints of the system, such as the desired motion range or the physical limitations of the components involved.