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Waves and Oscillations
Oscillations in Physics
Oscillations refer to periodic motions of a physical system about an equilibrium position, typically governed by a linear restoring force. They form the foundational mechanics for understanding wave phenomena, acoustics, and alternating current circuits.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Simple Harmonic Motion (SHM) kinematics, dynamics, energy conservation, simple pendulum, and spring-mass systems.
PGT extension
Damped and forced oscillations, resonance, phase portraits, superposition of orthogonal oscillations (Lissajous figures), and Fourier analysis of periodic waveforms.
Prerequisites
Newton's laws of motion and rotational dynamics
Hooke's law of elasticity
Differential equations of second order with constant coefficients
Kinematics of Simple Harmonic Motion
Displacement in SHM is given by x(t) = A sin(ωt + φ), where A is amplitude, ω is angular frequency, and φ is initial phase.
Velocity v(t) = dx/d$t = A\omega \cos(\omegat + \phi) leads$ to the velocity-displacement relation $v = \omega \sqrt{A^{2} - x^{2}}$.
Acceleration a(t) = dv/d$t = -\omega^{2}x demonstrates that acceleration$ is directly proportional to displacement and oppositely directed.
Energy in Simple Harmonic Motion
Potential energy (U) stored in a spring of stiffness k is $U = (1/2)kx^{2} = (1/2)m\omega^{2}A^{2} \sin^{2}(\omegat + \phi)$.
Kinetic energy (K) of the oscillating mass is $K = (1/2)mv^{2} = (1/2)m\omega^{2}A^{2} \cos^{2}(\omegat + \phi)$.
Total mechanical energy $E = U + K = (1/2)m\omega^{2}A^{2} remains constant throughout the motion$, confirming energy conservation in the absence of non-conservative forces.
Damped Harmonic Motion
Real physical systems experience a resistive force proportional to velocity, represented as Fd = -b(dx/dt).
The displacement equation for light damping (γ < ω0) is x(t) = A0 e-γt cos(ωdt + φ), where $\gamma = b/(2m)$ is the damping coefficient.
The damped angular frequency is given by ωd = √(ω02 - γ2).
Forced Oscillations and Resonance
A system driven by an external periodic force F(t) = F0 cos(ωt) oscillates at the driving frequency ω after transient terms decay.
The steady-state amplitude depends strongly on the driving frequency and reaches a maximum when ω approaches the natural frequency ω0.
Resonance sharpness is determined by the quality factor $Q = \omega_{0} / (2\gamma)$, where lower damping yields a higher peak amplitude.
Derivations
Time Period of a Simple Pendulum
Conditions: Amplitude of oscillation is small (θ << 1 radian) so that sin(θ) ≈ θ.
Write the torque about the point of suspension: $\tau = -mg L \sin(\theta)$.
Equate torque to angular acceleration using moment of inertia $I = mL^{2}: mL^{2} (d^{2}\theta/dt^{2}) = -mg L \sin(\theta)$.
Apply small angle approximation to obtain the linear second-order differential equation: d2θ/dt2 + (g/L)$\theta = 0$.
Compare with standard SHM equation d2θ/dt2 + ω2$\theta = 0$ to identify $\omega = \sqrt{g/L}$.
Result: $T = 2\pi \sqrt{L/g}$
Differential Equation of Damped Harmonic Oscillator
Conditions: System subject to restoring force -kx and viscous damping force -b(dx/dt).
Apply Newton's second law: m(d2x/dt2) = -kx - b(dx/dt).
Rearrange terms to form a homogeneous linear differential equation: m(d2x/dt2) + b(dx/dt) + kx = 0.
Divide by mass m and substitute ω20 = k/m and 2$\gamma = b/m: d^{2}x/dt^{2} + 2\gamma(dx/dt) + \omega^{2}_{0}x = 0$.
Result: d2x/dt2 + 2γ(dx/dt) + ω20x = 0
Formula sheet
Formula / law
Equation
Variables and units
Conditions
Angular Frequency of Spring-Mass System
$$\omega = \sqrt{\frac{k}{m}}$$
k = spring constant, m = mass rad/s
Ideal massless spring, linear elastic limit
Time Period of Simple Pendulum
$$T = 2pi \sqrt{\frac{L}{g}}$$
L = length of string, g = acceleration due to gravity s
Small angle approximation
Total Energy of SHM
$$E = \frac{1}{2} m \omega^2 A^2$$
m = mass, omega = angular frequency, A = amplitude J
A 0.5 kg mass is attached to a spring of stiffness 200 N/m and experiences a damping force with coefficient $b = 4.0 kg/s$. Determine the damped angular frequency.
Hooke's law assumes elasticity is linear and springs do not exceed their elastic limit.
Simple pendulum derivations assume small angular displacements where sin(θ) ≈ θ and ignore mass of the string and air resistance.
Common misconceptions
Common error: Acceleration in SHM is maximum when displacement is zero.
Correct: Acceleration is zero when displacement is zero, and maximum when displacement is at its peak amplitude.
Acceleration is given by $a = -\omega^{2}x$, making it directly proportional to displacement magnitude.
Common error: Total energy of a damped oscillator remains constant.
Correct: Total energy decreases continuously with time in damped oscillations due to work done against non-conservative viscous forces.
Viscous damping dissipates mechanical energy into thermal energy.
Quick revision
In SHM, displacement, velocity, and acceleration are successively out of phase by π/2 radians.
Angular Frequency of Spring-Mass System: omega = sqrt{\frac{k}{m}}
Time Period of Simple Pendulum: T = 2pi sqrt{\frac{L}{g}}
Must remember
Displacement in SHM is expressed as $x(t) = A \sin(\omega t + \phi)$, where $A$ is amplitude, $\omega$ is angular frequency, and $\phi$ is the initial phase.
Acceleration in SHM is directly proportional to displacement and oppositely directed, given by $a(t) = -\omega^2 x$.
Total mechanical energy $E = \frac{1}{2}m\omega^2 A^2$ remains constant throughout SHM in the absence of non-conservative forces.
The angular frequency of a spring-mass system is $\omega = \sqrt{k/m}$ and the time period of a simple pendulum is $T = 2\pi \sqrt{L/g}$.
In damped oscillations, the displacement decays exponentially as $x(t) = A_0 e^{-\gamma t} \cos(\omega_d t + \phi)$, where $\gamma = b/(2m)$.
Resonance occurs in forced oscillations when the driving frequency $\omega$ equals the natural frequency $\omega_0$, yielding maximum steady-state amplitude.
Glossary
Amplitude ($A$)The maximum displacement of a particle from its mean position in oscillatory motion.
Angular Frequency ($\omega$)The rate of change of phase of a sinusoidal waveform, measured in rad/s.
Damping Coefficient ($\gamma$)A parameter proportional to the resistive force per unit mass, defined as $\gamma = b/(2m)$.
Quality Factor ($Q$)A dimensionless parameter that describes the sharpness of resonance or how underdamped an oscillator is, $Q = \omega_0 / (2\gamma)$.
Torsional PendulumA rigid body suspended by a wire that oscillates about the wire's axis under a restoring couple per unit twist $C$.
Comparison of Simple Pendulum and Spring-Mass System
Parameter
Simple Pendulum
Spring-Mass System
Time Period Formula
$T = 2\pi \sqrt{L/g}$
$T = 2\pi \sqrt{m/k}$
Restoring Force Origin
Component of gravitational force ($mg \sin\theta$)
Elastic spring force ($kx$)
Dependence on Mass
Independent of mass ($m$)
Dependent on mass ($m$)
Practice Quiz
Choose an option to reveal the explanation.
1. What is the acceleration of a particle executing simple harmonic motion with amplitude $A$ and angular frequency $\omega$ when its displacement is $x$?
Explanation: Acceleration in SHM is given by $a(t) = \frac{d^2x}{dt^2} = -\omega^2 x$, showing it is proportional to displacement and oppositely directed.
2. At what displacement from the mean position is the kinetic energy of a simple harmonic oscillator equal to its potential energy?
Explanation: Kinetic energy is $K = \frac{1}{2}m\omega^2(A^2 - x^2)$ and potential energy is $U = \frac{1}{2}m\omega^2 x^2$. Setting $K = U$ gives $A^2 - x^2 = x^2$, which yields $x = \pm \frac{A}{\sqrt{2}}$.
3. The time period of a simple pendulum of length $L$ inside a freely falling elevator is:
Explanation: In a freely falling elevator, the effective acceleration due to gravity $g_{eff} = 0$. Since $T = 2\pi \sqrt{L/g_{eff}}$, the time period becomes infinite, meaning the pendulum does not oscillate.
4. Which of the following expressions represents the total mechanical energy of a spring-mass system oscillating with angular frequency $\omega$ and amplitude $A$?
Explanation: The total energy $E = U_{max} = \frac{1}{2}kA^2 = \frac{1}{2}m\omega^2A^2$, which remains conserved throughout the motion.
5. In underdamped harmonic motion, what is the expression for the damped angular frequency $\omega_d$ in terms of natural frequency $\omega_0$ and damping factor $\gamma$?
Explanation: The damped angular frequency is given by $\omega_d = \sqrt{\omega_0^2 - \gamma^2}$ for light damping where $\gamma < \omega_0$.
6. What is the quality factor $Q$ of a weakly damped oscillator with natural frequency $\omega_0$ and damping factor $\gamma$?
Explanation: The quality factor is defined as $Q = \frac{\omega_0}{2\gamma}$, representing the ratio of stored energy to dissipated energy per radian.
7. When does the velocity of a particle executing SHM attain its maximum magnitude?
Explanation: The velocity relation is $v = \omega \sqrt{A^2 - x^2}$. At the mean position ($x = 0$), velocity is maximum, equal to $\omega A$.
8. The time period of a physical pendulum oscillating about a fixed horizontal axis is given by:
Explanation: For a physical pendulum, $T = 2\pi \sqrt{\frac{I}{mgl}}$, where $I$ is the moment of inertia about the pivot and $l$ is the distance from the pivot to the center of mass.
9. What is the main consequence of viscous damping on the total mechanical energy of an oscillator over time?
Explanation: Viscous forces are non-conservative and continuously dissipate mechanical energy into thermal energy, causing the total energy to decay over time.
10. In forced oscillations, when the driving frequency $\omega$ matches the natural frequency $\omega_0$, what phenomenon is observed?
Explanation: When the driving frequency equals the natural frequency ($\omega = \omega_0$), the system experiences resonance, leading to a maximum steady-state amplitude.
11. What is the unit of angular frequency $\omega$?
Explanation: Angular frequency measures the angular displacement per unit time, expressed in rad/s.
12. The time period of a torsional pendulum depends on the moment of inertia $I$ and the restoring couple per unit twist $C$ as:
Explanation: The restoring torque is $\tau = -C\theta$, leading to the time period formula $T = 2\pi \sqrt{\frac{I}{C}}$.
13. If the mass of a spring-mass system is quadrupled while keeping the spring constant unchanged, its time period will:
Explanation: Since $T = 2\pi \sqrt{m/k}$, the time period is proportional to $\sqrt{m}$. Multiplying mass by 4 multiplies the time period by $\sqrt{4} = 2$.
14. Which statement is correct regarding acceleration in SHM?
Explanation: Since $a = -\omega^2 x$, the magnitude of acceleration is directly proportional to displacement, meaning it reaches its maximum value at the extremes ($x = \pm A$).
15. What does the phase angle $\phi$ in the displacement equation $x(t) = A \sin(\omega t + \phi)$ determine?
Explanation: The initial phase $\phi$ determines the position and velocity of the oscillator at time $t = 0$.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. A particle executes simple harmonic motion with an amplitude of $5\text{ cm}$ and a period of $2\text{ s}$. What is its maximum velocity?
Explanation: Maximum velocity is $v_{max} = A\omega$. Here $A = 5\text{ cm}$ and $\omega = \frac{2\pi}{T} = \frac{2\pi}{2} = \pi\text{ rad/s}$. Thus, $v_{max} = 5 \times \pi = 5\pi\text{ cm/s}$? Wait: $T = 2$ so $\omega = \pi$, $A = 5$, hence $5\pi$. Let's check options: option A is $5\pi \text{ cm/s}$, option B is $2.5\pi$. Correct index should be 0.
2. If a simple pendulum's length is increased by $300\%$, its time period will increase by what percentage?
Explanation: New length $L' = L + 3L = 4L$. The new time period $T' = 2\pi \sqrt{4L/g} = 2T$. The percentage increase is $\frac{T' - T}{T} \times 100\% = \frac{2T - T}{T} \times 100\% = 100\%.$
3. The potential energy of a particle in SHM at displacement $x$ is given by $U = \frac{1}{2}m\omega^2 x^2$. If the amplitude is doubled, the maximum potential energy becomes:
Explanation: Maximum potential energy is $U_{max} = \frac{1}{2}m\omega^2 A^2$. Since $U_{max} \propto A^2$, doubling the amplitude increases the maximum potential energy by a factor of $2^2 = 4$.
4. A body of mass $0.5\text{ kg}$ is attached to a vertical spring of spring constant $50\text{ N/m}$. What is its angular frequency of oscillation?
Explanation: Angular frequency is $\omega = \sqrt{\frac{k}{m}} = \sqrt{\frac{50}{0.5}} = \sqrt{100} = 10\text{ rad/s}$.
5. In an underdamped oscillator, the amplitude decreases to $1/e$ of its initial value in a time interval equal to:
Explanation: The amplitude envelope in damped motion is $A(t) = A_0 e^{-\gamma t}$. Setting $A(t) = A_0 / e$ gives $e^{-\gamma t} = e^{-1}$, which implies $\gamma t = 1$, so $t = \frac{1}{\gamma}$.
6. Which of the following differential equations correctly describes damped harmonic motion?
Explanation: The equation of motion for a damped oscillator including inertial, damping, and restoring terms is $\frac{d^2x}{dt^2} + 2\gamma \frac{dx}{dt} + \omega_0^2 x = 0$.
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. What is the phase difference between velocity and displacement of a particle executing simple harmonic motion?
Explanation: Displacement is $x = A \sin(\omega t + \phi)$ and velocity is $v = A\omega \cos(\omega t + \phi) = A\omega \sin(\omega t + \phi + \pi/2)$. Thus, the phase difference is $\frac{\pi}{2}$.
2. If a clock regulated by a simple pendulum is taken to a mountain top, what happens to the clock?
Explanation: On a mountain top, acceleration due to gravity $g$ decreases. Since $T = 2\pi \sqrt{L/g}$, time period increases, meaning each oscillation takes longer, so the clock loses time.
3. The total mechanical energy of a damped oscillator:
Explanation: Because energy is dissipated by viscous forces, the mechanical energy decreases over time, typically falling off exponentially as $E(t) = E_0 e^{-2\gamma t}$.
4. What is the condition for critical damping in an oscillator?
Explanation: Critical damping occurs when the damping factor equals the natural frequency ($\gamma = \omega_0$), allowing the system to return to equilibrium in the shortest possible time without oscillating.
5. In steady-state forced oscillations, the frequency of the oscillator is equal to:
Explanation: After transient terms die out, a forced oscillator oscillates entirely at the frequency of the external driving force ($\omega$).
6. The maximum acceleration of a particle in SHM is $a_{max}$ and maximum velocity is $v_{max}$. What is its angular frequency $\omega$?
Explanation: We have $v_{max} = A\omega$ and $a_{max} = A\omega^2$. Dividing $a_{max}$ by $v_{max}$ gives $\frac{A\omega^2}{A\omega} = \omega$.
7. A particle oscillates with SHM along a straight line with amplitude $A$. What fraction of the total energy is kinetic when the displacement is $A/2$?
Explanation: Potential energy at $x = A/2$ is $U = \frac{1}{2}m\omega^2 (A/2)^2 = \frac{1}{4} E_{total}$. Hence, kinetic energy $K = E_{total} - U = \frac{3}{4} E_{total}$.
8. Which of the following physical quantities remains constant during simple harmonic motion?
Explanation: In an ideal conservative system executing SHM, kinetic and potential energies continuously interconvert, but their sum—the total mechanical energy—remains constant.
9. If the amplitude of a forced oscillator is maximum at resonance, what is the phase relationship between the displacement and the driving force at resonance (assuming light damping)?
Explanation: At exact resonance ($\omega = \omega_0$), the phase lag of displacement relative to the driving force is exactly $\pi/2$ radians (or $90^{\circ}$)..
10. What happens to the resonant peak of a forced oscillator if the damping coefficient $\gamma$ is decreased?
Explanation: A lower damping coefficient increases the quality factor $Q$, resulting in a much sharper and higher amplitude peak at resonance.