Two simple harmonic motions, as shown below, are at right angles. They are combined to form lissajous figures. $x(t) = A \sin (at + \delta)$ $y(t) = B \sin (bt)$ Identify the correct match below :
Lissajous Conditions Explained
Lissajous figures are graphical representations formed by combining two simple harmonic motions (SHMs) that are perpendicular to each other. The resulting shape depends critically on the amplitudes ($A, B$), frequencies ($a, b$), and phase difference ($\delta$) of the two SHMs.
The given equations are:
- $x(t) = A \sin (at + \delta)$
- $y(t) = B \sin (bt)$
Deriving Ellipse Shape
To form an ellipse, specific relationships between the parameters ($A, B, a, b, \delta$) must hold. We examine the conditions that lead to an ellipse.
1. Condition Substitution
Consider the case where the angular frequencies are equal ($a=b$) and the phase difference is $\delta = \pi/2$. The amplitudes are $A$ and $B$, with $A \ne B$. Substituting these into the general equations:
- $x(t) = A \sin (at + \pi/2) = A \cos (at)$
- $y(t) = B \sin (at)$
2. Trigonometric Isolation
From the equations in Step 1, isolate the trigonometric terms:
- $\cos(at) = \frac{x(t)}{A}$
- $\sin(at) = \frac{y(t)}{B}$
3. Identity Application
Apply the fundamental trigonometric identity $\sin^2(\theta) + \cos^2(\theta) = 1$ (where $\theta = at$):
$ \left(\frac{y(t)}{B}\right)^2 + \left(\frac{x(t)}{A}\right)^2 = \sin^2(at) + \cos^2(at) $
$ \left(\frac{x}{A}\right)^2 + \left(\frac{y}{B}\right)^2 = 1 $
4. Result Interpretation
The equation $\left(\frac{x}{A}\right)^2 + \left(\frac{y}{B}\right)^2 = 1$ represents an ellipse centered at the origin. The semi-axes are $A$ and $B$. The condition $A \ne B$ ensures it is a non-circular ellipse.
Summary of Equal Frequency Cases ($a=b$)
When the frequencies are equal ($a=b$), the Lissajous figure depends on the phase difference ($\delta$) and amplitude ratio ($A:B$):
| Phase Difference ($\delta$) | Amplitude Ratio ($A:B$) | Figure |
|---|---|---|
| $0$ or $\pi$ | Any ($A \ne 0, B \ne 0$) | Straight Line |
| $\pi/2$ | $A = B$ | Circle |
| $\pi/2$ | $A \ne B$ | Ellipse |
Therefore, the condition $A \ne B, a = b, \delta = \pi/2$ correctly describes an ellipse.
Similar Questions
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Important Questions from Oscillations and Waves
- Two tuning forks $A$ and $B$ are sounded together giving rise to $8$ beats in $2\text{ s}$. When fork $A$ is loaded with wax, the beat frequency is reduced to $4$ beats in $2\text{ s}$. If the original frequency of tuning fork $B$ is $380\text{ Hz}$ then original frequency of tuning fork $A$ is _________ $\text{Hz}$.
- A simple pendulum of string length 30 cm performs 20 oscillations in 10 s. The length of the string required for the pendulum to perform 40 oscillations in the same time duration is ___________ cm. [Assume that the mass of the pendulum remains same.]
- The velocity of sound in air is doubled when the temperature is raised from $0^\circ\text{C}$ to $\alpha^\circ\text{C}$. The value of $\alpha$ is ________.
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