Question

What is the relationship between amplitude and period in a simple harmonic oscillator? Is this relationship dependent on damping? Why or why not?

Answer 1

In a simple harmonic oscillator, the relationship between amplitude and period is independent of damping. The period of a simple harmonic oscillator refers to the time taken for one complete oscillation, while the amplitude is the maximum displacement of the oscillator from its equilibrium position.

The relationship between amplitude and period can be understood by considering the equation of motion for a simple harmonic oscillator:

$\ddot{x}+{\omega }^{2}x=0$

where $\ddot{x}$ is the acceleration of the oscillator, $x$ is the displacement from equilibrium, and $\omega$ is the angular frequency.

In the absence of damping, the solution to this equation is given by:

$x(t)=A\cos (\omega t+\varphi )$

where $A$ is the amplitude, $\omega$ is the angular frequency, $t$ is time, and $\varphi$ is the phase constant.

The period, $T$, is the time taken for one complete oscillation. It can be determined by finding the time interval between two consecutive instances when the oscillator has the same displacement and velocity. In other words, it is the time it takes for the cosine function to complete one full cycle. The period is related to the angular frequency by the equation:

$T=\frac{2\pi }{\omega }$

From this relationship, it is evident that the period and the angular frequency are inversely proportional. Therefore, as the period increases, the angular frequency decreases, and vice versa. However, the amplitude, $A$, does not affect the period; it only determines the extent of displacement.

When damping is introduced in the system, the situation becomes more complex. Damping causes energy dissipation, and the amplitude of the oscillations decreases over time. However, the relationship between amplitude and period remains unchanged. The presence of damping affects the rate at which the amplitude decreases, but it does not alter the basic relationship between amplitude and period.

In summary, the relationship between amplitude and period in a simple harmonic oscillator is independent of damping. The period is determined solely by the angular frequency, while the amplitude represents the maximum displacement of the oscillator from its equilibrium position. Damping affects the amplitude but does not alter the fundamental relationship between amplitude and period.