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:
mdt2d2x+kx=0
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:
Fdamping=−bv=−bdtdx
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:
mdt2d2x+bdtdx+kx=0
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:
x(t)=A0e−2mbtcos(ω′t+ϕ)
Here is what each piece means:
A0 is the initial amplitude.
e−2mbt is the exponential decay factor. As time t increases, this factor shrinks from 1 toward 0. The quantity 2mb is often written as γ (the damping constant) or β.
cos(ω′t+ϕ) is the oscillatory part, with a new angular frequency ω′ that is slightly less than the natural frequency ω0=k/m:
ω′=ω02−(2mb)2
Important
The amplitude of a damped oscillation decays as A(t)=A0e−2mbt. The energy, which is proportional to amplitude squared, decays as E(t)=E0e−mbt.
Three Regimes of Damping
Not all damped systems oscillate. The value of b relative to the critical value bc=2km decides the behaviour:
Regime
Condition
Behaviour
Underdamped
b<2km
Oscillates with decaying amplitude
Critically damped
b=2km
Returns to equilibrium fastest, no oscillation
Overdamped
b>2km
Returns slowly, no oscillation
Note
Critical damping is the sweet spot for things like door closers and car shock absorbers — you want the system to settle to zero as quickly as possible without bouncing.
Damped oscillations are oscillations whose amplitude progressively decreases with time because energy is continuously dissipated (by friction, air resistance, etc.). Undamped oscillations are idealised oscillations in which no energy is lost, so the amplitude remains constant forever. …
Damped oscillations lose amplitude over time to dissipative forces; undamped oscillations keep constant amplitude forever (an idealisation).
Undamped oscillations: In an ideal oscillating system with no resistive or dissipative forces acting on it (no friction, no air resistance, no internal losses), the total mechanical energy remains constant. As a result, the amplitude of oscillation stays exactly the same, cycle after cycle, for all time. Simple harmonic motion as normally analysed (e.g. an ideal spring-mass system in vacuum) is undamped.
Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.
CBSE 2020Set ANNUAL1 markMCQ
Q.When a damped harmonic oscillator completes 100 oscillations, its amplitude is reduced to 1/3 of its initial value. What will be its amplitude when it completes 200 oscillations?
(a) 1/5
(b) 2/3
(c) 1/6
(d) 1/9
›Reveal solutionSolution
Amplitude decays exponentially with time, so if it becomes 1/3 after 100 oscillations, it becomes (1/3)^2 = 1/9 after 200 oscillations (twice as many).
For a damped harmonic oscillator, the amplitude decreases exponentially with time:
A(t) = A0 e^(-bt/2m)
Since the period of oscillation is essentially constant for light damping, the time elapsed is proportional to the number of oscillations completed. So the amplitude after N oscillations can be written as:
Q.In a damped harmonic oscillator, periodic oscillations have ______ amplitude.
(a) gradually increasing
(b) suddenly increasing
(c) suddenly decreasing
(d) gradually decreasing
›Reveal solutionSolution
A damped oscillator loses energy continuously to the resistive/dissipative medium, so its amplitude falls off with time.
In a damped harmonic oscillator, besides the restoring force there is a resistive (damping) force, usually proportional to velocity, which removes mechanical energy from the system as heat. The displacement is of the form