CUET PG Physics: Complete Notes on Oscillations, Waves & Optics
This guide covers all essential topics from the Oscillations, Waves, and Optics section of CUET PG Physics, including detailed theory, formulas, and solved previous year questions (PYQs). Perfect for last-minute revision and concept clarity.
1. Simple Harmonic Oscillator
1.1 Differential Equation
For a mass-spring system: \(F = -kx\). From Newton's law:
1.2 General Solution
\(x(t) = A \sin(\omega t + \phi)\) or \(x(t) = A \cos(\omega t + \phi')\).
1.3 Key Formulas
| Quantity | Formula |
|---|---|
| Velocity | \(v(t) = \omega A \cos(\omega t + \phi)\) |
| Acceleration | \(a(t) = -\omega^2 x(t)\) |
| Total energy | \(E = \frac{1}{2}m\omega^2 A^2\) |
Solution: \(\omega = 100\pi\) → \(f = \frac{100\pi}{2\pi} = 50\) Hz.
2. Superposition of Harmonic Oscillators
2.1 Same Frequency, Same Direction
Resultant amplitude: \(A = \sqrt{A_1^2 + A_2^2 + 2A_1A_2\cos(\phi_1-\phi_2)}\)
2.2 Beats (Different Frequencies)
3. Lissajous Figures
For perpendicular oscillations: \(x = A_1\sin(\omega_1 t+\phi_1)\), \(y = A_2\sin(\omega_2 t+\phi_2)\).
Equal frequencies
| \(\delta\) | Figure |
|---|---|
| 0 or \(\pi\) | Straight line |
| \(\pi/2\) | Ellipse (circle if \(A_1=A_2\)) |
4. Damped and Forced Oscillators, Resonance
4.1 Damped oscillator
Underdamped: \(x = A_0 e^{-\beta t}\cos(\omega_d t+\phi)\), \(\omega_d = \sqrt{\omega_0^2-\beta^2}\).
4.2 Forced oscillator
Amplitude: \(A = \frac{f_0}{\sqrt{(\omega_0^2-\omega^2)^2 + (2\beta\omega)^2}}\)
Phase: \(\tan\delta = \frac{2\beta\omega}{\omega_0^2-\omega^2}\)
4.3 Resonance
At \(\omega = \omega_0\): \(A_{\text{max}} = \frac{f_0}{2\beta\omega_0}\), \(\delta = \pi/2\).
4.4 Quality factor
20. Quick Reference Formula Sheet
| Topic | Formula |
|---|---|
| SHM | \(\frac{d^2x}{dt^2} + \omega^2 x = 0\) |
| Damped oscillator | \(\frac{d^2x}{dt^2} + 2\beta\frac{dx}{dt} + \omega_0^2 x = 0\) |
| Quality factor | \(Q = \omega_0/(2\beta)\) |
| Wave equation | \(\frac{\partial^2 y}{\partial t^2} = v^2 \frac{\partial^2 y}{\partial x^2}\) |
| Doppler effect (sound) | \(f' = f\frac{v\pm v_o}{v\mp v_s}\) |
| Lens formula | \(\frac{1}{f} = \frac{1}{v} - \frac{1}{u}\) |
| YDSE fringe width | \(\beta = \frac{\lambda D}{d}\) |
| Single slit minima | \(a\sin\theta = n\lambda\) |
| Grating equation | \(d\sin\theta = m\lambda\) |
| Grating resolving power | \(R = mN\) |
| Specific rotation | \([S] = \theta/(l c)\) |
