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Waves and Oscillations
Sharpness of Resonance and Quality Factor
The sharpness of resonance quantifies how rapidly the amplitude or energy of a driven damped oscillator drops as the driving frequency deviates from the natural frequency. It is inversely proportional to the damping coefficient and directly proportional to the quality factor Q.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Definition of resonance, condition for amplitude resonance, effect of damping on resonance curves, and fundamental frequency matching.
PGT extension
Rigorous derivation of amplitude and velocity resonance frequencies, half-power bandwidth, quality factor Q, phase lag variation, and sharpness calculations in LCR circuits and mechanical systems.
Prerequisites
Damped harmonic motion and differential equations
Forced oscillations and driving forces
Complex numbers and phasor representation
Amplitude Resonance and Frequency
In a forced damped oscillator governed by m(d²x/dt²) + r(dx/dt) + s$x = F_0 \cos(\omegat)$, the steady-state amplitude depends on the driving frequency ω.
The amplitude resonance frequency ω_r is given by ω_$r = \omega_0^{2} - 2(\gamma/m)^{2}$, where ω_0 is the undamped natural frequency and $\gamma = r/2m$ is the damping factor.
For very light damping (γ << ω_0), the amplitude resonance frequency approximately equals the natural frequency ω_0.
Sharpness and Quality Factor Q
The sharpness of resonance is measured by the Quality Factor (Q), defined as $Q = \omega_r / (\omega_2 - \omega_1)$, where (ω_2 - ω_1) is the full width at half maximum (bandwidth) of the power-frequency curve.
Higher Q-factor implies lower damping, leading to a narrower resonance peak and a sharper resonance.
Q can also be expressed as 2π (Energy stored / Energy dissipated per cycle).
Phase Relations at Resonance
At exact resonance ($\omega = \omega_0)$, the phase difference between the driving force and the displacement is exactly π/2 radians (90°).
Below resonance (ω < ω_0), the oscillator is stiffness-controlled and the phase lag approaches 0°.
Above resonance (ω > ω_0), the oscillator is mass-controlled and the phase lag approaches π radians (180°).
Velocity Resonance vs Amplitude Resonance
Velocity resonance occurs strictly when the driving frequency equals the natural frequency ($\omega = \omega_0)$, independent of the damping coefficient.
At velocity resonance, the impedance of the system is purely resistive, and the velocity is in phase with the driving force.
Amplitude resonance occurs at a slightly lower frequency than velocity resonance due to the frequency-dependent stiffness term.
Derivations
Derivation of Resonance Amplitude
Conditions: Steady-state forced oscillations with damping coefficient r and driving force F_0 cos(ωt)
Write the equation of motion: m(d²x/dt²) + r(dx/dt) + s$x = F_0 \cos(\omegat)$.
Assume a steady-state solution x(t) = A cos(ωt - φ), substitute into the equation, and equate coefficients of sin(ωt) and cos(ωt).
Solve for the amplitude A to obtain $A = (F_0/m) / \sqrt((\omega_0^2 - \omega^2)^2 + 4gamma^2omega^2)$, where $\gamma = r/2m$ and ω_0^2 = s/m$.
Differentiate A with respect to ω and set dA/d$\omega = 0$ to find the frequency of maximum amplitude: ω_$r = \sqrt(\omega_0^2 - 2gamma^2)$.
$v_{max} = maximum velocity amplitude at \omega = \omega_0$, r = damping coefficient m s^-1
At exact natural frequency $\omega = \omega_0$
Series LCR Circuit Impedance at Resonance
$$Z_0 = R$$
R = resistance in the circuit Ω (ohms)
ω$L = 1/(\omega C)$
LCR Circuit Quality Factor
$$Q = \frac{1}{R}\sqrt{\frac{L}{C}}$$
L = inductance, C = capacitance, R = resistance Dimensionless
Series resonance configuration
Graphs and experiments
Resonance Curves for Different Damping Levels
Draw: A plot of Amplitude A versus Driving Frequency ω featuring three distinct curves for low, medium, and high damping (γ_1 < γ_2 < γ_3). Show sharp peaks at low damping and broader, lower peaks shifted to the left for higher damping.
Interpret: As damping increases, the peak height decreases, the resonance frequency shifts to lower values, and the resonance curve becomes wider (less sharp).
Exam use: Frequently asked in MCQs to identify which curve corresponds to minimum damping or highest Q-factor.
Phase Lag vs Driving Frequency
Draw: A sigmoidal curve plotting phase φ against ω/ω_0. The curve starts at 0° at low frequencies, passes through exactly 90° (π/2) at $\omega = \omega_0$, and approaches 180° (π) at high frequencies.
Interpret: Demonstrates the transition from stiffness control to mass control, with the steepest slope occurring for small damping values.
Exam use: Tested via conceptual questions regarding the phase relationship at resonance.
Draw: Experimental setup showing a tuning fork placed over a vertical water column tube, noting resonance lengths l_1 and l_2 for fundamental and overtone frequencies.
Interpret: Allows determination of sound velocity in air and observation of sharp acoustic resonance peaks when air column length matches quarter-wavelength multiples.
Exam use: Laboratory physics practical questions and numericals on end corrections.
Solved numericals
TGT core
A particle of mass 0.5 kg executes forced oscillations under a periodic force. If the natural frequency is 10 rad/s and the damping constant is 0.2 kg/s, calculate the quality factor Q of the system.
Identify given parameters: m = 0.5 kg, ω_$0 = 10 rad/s$, $r = 0.2 kg/s$.
Recall the formula for Q-factor in terms of mass, natural frequency, and damping coefficient: $Q = (\omega_0 m) / r$.
An LCR series circuit has inductance L = 2 H, capacitance $C = 2 \muF$, and resistance $R = 10 \Omega$. Calculate the resonance frequency, the quality factor Q, and the half-power bandwidth in Hz.
A mechanical oscillator has a natural frequency of 200 rad/s and a bandwidth of 4 rad/s. If the driving force amplitude is 10 N and the mass is 1 kg, find the maximum amplitude at resonance.
Identify given parameters: ω_$0 = 200 rad/s$, Δ$\omega = 4 rad/s$, $F_0 = 10 N$, m = 1 kg.
Find the damping constant r using bandwidth: Δ$\omega = r / m \Rightarrow r = m \times \Delta\omega = 1 \times 4 = 4 kg/s$.
Use the maximum amplitude formula at resonance: $A_{max} = F_0 / (2mgammasqrt{\omega_0^2 - \gamma^2})$. Since γ << ω_0, approximate A_{max} approx F_0 / (2mgammaomega_0) or use exact formula $A_{max} = 10 / (2 \times 1 \times 2 \times \sqrt{200^2 - 2^2}) \approx 10 / (4 \times 200) = 10 / 800 = 0.0125 m$.
Answer: 0.0125 m (or 1.25 cm)
Assumptions and limitations
Linearity assumption: Restoring force is strictly proportional to displacement (Hooke's law) and damping force is linear with velocity ($F_d = -r v)$.
Steady-state condition: Transient terms have completely decayed, leaving only the steady forced oscillation response.
Light damping approximation (γ << ω_0) is frequently invoked for analytical simplification of resonance width and Q-factor relations.
Common misconceptions
Common error: Amplitude resonance occurs exactly at the natural frequency ω_0 for all damped systems.
Correct: Amplitude resonance occurs at ω_$r = \sqrt(\omega_0^2 - 2gamma^2)$, which is slightly lower than ω_0.
The damping term introduces a frequency shift in the denominator of the amplitude response function.
Common error: A higher Q-factor means a wider resonance peak covering a broader frequency range.
Correct: A higher Q-factor means a narrower, sharper resonance peak with a smaller bandwidth.
Q is inversely proportional to bandwidth ($Q = \omega_0 / \Delta\omega)$.
Common error: At resonance, the phase difference between force and displacement is zero.
Correct: At resonance ($\omega = \omega_0)$, the phase difference is exactly 90° (π/2 radians).
Displacement lags velocity by 90° and velocity is in phase with force at resonance, making displacement lag force by 90°.
Quick revision
Sharpness of resonance is directly quantified by the Quality Factor $Q = \omega_0 / \Delta\omega$.
Amplitude resonance frequency is ω_$r = \sqrt(\omega_0^2 - 2gamma^2)$, whereas velocity resonance occurs strictly at ω_ = ω_0.
At resonance, phase lag between driving force and displacement is π/2.
Half-power bandwidth is equal to 2γ or r/m for light damping.
Must remember
The amplitude resonance frequency is given by $\omega_r = \sqrt{\omega_0^2 - 2\gamma^2}$, which is slightly lower than the natural frequency $\omega_0$.
Velocity resonance occurs strictly at $\omega = \omega_0$ independent of the damping coefficient, where impedance is purely resistive.
The Quality Factor $Q$ measures the sharpness of resonance and is defined as $Q = \frac{\omega_0}{2\gamma} = \frac{\omega_r}{\omega_2 - \omega_1}$.
At exact resonance ($\omega = \omega_0$), the phase difference between the driving force and displacement is exactly $\pi/2$ radians ($90^\circ$).
For a series LCR circuit, the quality factor is given by $Q = \frac{1}{R}\sqrt{\frac{L}{C}}$ and bandwidth is $\Delta\omega = \frac{R}{L}$.
Glossary
Quality Factor ($Q$)A dimensionless parameter that characterizes the sharpness of resonance of an oscillator or a resonant circuit, equal to $2\pi \times (\text{Energy stored} / \text{Energy dissipated per cycle})$.
Damping Factor ($\gamma$)The damping coefficient per unit mass, expressed as $\gamma = r / (2m)$.
Half-Power Bandwidth ($\Delta\omega$)The frequency interval $\omega_2 - \omega_1$ between the two half-power points where the power drops to half of its maximum value.
Amplitude Resonance Frequency ($\omega_r$)The frequency at which the steady-state displacement amplitude of a forced damped oscillator is maximum.
Velocity Resonance FrequencyThe frequency at which the velocity amplitude is maximum, occurring strictly at the natural frequency $\omega_0$.
Amplitude Resonance vs Velocity Resonance
Feature
Amplitude Resonance
Velocity Resonance
Condition for Peak
$\omega_r = \sqrt{\omega_0^2 - 2\gamma^2}$
$\omega = \omega_0$
Damping Dependency
Depends on damping $\gamma$
Independent of damping
Impedance State
Reactance is non-zero
Purely resistive (reactances cancel)
Phase Lag with Force
Depends on frequency (not $\pi/2$ at peak)
Exactly $\pi/2$ ($90^\circ$)
Practice Quiz
Choose an option to reveal the explanation.
1. What is the condition for amplitude resonance frequency $\omega_r$ in a forced damped oscillator?
Explanation: The amplitude resonance frequency accounts for damping and is given by $\omega_r = \sqrt{\omega_0^2 - 2\gamma^2}$, making it slightly lower than the undamped natural frequency $\omega_0$.
2. At velocity resonance, the frequency of the driving force is equal to:
Explanation: Velocity resonance occurs strictly when the driving frequency equals the natural frequency ($\omega = \omega_0$), independent of the damping coefficient.
3. What is the phase difference between the driving force and the displacement at exact resonance (\(\omega = \omega_0\))?
Explanation: At exact resonance ($\omega = \omega_0$), the velocity is in phase with the driving force, and since displacement lags velocity by $\pi/2$, the displacement lags the driving force by exactly $\pi/2$ radians ($90^\circ$).
4. How is the Quality Factor $Q$ related to the half-power bandwidth (\(\Delta\omega = \omega_2 - \omega_1\)) and resonance frequency?
Explanation: The Quality Factor is defined as the ratio of the resonance frequency to the full width at half maximum (bandwidth): $Q = \frac{\omega_0}{\Delta\omega}$.
5. For a series LCR circuit, what is the expression for the Quality Factor $Q$?
Explanation: The Quality Factor of a series LCR circuit is given by $Q = \frac{\omega_0 L}{R} = \frac{1}{R}\sqrt{\frac{L}{C}}$.
6. A higher Q-factor in a resonant system implies:
Explanation: Higher Q-factor implies lower damping, leading to a narrower resonance peak and a sharper resonance with a smaller bandwidth.
7. Below resonance ($\omega < \omega_0$), the forced oscillator is predominantly:
Explanation: Below resonance, the stiffness term dominates the denominator of the response, making the system stiffness-controlled with phase lag approaching $0^\circ$.
8. Above resonance ($\omega > \omega_0$), the forced oscillator is predominantly:
Explanation: Above resonance, the inertial mass term dominates, making the system mass-controlled with phase lag approaching $\pi$ ($180^\circ$).
9. What is the half-power bandwidth (\(\Delta\omega\)) in terms of the damping factor $\gamma$ for light damping?
Explanation: Under the light damping approximation, the half-power bandwidth is $\Delta\omega = \omega_2 - \omega_1 = \frac{r}{m} = 2\gamma$.
10. At velocity resonance, what is the impedance of a series LCR circuit?
Explanation: At velocity resonance ($\omega = \omega_0$), the inductive and capacitive reactances cancel out (${\omega L = 1/\omega C}$), making the impedance purely resistive ($Z_0 = R$).
11. What is the maximum velocity amplitude ($v_{max}$) at velocity resonance for a driving force of amplitude $F_0$ and damping coefficient $r$?
Explanation: At velocity resonance, the velocity amplitude reaches its maximum value given by $v_{max} = \frac{F_0}{r}$.
12. Which of the following correctly defines the Quality Factor in terms of energy?
Explanation: The Quality Factor is fundamentally defined as $Q = 2\pi \left(\frac{\text{Energy stored}}{\text{Energy dissipated per cycle}}\right)$.
13. What happens to the sharpness of resonance as the damping coefficient $r$ increases?
Explanation: Increasing damping increases energy dissipation, lowers the Q-factor, broadens the bandwidth, and makes the resonance peak wider and less sharp.
14. What is the unit of the Quality Factor $Q$?
Explanation: Quality Factor $Q$ is a dimensionless parameter representing the ratio of energies or frequencies.
15. For very light damping (\(\gamma \ll \omega_0\)), what can be said about the amplitude resonance frequency $\omega_r$?
Explanation: When damping is extremely light ($\gamma \ll \omega_0$), the correction term $2\gamma^2$ becomes negligible, so $\omega_r = \sqrt{\omega_0^2 - 2\gamma^2} \approx \omega_0$.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. In a forced damped mechanical oscillator, the amplitude resonance frequency $\omega_r$ is given by:
Explanation: The derivation for steady-state amplitude maximum yields $\omega_r = \sqrt{\omega_0^2 - 2\gamma^2}$.
2. The Quality Factor $Q$ of a series LCR circuit with inductance $L$, capacitance $C$, and resistance $R$ is expressed as:
Explanation: By substituting circuit parameters into the general Q definition ($Q = \frac{\omega_0 L}{R}$), we get $Q = \frac{1}{R}\sqrt{\frac{L}{C}}$ since $\omega_0 = \frac{1}{\sqrt{LC}}$.
3. At the half-power frequencies of a resonance curve, the power absorbed by the oscillator is what fraction of its maximum value at resonance?
Explanation: By definition, the half-power frequencies (or $-3\text{ dB}$ points) are the frequencies where the average power is half ($1/2$) of the maximum power at resonance.
4. What is the phase lag of the displacement behind the driving force at exact resonance (\(\omega = \omega_0\))?
Explanation: At resonance ($\omega = \omega_0$), the velocity is in phase with the driving force. Since displacement lags velocity by $\pi/2$, the displacement lags the driving force by $\pi/2$ radians ($90^\circ$).
5. If the damping constant $r$ of a resonant system is doubled while keeping $m$ and $s$ constant, the quality factor $Q$ will:
Explanation: Since $Q = \frac{\sqrt{ms}}{r}$, $Q$ is inversely proportional to the damping coefficient $r$. Doubling $r$ will halve the quality factor $Q$.
6. Velocity resonance in a forced oscillator occurs strictly when:
Explanation: Velocity resonance condition requires the reactive impedance to vanish, which happens exactly at $\omega = \omega_0$, independent of damping.
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. The sharpness of resonance in an electrical or mechanical system is directly indicated by:
Explanation: The Quality Factor $Q$ is the standard metric used to quantify the sharpness of resonance and the narrowness of the bandwidth.
2. In a series LCR circuit, if $L = 2\text{ H}$, $C = 2\mu\text{F}$, and $R = 20\ \Omega$, what is the Quality Factor $Q$?
3. What is the bandwidth (\(\Delta\omega\)) of a system with natural frequency $1000\text{ rad s}^{-1}$ and Quality Factor $Q = 50$?
Explanation: Since $Q = \frac{\omega_0}{\Delta\omega}$, the bandwidth is $\Delta\omega = \frac{\omega_0}{Q} = \frac{1000}{50} = 20\text{ rad s}^{-1}$.
4. Which term dominates the denominator of the forced oscillator amplitude expression when the driving frequency is much higher than the natural frequency (\(\omega \gg \omega_0\))?
Explanation: When $\omega \gg \omega_0$, the frequency term $\omega^2$ in $(\omega_0^2 - \omega^2)^2$ dominates, making the system mass-controlled.
5. If the damping factor $\gamma$ approaches zero in a forced oscillator, the amplitude at resonance approaches:
Explanation: In the absence of damping ($\gamma \to 0$), the denominator of the amplitude formula at $\omega = \omega_0$ vanishes, causing the resonance amplitude to approach infinity.
6. The phase angle $\phi$ of displacement response satisfies $\tan(\phi) = \frac{2\gamma\omega}{\omega_0^2 - \omega^2}$. At frequencies well above resonance (\(\omega \to \infty\)), what is the phase lag $\phi$?
Explanation: As $\omega \to \infty$, $\tan(\phi) \to 0^-$ from the negative side, meaning the phase lag approaches $\pi$ radians ($180^\circ$). inductance/mass dominates completely.
7. Why does amplitude resonance occur at a frequency slightly lower than velocity resonance?
Explanation: Differentiation of the amplitude response function includes terms involving $2\gamma^2$, shifting the peak amplitude to $\omega_r = \sqrt{\omega_0^2 - 2\gamma^2}$ which is less than $\omega_0$.
8. In an LCR circuit, if the resistance $R$ is decreased while keeping $L$ and $C$ constant, what is the effect on the resonance curve?
Explanation: Decreasing $R$ decreases damping, which increases the Q-factor ($Q = \frac{1}{R}\sqrt{\frac{L}{C}}$), resulting in a taller, narrower, and sharper resonance peak.
9. What is the physical significance of the half-power frequencies $\omega_1$ and $\omega_2$?
Explanation: At $\omega_1$ and $\omega_2$, the power absorbed by the system drops to $50\%$ ($1/2$) of the peak power value at resonance.
10. For a lightly damped system, if the natural frequency is $200\text{ rad s}^{-1}$ and the damping constant $\gamma = 5\text{ s}^{-1}$, what is the amplitude resonance frequency $\omega_r$?
Explanation: Using $\omega_r = \sqrt{\omega_0^2 - 2\gamma^2} = \sqrt{200^2 - 2(5)^2} = \sqrt{40000 - 50} = \sqrt{39950}\text{ rad s}^{-1}$.