An elementary proof of the amplitude’s exponential decrease in damped oscillations

Abstract

A damped oscillation is characterized by the diminishing amplitude of an oscillating system resulting from the dissipation of energy. A crucial example of damped oscillations involves a block of mass attached to the end of a linear spring, experiencing a damping force proportional to the object’s velocity and acting in opposition to the direction of its motion. For small values of the damping constant, the amplitude decreases exponentially over time. Typically, this behavior is introduced at the secondary education level without providing a justification. In this paper, a new approach tailored for the secondary education level is introduced to explain the aforementioned exponential decrease without relying on advanced mathematical tools. In addition, utilizing this analysis simplifies the demonstration of why the exponential decrease in amplitude is applicable only for small damping forces. It is also worth noting that the methods used to derive this result can be applied in various areas, including the derivation of the equation that connects the root mean square value of the AC sinusoidal current to its maximum value.