Q.A pendulum oscillates in air, so it experiences air resistance (a damping force). Consider how its total mechanical energy $E$ changes with time. Which of the following $E$-versus-time curves correctly represents this variation?

Damped Oscillations: When Motion Fades

Imagine pushing a child on a swing. You give one big push, then step back. The swing goes high, then lower, then lower still, until eventually it stops. That is a damped oscillation in everyday life. The swing wants to keep swinging forever — that would be an ideal, undamped oscillation — but something is stealing its energy. Air resistance, friction at the pivot, even the slight bending of the ropes all act as a brake.

The key intuition: the system still oscillates, but each swing is a little smaller than the last. The amplitude does not drop suddenly; it shrinks in a smooth, predictable way — exponentially.

The Physics: Where Does the Energy Go?

In an ideal oscillator (like a mass on a spring with no friction), the total mechanical energy is constant. Kinetic energy converts to potential energy and back, forever. The equation of motion is:

where $m$ is mass, $k$ is spring constant, and $x$ is displacement.

Now add a resistive force. The simplest model is a force proportional to velocity, like air drag at low speeds or the friction in a dashpot (a piston in oil). That force is:

where $b$ is the damping coefficient — a positive number that measures how strong the resistive force is. The minus sign means the force always opposes the motion.

Newton's second law then becomes:

That is the damped harmonic oscillator equation. It is the precise statement.

The Solution: Exponential Decay of Amplitude

The solution to this differential equation depends on how strong the damping is. For the most common case — underdamping — the system still oscillates, and the displacement is:

Here is what each piece means:

• $A_0$ is the initial amplitude.
• $e^{-\frac{b}{2m}t}$ is the exponential decay factor. As time $t$ increases, this factor shrinks from 1 toward 0. The quantity $\frac{b}{2m}$ is often written as $\gamma$ (the damping constant) or $\beta$.
• $\cos(\omega' t + \phi)$ is the oscillatory part, with a new angular frequency $\omega'$ that is slightly less than the natural frequency $\omega_0 = \sqrt{k/m}$:

Three Regimes of Damping

Not all damped systems oscillate. The value of $b$ relative to the critical value $b_c = 2\sqrt{km}$ decides the behaviour:

Why Exponential? The Intuition …