Animated Solution for Physics - Oscillations: In forced oscillation of a particle, the amplitude is maximum for a frequency ${\omega }_1$ of the force, while the energy is maximum for a frequency ${\omega }_2$ of the force, then

The phenomenon of resonance is one of the most beautiful and powerful concepts in all of physics. Whether it is the tuning of a radio to catch your favorite station, the shattering of a wine glass by a trained opera singer, or the rhythmic pushing of a child on a playground swing, resonance is the invisible hand maximizing the transfer of energy.

In this problem, we are exploring the behavior of a particle undergoing forced oscillations. Let's dive deep into the mechanics of what happens when an external periodic force dictates the motion of a system.

Imagine you are sitting on a swing, and a friend is pushing you. If your friend pushes you randomly, your swinging motion will be erratic and small. However, if your friend times their pushes perfectly—matching the natural rhythm of the swing—you will go higher and higher.

In physics terms, the swing has a natural frequency (${\omega }_0$). The pushes from your friend represent an external driving force with a driving frequency ($\omega$). When a system is subjected to this external periodic force, it eventually forgets its own initial conditions and begins to oscillate entirely at the driving frequency $\omega$. This state is known as a forced oscillation.

Amplitude Resonance

The amplitude of a forced oscillator is highly sensitive to the driving frequency. If $\omega$ is very small (slow pushes) or very large (rapid, jittery pushes), the amplitude remains relatively small.

However, as the driving frequency $\omega$ approaches the natural frequency ${\omega }_0$, something magical happens. The system becomes incredibly responsive. The amplitude of oscillation swells, reaching a dramatic peak exactly when $\omega ={\omega }_0$. This condition, where the amplitude is maximized, is universally known as amplitude resonance.

According to our problem, the frequency at which the amplitude is maximum is denoted as ${\omega }_1$. Therefore, we can confidently state that:

Energy Resonance

Now, let's think about the energy. The total mechanical energy of an oscillating system is directly tied to its velocity and amplitude. For the system to gain maximum energy, the external force must do the maximum possible work on it.

Work is done most efficiently when the driving force is perfectly in phase with the velocity of the particle. This perfect synchronization also occurs precisely at the natural frequency ${\omega }_0$. At this sweet spot, the power absorbed by the oscillator from the driving force hits its absolute maximum, leading to what we call energy resonance (or velocity resonance).

The problem states that the energy is maximum at a frequency ${\omega }_2$. Thus, we find that:

By analyzing both the amplitude and the energy of the forced oscillator, we see that they are two sides of the same resonant coin. Both the physical displacement (amplitude) and the power transfer (energy) reach their zenith when the external driving frequency perfectly matches the system's inherent natural frequency.

Since both ${\omega }_1$ and ${\omega }_2$ are equal to the natural frequency ${\omega }_0$, it logically follows that: