Oscillations L11 Superposition of SHMs, Phasors & Lissajous Figures скачать в хорошем качестве

Oscillations L11 Superposition of SHMs, Phasors & Lissajous Figures

In Oscillations Lecture 11, we study how multiple simple harmonic motions (SHMs) combine, and how to handle them using phasor (vector) methods. Covered in this lecture: Superposition of two SHMs along the same line (same frequency) Let [ x_1 = A_1\sin\omega t,\quad x_2 = A_2\sin(\omega t + \theta). ] Then the resultant motion is also SHM with the same angular frequency (\omega): [ x = A\sin(\omega t + \phi). ] Resultant amplitude and phase: [ A = \sqrt{A_1^2 + A_2^2 + 2A_1A_2\cos\theta}, ] [ \tan\phi = \frac{A_2\sin\theta}{A_1 + A_2\cos\theta}. ] Special cases: (\theta = 0): in‑phase, (A = A_1 + A_2) (maximum addition). (\theta = \pi): out of phase, (A = |A_1 - A_2|) (cancellation). Phasor (vector) method for SHM superposition Treat each SHM’s amplitude as a rotating vector (phasor). Phase difference = angle between phasors. Resultant amplitude and phase are obtained by vector addition. This provides an intuitive and fast way to add SHMs, and is the basis for AC phasor diagrams. Different frequencies case If [ x_1 = A_1\sin\omega_1 t,\quad x_2 = A_2\sin\omega_2 t,\quad \omega_1
eq \omega_2, ] then the sum (x = x_1 + x_2) is not a single simple harmonic motion. Resultant motion can show more complicated behaviour (beats, modulation etc.). In this lecture we focus on the neat, same‑frequency case. Perpendicular SHMs and Lissajous figures Two SHMs at right angles: [ x = A_1\sin\omega t,\quad y = A_2\sin(\omega t + \phi). ] Eliminating (t) gives: [ \frac{x^2}{A_1^{2}} + \frac{y^2}{A_2^{2}} \frac{2xy\cos\phi}{A_1A_2} = \sin^{2}\phi, ] which is a general ellipse: a Lissajous figure. Special cases: (\phi = 0): straight line (y = (A_2/A_1)x). (\phi = 90^\circ): (\dfrac{x^2}{A_1^{2}} + \dfrac{y^2}{A_2^{2}} = 1) (ellipse). (\phi = 90^\circ) and (A_1 = A_2 = A): (x^2 + y^2 = A^2) (circle). Algebraic identity: combining sine and cosine into one SHM Any expression of the form: [ y = A\sin\omega t + B\cos\omega t ] can be written as a single SHM: [ y = R\sin(\omega t + \phi), ] with [ R = \sqrt{A^2 + B^2},\quad \tan\phi = \frac{B}{A}. ] This trick is used repeatedly in waves, oscillations, and AC circuits. Worked examples Finding resultant amplitude when two SHMs of given amplitudes and phase difference act along the same line. Rewriting given motions (e.g. (y = 10\sin(\frac{\pi}{4}(12t + 1))) or (y = 5(\sin 3\pi t + \sqrt{3}\cos 3\pi t))) into standard form (A\sin(\omega t + \phi)), and extracting amplitude and time period. This lecture gives you the superposition toolkit for SHM: phasors, resultant amplitude and phase, and Lissajous figures – all of which are essential later in waves, interference, and AC circuits.