Additional: Superposition of SHM
Combining Two SHMs in the same direction
When an object is simultaneously subjected to two or more conditions that would individually cause it to undergo Simple Harmonic Motion in the same direction, the resulting motion is the superposition of these individual SHMs. The principle of superposition states that the net displacement at any time is the vector sum of the displacements due to each individual SHM acting alone.
Consider a particle undergoing two simultaneous SHMs along the x-axis. Let the individual displacements be:
$ x_1(t) = A_1 \sin(\omega_1 t + \phi_1) $
$ x_2(t) = A_2 \sin(\omega_2 t + \phi_2) $
By the principle of superposition, the resultant displacement is:
$ x(t) = x_1(t) + x_2(t) = A_1 \sin(\omega_1 t + \phi_1) + A_2 \sin(\omega_2 t + \phi_2) $
Case 1: Two SHMs with the same frequency ($\omega_1 = \omega_2 = \omega$)
If the two SHMs have the same angular frequency $\omega$, the resultant motion is also a Simple Harmonic Motion with the same frequency $\omega$. The amplitude and initial phase of the resultant SHM depend on the amplitudes and phases of the individual SHMs.
$ x(t) = A_1 \sin(\omega t + \phi_1) + A_2 \sin(\omega t + \phi_2) $
Expanding the terms using $\sin(A+B) = \sin A \cos B + \cos A \sin B$:
$ x(t) = A_1 (\sin(\omega t)\cos\phi_1 + \cos(\omega t)\sin\phi_1) + A_2 (\sin(\omega t)\cos\phi_2 + \cos(\omega t)\sin\phi_2) $
$ x(t) = (\boldsymbol{A_1 \cos\phi_1 + A_2 \cos\phi_2})\sin(\omega t) + (\boldsymbol{A_1 \sin\phi_1 + A_2 \sin\phi_2})\cos(\omega t) $
Let $R \cos\Phi = A_1 \cos\phi_1 + A_2 \cos\phi_2$ and $R \sin\Phi = A_1 \sin\phi_1 + A_2 \sin\phi_2$. Then:
$ x(t) = R \cos\Phi \sin(\omega t) + R \sin\Phi \cos(\omega t) = R (\sin(\omega t)\cos\Phi + \cos(\omega t)\sin\Phi) $
$ x(t) = R \sin(\omega t + \Phi) $
This is the equation for SHM with amplitude $R$ and initial phase $\Phi$. The resultant amplitude $R$ can be found by squaring and adding the expressions for $R \cos\Phi$ and $R \sin\Phi$:
$ R^2 \cos^2\Phi + R^2 \sin^2\Phi = (A_1 \cos\phi_1 + A_2 \cos\phi_2)^2 + (A_1 \sin\phi_1 + A_2 \sin\phi_2)^2 $
$ R^2 (\cos^2\Phi + \sin^2\Phi) = A_1^2\cos^2\phi_1 + 2A_1A_2\cos\phi_1\cos\phi_2 + A_2^2\cos^2\phi_2 + A_1^2\sin^2\phi_1 + 2A_1A_2\sin\phi_1\sin\phi_2 + A_2^2\sin^2\phi_2 $
$ R^2 (1) = A_1^2(\cos^2\phi_1 + \sin^2\phi_1) + A_2^2(\cos^2\phi_2 + \sin^2\phi_2) + 2A_1A_2(\cos\phi_1\cos\phi_2 + \sin\phi_1\sin\phi_2) $
$ R^2 = A_1^2 (1) + A_2^2 (1) + 2A_1A_2 \cos(\phi_1 - \phi_2) $
$ R = \sqrt{A_1^2 + A_2^2 + 2A_1A_2 \cos(\Delta\phi)} $
where $\Delta\phi = \phi_1 - \phi_2$ is the phase difference between the two SHMs. The resultant amplitude $R$ depends on the individual amplitudes and their phase difference. This is analogous to the amplitude of the resultant wave in interference.
The resultant phase $\Phi$ can be found from $\tan\Phi = \frac{R \sin\Phi}{R \cos\Phi} = \frac{A_1 \sin\phi_1 + A_2 \sin\phi_2}{A_1 \cos\phi_1 + A_2 \cos\phi_2}$.
Special Cases:
- In-phase: If $\Delta\phi = 0$ or $2\pi n$, $\cos(0) = 1$. $R = \sqrt{A_1^2 + A_2^2 + 2A_1A_2} = \sqrt{(A_1+A_2)^2} = A_1+A_2$. Amplitudes add up.
- Out-of-phase: If $\Delta\phi = \pi$ or $\pi + 2\pi n$, $\cos(\pi) = -1$. $R = \sqrt{A_1^2 + A_2^2 - 2A_1A_2} = \sqrt{(A_1-A_2)^2} = |A_1-A_2|$. Amplitudes subtract.
- $90^\circ$ phase difference: If $\Delta\phi = \pm \pi/2$, $\cos(\pm \pi/2) = 0$. $R = \sqrt{A_1^2 + A_2^2}$. Amplitudes add vectorially (like perpendicular vectors).
Case 2: Two SHMs with different frequencies ($\omega_1 \ne \omega_2$)
If the two SHMs have different angular frequencies, the resultant motion is not simple harmonic motion. The motion is still periodic if the frequencies are integer multiples of a fundamental frequency (i.e., commensurate), but the resulting motion is generally more complex and might exhibit phenomena like beats if the frequencies are close to each other.
$ x(t) = A_1 \sin(\omega_1 t + \phi_1) + A_2 \sin(\omega_2 t + \phi_2) $
This sum of two sinusoids with different frequencies is not a single sinusoid. The motion is typically complex and repeating only if there's a common period for both frequencies.
Example: A system subjected to a restoring force $F=-kx$ (causing SHM at frequency $\omega_0 = \sqrt{k/m}$) is also driven by an external force $F_{driving}(t) = F_0 \sin(\omega_d t)$ (causing forced oscillation at frequency $\omega_d$). The resultant motion is a superposition of the natural oscillation (transient, damped) and the forced oscillation (steady-state, at $\omega_d$). If $\omega_d = \omega_0$, resonance occurs, leading to large amplitudes (in the presence of damping).
Lissajous Figures (Combining Perpendicular SHMs)
When a particle undergoes two simultaneous Simple Harmonic Motions that are perpendicular to each other, the resultant path traced by the particle is a curve called a Lissajous figure (or Bowditch curve). The shape of the Lissajous figure depends on the amplitudes, frequencies, and initial phases of the two perpendicular SHMs.
Consider a particle undergoing SHM along the x-axis and simultaneously SHM along the y-axis, both centered at the origin.
$ x(t) = A_x \sin(\omega_x t + \phi_x) $
$ y(t) = A_y \sin(\omega_y t + \phi_y) $
The resulting motion is the path traced by the point $(x(t), y(t))$ in the xy-plane as time progresses.
Case 1: Perpendicular SHMs with the same frequency ($\omega_x = \omega_y = \omega$)
If the frequencies are the same, the resultant path is an ellipse, a circle, or a straight line, depending on the amplitudes and the phase difference between the two SHMs.
$ x(t) = A_x \sin(\omega t + \phi_x) $
$ y(t) = A_y \sin(\omega t + \phi_y) = A_y \sin(\omega t + \phi_x + \Delta\phi) $, where $\Delta\phi = \phi_y - \phi_x$ is the phase difference.
Expanding $y(t) = A_y (\sin(\omega t + \phi_x)\cos(\Delta\phi) + \cos(\omega t + \phi_x)\sin(\Delta\phi))$.
From $x(t) = A_x \sin(\omega t + \phi_x)$, $\sin(\omega t + \phi_x) = x/A_x$. From $y(t) = A_y \sin(\omega t + \phi_y)$, we can also use $\cos(\omega t + \phi_x) = \sqrt{1 - \sin^2(\omega t + \phi_x)} = \sqrt{1 - (x/A_x)^2}$.
The general equation for the ellipse is obtained by eliminating $t$. This results in:
$ \left(\frac{x}{A_x}\right)^2 + \left(\frac{y}{A_y}\right)^2 - 2\left(\frac{x}{A_x}\right)\left(\frac{y}{A_y}\right)\cos(\Delta\phi) = \sin^2(\Delta\phi) $
This is the equation of an ellipse centered at the origin. The specific shape and orientation of the ellipse depend on $A_x$, $A_y$, and $\Delta\phi$.
Special Cases (Same Frequency):
- In-phase ($\Delta\phi = 0$ or $\pi$): $\cos(\Delta\phi) = \pm 1$, $\sin(\Delta\phi) = 0$.
$ (\frac{x}{A_x})^2 + (\frac{y}{A_y})^2 - 2(\frac{x}{A_x})(\frac{y}{A_y})(\pm 1) = 0 $
$ (\frac{x}{A_x} \mp \frac{y}{A_y})^2 = 0 \implies \frac{x}{A_x} = \pm \frac{y}{A_y} $
This is the equation of a straight line passing through the origin with slope $\pm A_y/A_x$. The motion is along a diagonal line segment.
(Image Placeholder: An xy-plot showing a straight line from bottom-left to top-right (or top-left to bottom-right) passing through the origin, bounded by a rectangle defined by axes from -Ax to Ax and -Ay to Ay.)
- $90^\circ$ Phase difference ($\Delta\phi = \pm \pi/2$): $\cos(\Delta\phi) = 0$, $\sin(\Delta\phi) = \pm 1$.
$ (\frac{x}{A_x})^2 + (\frac{y}{A_y})^2 - 0 = (\pm 1)^2 = 1 $
$ \frac{x^2}{A_x^2} + \frac{y^2}{A_y^2} = 1 $
This is the equation of an ellipse aligned with the x and y axes.
- If $A_x = A_y = A$, then $x^2 + y^2 = A^2$, which is the equation of a circle. Circular motion is a special case of two perpendicular SHMs with equal amplitudes and a 90° phase difference.
(Image Placeholder: An xy-plot showing a circle centered at the origin.)
- If $A_x \ne A_y$, the path is an ellipse with semi-axes $A_x$ and $A_y$ along the x and y axes.
(Image Placeholder: An xy-plot showing an ellipse centered at the origin, aligned with the x and y axes, with semi-axes Ax and Ay.)
- If $A_x = A_y = A$, then $x^2 + y^2 = A^2$, which is the equation of a circle. Circular motion is a special case of two perpendicular SHMs with equal amplitudes and a 90° phase difference.
Case 2: Perpendicular SHMs with different frequencies ($\omega_x \ne \omega_y$)
If the frequencies are different, the resultant path is a more complex Lissajous figure. The shape depends on the ratio of the frequencies ($\omega_x/\omega_y$) and the phase difference ($\Delta\phi$).
If the ratio of frequencies is a rational number ($\omega_x/\omega_y = p/q$, where p and q are integers), the figure is a closed curve that the particle traces repeatedly. The shape can be quite intricate, with multiple loops. The number of loops is related to the integers p and q.
If the ratio of frequencies is irrational, the figure is not a closed curve, and the path never repeats, eventually filling the entire rectangle defined by the amplitudes $A_x$ and $A_y$.
Lissajous figures can be demonstrated using oscilloscopes or mechanical setups combining pendulums oscillating in perpendicular directions. They are used in physics to analyse oscillating systems and determine frequency ratios and phase differences.
(Image Placeholder: A collage of different Lissajous figures showing complex patterns for various frequency ratios (e.g., 1:2, 2:3) and phase differences.)