Does the amplitude of a damped oscillator decay exponentially or linearly?

Exponentially. The amplitude follows A(t) = A e^(−bt/2m), so equal time intervals reduce the amplitude by equal factors, never by equal amounts. The mechanical energy, being proportional to amplitude squared, decays as E(t) = ½kA²e^(−bt/m) — twice as fast in the exponent. A common NEET trap pairs an exponential amplitude with a linear energy decay; both must be exponential.

Is the angular frequency of a damped oscillator larger or smaller than the natural frequency?

Smaller. The damped angular frequency is ω′ = √(k/m − b²/4m²), which is always less than the natural frequency ω₀ = √(k/m) for any non-zero damping b. Damping slows the oscillation slightly and lengthens the period. Only when b = 0 do ω′ and ω₀ coincide.

Why does less damping make resonance sharper?

The steady-state amplitude near resonance is limited only by the damping term. With small b, the amplitude at ω_d = ω₀ rises to a very tall, narrow peak; with large b, the peak is short and broad, and its maximum shifts slightly below ω₀. A lightly damped system therefore responds strongly only in a narrow band of frequencies — that selectivity is exactly what a radio tuner exploits.

Is damped harmonic motion still simple harmonic motion?

Not strictly. NCERT notes that damped motion is only approximately simple harmonic, and only for time intervals much shorter than 2m/b. The presence of the cosine factor cos(ω′t + φ) keeps it oscillatory, but the shrinking amplitude envelope means it never repeats exactly — a true SHM has constant amplitude.

What do underdamping, critical damping and overdamping mean?

They describe how strongly the resistive force opposes motion. Underdamping (small b) lets the system oscillate while the amplitude decays — the case described by ω′. Critical damping returns the system to equilibrium in the shortest time without oscillating. Overdamping (large b) also avoids oscillation but returns the system to equilibrium more slowly, sluggishly. Door closers and analogue meter needles are designed near critical damping.

Damped, Forced Oscillations & Resonance — NEET Physics

Q: What is the difference between forced oscillation and resonance?

Forced oscillation is the general case: a system driven by an external periodic force eventually oscillates at the driving frequency ω_d, not its own natural frequency. Resonance is the special case of forced oscillation in which the driving frequency is tuned close to the natural frequency, ω_d ≈ ω₀, so the steady-state amplitude becomes very large. Resonance is forced oscillation at the right frequency.

Free oscillations and natural frequency

When the moving part of an oscillatory system is displaced from equilibrium and then set free, it oscillates to and fro at a frequency fixed entirely by its own physical parameters. Those are free oscillations, and the frequency at which they happen is the system's natural frequency. For a spring-mass system the natural angular frequency is

$$\omega_0 = \sqrt{\frac{k}{m}},$$

and for a simple pendulum it is $\omega_0 = \sqrt{g/L}$. In an idealised free oscillation no energy leaves the system, so the amplitude stays constant and the motion is genuine simple harmonic motion. That idealisation is the starting point. Two corrections take us to the real world: a resistive force that removes energy (damping), and an external periodic force that supplies energy (driving).

Foundation

The natural frequency comes straight from the restoring law $F=-kx$. If that is not yet automatic, revisit simple harmonic motion and the force law for SHM.

The damping force and its equation

Every oscillating system is normally surrounded by a viscous medium — air, water, oil at a hinge. As the body moves, that medium pushes back. For slow motion this resistive force is, to good approximation, proportional to the speed and directed opposite to the velocity:

$$F_{\text{damp}} = -b\,v = -b\frac{dx}{dt},$$

where $b$ is the positive damping constant, measured in $\text{kg s}^{-1}$. A damped oscillator therefore feels two internal forces: the restoring force $-kx$ and the resistive force $-b\,dx/dt$. Newton's second law gives the defining equation of damped harmonic motion:

$$m\frac{d^2x}{dt^2} + b\frac{dx}{dt} + kx = 0.$$

The middle term is the only newcomer compared with undamped SHM. It is what removes energy from the system, oscillation after oscillation, and dissipates it as heat.

Amplitude decay and reduced frequency

For weak damping the solution of that equation is an oscillation whose amplitude is no longer constant but shrinks exponentially:

$$x(t) = A\,e^{-bt/2m}\cos(\omega' t + \phi).$$

Two pieces deserve attention. The cosine factor keeps the motion oscillatory. The exponential factor $A\,e^{-bt/2m}$ is a steadily shrinking envelope — the effective amplitude at time $t$ is

$$A(t) = A\,e^{-bt/2m}.$$

The angular frequency of the damped oscillation is not the natural frequency. It is the reduced angular frequency

$$\omega' = \sqrt{\frac{k}{m} - \frac{b^2}{4m^2}},$$

which is always smaller than $\omega_0 = \sqrt{k/m}$ for any non-zero $b$. Damping not only bleeds away amplitude, it also slows the oscillation slightly and lengthens the period. As NCERT notes, damped motion is only approximately simple harmonic, and only for time intervals much shorter than $2m/b$.

Figure 1 — Exponentially decaying amplitude. The teal curve is $x(t)=A\,e^{-bt/2m}\cos(\omega' t+\phi)$. The dashed coral curves are the envelopes $\pm A\,e^{-bt/2m}$; the oscillation always touches but never crosses them. Equal time intervals shrink the amplitude by equal factors, not equal amounts.

How the energy decays

Mechanical energy of an oscillator scales with the square of its amplitude, $E \propto A^2$. Replace the constant amplitude of free SHM with the shrinking envelope $A\,e^{-bt/2m}$ and the energy at time $t$ becomes

$$E(t) = \tfrac{1}{2}kA^2 e^{-bt/m}.$$

The squaring of the amplitude doubles the exponent: amplitude carries $e^{-bt/2m}$, energy carries $e^{-bt/m}$. Energy therefore drains twice as fast in the exponent as the amplitude does. This single fact is one of the most heavily examined points in the topic.

Worked example

A damped oscillator of mass $m$ and damping constant $b$ is released. After what time does (a) the amplitude fall to half its initial value, and (b) the mechanical energy fall to half its initial value?

(a) Amplitude. Set $A\,e^{-bt/2m} = A/2$. Then $e^{-bt/2m} = \tfrac12$, so $\dfrac{bt}{2m} = \ln 2$ and $t_{1/2}^{\,A} = \dfrac{2m\ln 2}{b}$.

(b) Energy. Set $\tfrac12 kA^2 e^{-bt/m} = \tfrac12\!\left(\tfrac12 kA^2\right)$. Then $e^{-bt/m} = \tfrac12$, so $\dfrac{bt}{m} = \ln 2$ and $t_{1/2}^{\,E} = \dfrac{m\ln 2}{b}$.

Observation. The energy halves in exactly half the time the amplitude takes to halve, $t_{1/2}^{\,E} = \tfrac12\, t_{1/2}^{\,A}$ — a direct consequence of the doubled exponent.

Under, critical and over damping

The size of $b$ relative to $\sqrt{km}$ decides whether the system oscillates at all. There are three regimes, distinguished by the sign of the quantity under the root in $\omega'$.

Regime	Condition	Behaviour	Example
Underdamping	$b^2 < 4mk$ (small $b$)	System oscillates with frequency $\omega'$; amplitude decays as $e^{-bt/2m}$	Pendulum in air; struck guitar string
Critical damping	$b^2 = 4mk$	No oscillation; returns to equilibrium in the shortest possible time	Door closers; analogue galvanometer needle
Overdamping	$b^2 > 4mk$ (large $b$)	No oscillation; returns to equilibrium slowly, sluggishly	A spoon pulled through thick honey

Figure 2 — Three damping regimes. Each curve starts from the same initial displacement and returns toward equilibrium. The underdamped (teal) case overshoots and oscillates; the critically damped (purple) case returns fastest without overshoot; the overdamped (amber) case returns slowest. Critical damping is the boundary that gives the quickest settle.

Forced (driven) oscillations

Left alone, a damped oscillator eventually stops. To keep it going we apply an external periodic force, say $F_0\cos\omega_d t$, where $\omega_d$ is the driving frequency imposed from outside. The system now does something distinctive: after the brief transient settles, it oscillates not at its own natural frequency $\omega_0$, nor at $\omega'$, but at the driving frequency $\omega_d$ of whatever is pushing it. This is a forced oscillation.

The system that has the oscillations impressed on it is the driven; the agency supplying the periodic force is the driver. In NIOS Activity 13.3, a heavy pendulum B set swinging forces three neighbouring pendulums C, D and A to oscillate — and C and D, despite their different lengths, are compelled to swing at B's frequency, not their own.

The crucial result is that the steady-state amplitude of the driven body depends on how close the driving frequency is to the natural frequency — on the difference $(\omega_d - \omega_0)$. Drive far from $\omega_0$ and the response is small. Drive near $\omega_0$ and the amplitude swells. That swelling is resonance.

Resonance and its sharpness

Resonance is the special case of forced oscillation in which the driving frequency is tuned close to the natural frequency, $\omega_d \approx \omega_0$. In NIOS Activity 13.3 it is pendulum A — the one whose natural frequency matches the driver B — that builds up a large amplitude while the others stay modest. At resonance the driver and the driven reinforce each other in step, and the amplitude reaches its maximum.

How tall and how narrow that resonance peak is depends on the damping. With small damping the peak is very tall and very narrow: the system responds violently, but only over a thin band of frequencies. With heavy damping the peak is short and broad, and its maximum slips slightly below $\omega_0$. Less damping therefore means a sharper resonance.

Figure 3 — Resonance peak sharpens with less damping. Steady-state amplitude $A$ versus driving frequency $\omega_d$. Each curve peaks near the natural frequency $\omega_0$. Small damping (teal) gives a tall, narrow peak; large damping (amber) gives a short, broad peak whose maximum drifts slightly below $\omega_0$.

Everyday resonance

Resonance is not an abstraction; it governs ordinary experience. A child on a swing is a driven oscillator: pushing once every period — that is, at the swing's natural frequency — builds a large amplitude, while pushing out of rhythm achieves little. A radio tuner is an electrical oscillator whose frequency you adjust until it matches the broadcast frequency; at that match resonance lets the antenna pick out one station from many. Singers can shatter a glass by sounding the note that matches the glass's natural frequency.

The same phenomenon turns destructive at scale. The Tacoma Narrows suspension bridge in Washington collapsed in 1940 when wind gusting at a frequency near the bridge's natural frequency drove it to ever-larger sway until the structure failed. For the same reason soldiers are ordered to break step while crossing a bridge: a column marching in cadence could drive the bridge at its natural frequency. Factory chimneys and cooling towers have likewise been set oscillating, and damaged, by wind.

Free vs damped vs forced

The three oscillation types differ in one question: what is the energy doing? Free oscillation conserves it, damped oscillation loses it, forced oscillation has it replenished from outside.

Feature	Free	Damped	Forced
External periodic force	None	None	Present, $F_0\cos\omega_d t$
Resistive force	Ignored	$-b\,v$ present	Usually present too
Amplitude with time	Constant	Decays as $e^{-bt/2m}$	Constant in steady state
Energy with time	Conserved	Decays as $e^{-bt/m}$	Steady (input balances loss)
Oscillation frequency	$\omega_0=\sqrt{k/m}$	$\omega'=\sqrt{k/m-b^2/4m^2}$	$\omega_d$ (the driver's)
Largest response	Set by initial push	Always shrinking	Resonance at $\omega_d\approx\omega_0$

Related drill

The energy bookkeeping behind $E\propto A^2$ is built in energy in SHM — the same $\tfrac12 kA^2$ total that the exponential factor erodes.

Quick recap

Damped, forced and resonance in one breath

Free oscillation: no damping, no driver. Constant amplitude, $\omega_0=\sqrt{k/m}$.
Damping force $F=-bv$ gives $m\ddot x + b\dot x + kx = 0$, solved by $x=A\,e^{-bt/2m}\cos(\omega' t+\phi)$.
Reduced frequency $\omega'=\sqrt{k/m-b^2/4m^2} < \omega_0$. Damping lengthens the period.
Amplitude decays as $e^{-bt/2m}$; energy decays twice as fast, $E=\tfrac12 kA^2 e^{-bt/m}$.
Underdamped oscillates; critical damping settles fastest; overdamped settles slowly.
Forced oscillation: the driven body ends up oscillating at the driver's frequency $\omega_d$.
Resonance: amplitude is maximum when $\omega_d\approx\omega_0$; the peak is sharper for smaller damping.

Damped, Forced Oscillations & Resonance