Resonance — mechanical and electrical, conditions and applications

Question

What is resonance, when does it occur in mechanical and electrical systems, and what are the conditions and practical applications?

Solution — Step by Step

Resonance occurs when a system is driven at a frequency equal to (or very close to) its natural frequency. At resonance, the system absorbs maximum energy from the driving force, and the amplitude of oscillation becomes maximum.

The natural frequency is the frequency at which the system would oscillate if disturbed and left alone (no driving force, no damping).

For a mass-spring system or a pendulum driven by a periodic force:

Natural frequency: $f_0 = \frac{1}{2\pi}\sqrt{\frac{k}{m}}$ (spring-mass)

At resonance ( $f_{driving} = f_0$ ):

Amplitude reaches maximum
The system oscillates with a phase lag of $\pi/2$ (90 degrees) behind the driving force
Energy transfer is most efficient

Damping effect: With damping, the resonance peak broadens and the maximum amplitude decreases. The resonance frequency shifts slightly below $f_0$ .

In a series LCR circuit driven by an AC source:

Resonance condition: $X_L = X_C$

\omega L = \frac{1}{\omega C}

\omega_0 = \frac{1}{\sqrt{LC}} \qquad f_0 = \frac{1}{2\pi\sqrt{LC}}

At resonance:

Impedance is minimum: $Z = R$ (purely resistive)
Current is maximum: $I_{max} = \varepsilon_0 / R$
Voltage across L and C are equal and opposite (they cancel)
Power factor = 1 (purely resistive behaviour)

Q = \frac{\omega_0 L}{R} = \frac{1}{R}\sqrt{\frac{L}{C}}

Higher $Q$ means sharper resonance peak, more selective tuning.

Feature	Mechanical	Electrical (LCR)
Natural frequency	$\frac{1}{2\pi}\sqrt{k/m}$	$\frac{1}{2\pi\sqrt{LC}}$
Condition	$f_{drive} = f_0$	$X_L = X_C$
Damping element	Friction / viscosity	Resistance $R$
At resonance	Max amplitude	Max current
Sharpness controlled by	Damping coefficient	$Q$ factor (lower $R$ = sharper)

flowchart TD
    A["Resonance"] --> B["Mechanical"]
    A --> C["Electrical"]
    B --> D["Condition: f_drive = f_natural"]
    B --> E["Result: maximum amplitude"]
    C --> F["Condition: XL = XC"]
    C --> G["Result: maximum current, Z = R"]
    B --> H["Damping: friction reduces peak, broadens curve"]
    C --> I["Damping: R reduces peak, broadens curve"]
    C --> J["Q factor = omega0 L / R"]

Why This Works

Resonance is energy matching. At the natural frequency, the driving force is perfectly in sync with the system’s tendency to oscillate. Energy pumped in during each cycle adds constructively to the existing oscillation, like pushing a child on a swing at just the right moment.

In LCR circuits, at resonance the energy stored in the inductor’s magnetic field and the capacitor’s electric field oscillate back and forth with zero net reactance — so the source only needs to supply energy to overcome the resistance.

Alternative Method

For finding the resonance frequency of an LCR circuit, use the impedance minimization approach. Since $Z = \sqrt{R^2 + (X_L - X_C)^2}$ , $Z$ is minimum when $X_L = X_C$ . This directly gives the resonance condition without needing to remember a separate formula.

Common Mistake

Students often think the voltage across the capacitor or inductor is maximum AT resonance. While the current is maximum at resonance, the voltage across $C$ is maximum at a frequency slightly below $\omega_0$ , and the voltage across $L$ is maximum slightly above $\omega_0$ . For most CBSE and NEET problems this distinction does not matter, but JEE Advanced has tested it.

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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