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Waves and Oscillations

Lissajous Figures

Lissajous figures are the resultant geometric patterns formed by the superposition of two harmonic oscillations acting at right angles to each other. They provide a precise graphical method for comparing frequencies, phase differences, and amplitudes of orthogonal vibrations.

UP TGT ScienceUP PGT PhysicsStudy Notes + Practice

Scope and prerequisites

TGT core

Understanding the graphical shapes formed for frequency ratios of 1:1 and 1:2 with various phase differences, and identification of straight lines, ellipses, and circles.

PGT extension

General parametric equations, analytical determination of frequency ratios using tangent counting methods, phase angle determination from axial intercepts, and applications in cathode-ray oscilloscopes.

Prerequisites

Simple Harmonic Motion (SHM)
Superposition principle of orthogonal oscillations
Trigonometric identities for multiple and compound angles

Definition and Principle of Superposition

Lissajous figures are formed when a particle is subjected simultaneously to two mutually perpendicular simple harmonic motions.
Let the motions be along the X-axis and Y-axis with frequencies ω_x and ω_y respectively.
The resultant trajectory traced by the particle in the X-Y plane depends on the amplitude ratio, frequency ratio, and initial phase difference.

Case of Equal Frequencies (ωx = ωy = ω)

When the frequencies are equal, the general path is an ellipse inscribed in a rectangle of sides 2a and 2b, where a and b are the amplitudes.
When the phase difference $\delta = 0 or \pi$, the trajectory degenerates into a straight line sloping upwards or downwards.
When $\delta = \pi/2 or 3\pi/2$ and amplitudes are equal (a = b), the ellipse becomes a circle.

Case of Harmonic Frequencies (ωy / ωx = n / m)

When the frequencies are in the ratio of small integers, closed open curves or loops are formed.
The number of horizontal loops equals the frequency of the Y-vibration, and the number of vertical loops equals the frequency of the X-vibration.
Tangents drawn to the curve parallel to the X-axis and Y-axis help determine the frequency ratio by the intersection method.

Experimental Determination using Cathode-Ray Oscilloscope (CRO)

A CRO can display Lissajous figures when sinusoidal signals are applied to the X-plates and Y-plates simultaneously.
By adjusting the frequency of a standard oscillator on one channel, unknown frequencies can be measured accurately.
A stationary, non-rotating pattern confirms a precise rational integer ratio between the two applied frequencies.

Derivations

Derivation of the General Cartesian Equation for Equal Frequencies

Conditions: Two mutually perpendicular SHMs with equal frequency ω and phase difference δ.

Write the displacement equations: $x = a \sin(\omegat)$ and $y = b \sin(\omegat + \delta)$.
Expand the second equation: $y = b [\sin(\omegat)\cos(\delta) + \cos(\omegat)\sin(\delta)]$.
Substitute sin(ωt) = x/a and cos(ωt) = ±√(1 - x²/a²) into the expanded equation.
Rearrange terms to isolate the radical: y - (x/a)b cos(δ) = b ±√(1 - x²/a²) sin(δ).
Square both sides and simplify to obtain the general equation of an ellipse: (x²/a²) + (y²/b²) - (2xy/ab)cos(δ) = sin²(δ).

Result: (x^2 / a^2) + (y^2 / b^2) - (2xy / ab) \cos(\delta) = \sin^2(\delta)

Determination of Phase Difference from Axial Intercepts

Conditions: Equal frequencies (ω$x = \omegay)$, elliptical trace with axes inclined to coordinate axes.

Consider the equation of the ellipse when $x = 0; the Y-intercept$ is given by y₀ = ± b sin(δ).
Consider the maximum amplitude along the Y-axis, which is y_max = b when sin(ωt + δ) = 1.
Take the ratio of the intercept to the maximum amplitude: y₀ / y_max = [b sin(δ)] / $b = \sin(\delta)$.
Calculate the phase difference using the inverse sine function: $\delta = \sin^{-1}(y_{0} / y_{max})$.

Result: $\delta = \sin^{-1}\left(\frac{y_0}{y_{max}}\right)$

Formula sheet

Formula / law	Equation	Variables and units	Conditions
General displacement along X	$$x = a \sin(\omega_x t)$$	$a = amplitude of X-motion$, ω_x = angular frequency, t = time m	SHM along X-axis
General displacement along Y	$$y = b \sin(\omega_y t + \delta)$$	$b = amplitude of Y-motion$, $\delta = phase difference$ m	SHM along Y-axis
Cartesian equation for equal frequencies	$$\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{2xy}{ab} \cos(\delta) = \sin^2(\delta)$$	a, b = amplitudes, $\delta = phase difference$ dimensionless	ω_x = ω_y
Straight line condition	$$\delta = n\pi \quad (n = 0, 1, 2, ...)$$	n = integer rad	Same frequency, in-phase or out-of-phase
Circle condition	$$a = b \quad \text{and} \quad \delta = \frac{\pi}{2} \pm n\pi$$	a, b = amplitudes m, rad	Equal frequencies, orthogonal phase
Frequency ratio from tangent counts	$$\frac{\omega_y}{\omega_x} = \frac{N_x}{N_y}$$	N_x = number of tangents touching the top/bottom, N_y = number of tangents touching the sides dimensionless	Closed Lissajous figure
Phase angle via intercepts	$$\sin(\delta) = \frac{y_{(x=0)}}{y_{max}}$$	y_($x=0) = intercept on Y-axis$, y_max = peak Y amplitude dimensionless	ω_x = ω_y, centered ellipse
Tangent of orientation angle for principal axes	$$\tan(2\theta) = \frac{2ab \cos(\delta)}{a^2 - b^2}$$	$\theta = angle of tilt of ellipse axes with coordinate axes$ rad	General elliptical Lissajous figure

Graphs and experiments

Lissajous Figure for δ = 0 (ωx = ωy)

Draw: A straight line passing through the origin with a positive slope $y = (b/a)x$.

Interpret: The two orthogonal oscillations are completely in phase, resulting in rectilinear motion.

Exam use: Direct question in TGT on identifying phase difference from a linear trace.

Lissajous Figure for δ = π/2, a = b (ωx = ωy)

Draw: A perfect circle inscribed in a square of side 2a.

Interpret: Equal amplitudes and a phase difference of π/2 create uniform circular motion.

Exam use: Frequently asked PGT condition for obtaining circular polarization or circular Lissajous figures.

Lissajous Figure for Frequency Ratio 1:2

Draw: A figure-eight (or parabola-like looped curve) enclosed in a rectangular box with 1 horizontal tangent and 2 vertical tangents.

Interpret: The frequency of the vertical oscillation is twice that of the horizontal oscillation (ω_y = 2ω_x).

Exam use: Standard diagram-based question for PGT to test frequency ratio identification.

Solved numericals

TGT core

Two perpendicular SHMs are given by $x = 3 \sin(\omegat)$ and $y = 3 \sin(\omegat + \pi/2)$. What is the resulting Lissajous figure?

Identify the amplitudes: a = 3 and b = 3.
Identify the frequencies: ω_x = ω_y = ω.
Identify the phase difference: $\delta = \pi/2$.
Since amplitudes are equal and phase difference is π/2, the equation x² + y² = a² represents a circle.

Answer: A circle of radius 3 units

PGT extension

A Lissajous figure formed by two orthogonal SHMs has 3 horizontal tangents and 2 vertical tangents. What is the frequency ratio ω_y / ω_x?

Recall the relation between frequency ratio and number of tangents: ω_y / ω_x = N_x / N_y.
Here, N_x (number of intersections or tangents touching horizontal boundaries / parallel to X-axis) = 3.
N_y (number of tangents touching vertical boundaries / parallel to Y-axis) = 2.
Substitute the values: ω_y / ω_x = 3 / 2.

Answer: 3:2

PGT extension

An elliptical Lissajous figure with equal amplitudes $a = b has a Y-intercept of 2 cm$ and a maximum Y-amplitude of 4 cm. Calculate the phase difference δ.

Use the formula relating phase difference to intercepts: sin(δ) = y₀ / y_max.
Substitute the given values: sin(δ) = 2 / 4 = 0.5.
Determine the angle: $\delta = \sin^{-1}(0.5) = \pi/6 radians (30°) or 5\pi/6 radians (150°)$.

Answer: 30° (π/6 rad) or 150° (5π/6 rad)

Assumptions and limitations

Assumes the orthogonal harmonic motions have constant amplitudes and stable frequencies over the observation interval.
Limitations include difficulty in resolving complex ratios when frequency drifts occur or when harmonic distortion is present in the input signals.

Common misconceptions

Common error: A straight-line Lissajous figure always means the phase difference is zero.

Correct: A straight line is obtained whenever the phase difference is an integral multiple of π ($\delta = n\pi)$, which includes both 0 and π.

When $\delta = \pi$, the slope of the line is negative, whereas for $\delta = 0$, the slope is positive.

Common error: The number of loops in a Lissajous figure directly equals the frequency of the wave.

Correct: The number of loops gives the ratio of the frequencies along the two axes, not the absolute frequency values.

Counting horizontal and vertical tangents yields only the relative integer ratio N_x:N_y = ω_y:ω_x.

Quick revision

Equal frequencies with $\delta = 0 or \pi give straight lines; \delta = \pi/2 with equal amplitudes gives a circle$.
General displacement along X: $x = a \sin(\omega_x t)$
General displacement along Y: $y = b \sin(\omega_y t + \delta)$

Must remember

Lissajous figures are formed by the superposition of two mutually perpendicular simple harmonic motions.
When frequencies are equal ($\omega_x = \omega_y$) and phase difference $\delta = 0$ or $\pi$, the resulting path is a straight line.
A circle is formed when amplitudes are equal ($a = b$) and the phase difference is an odd multiple of $\pi/2$.
The frequency ratio $\omega_y / \omega_x$ can be determined by counting the number of tangential intersections along the vertical and horizontal axes.
A Cathode-Ray Oscilloscope (CRO) uses sinusoidal signals applied to X and Y plates to display stationary Lissajous figures for precise frequency comparison.

Glossary

Lissajous FiguresThe trajectory traced by a particle subjected simultaneously to two mutually perpendicular simple harmonic motions.

Phase Difference ($\delta$)The difference in phase between the orthogonal simple harmonic motions along the X-axis and Y-axis.

Harmonic FrequenciesFrequencies that are in the ratio of small integers, producing closed open loops or curves.

Cathode-Ray Oscilloscope (CRO)An electronic test instrument used to visually display and measure electrical signals and Lissajous figures.

Characteristics of Lissajous Figures for Equal Frequencies

Phase Difference ($\delta$)	Condition for Amplitudes	Resultant Trajectory
$0$ or $\pi$	Any $a, b$	Straight line sloping upwards or downwards
$\pi/2$ or $3\pi/2$	$a = b$	Circle
$\pi/2$ or $3\pi/2$	$a \neq b$	Ellipse with axes aligned with coordinate axes
Arbitrary $\delta$	Any $a, b$	Inclined ellipse inscribed in rectangle $2a \times 2b$

Practice Quiz

Choose an option to reveal the explanation.

1. What trajectory is formed when two mutually perpendicular simple harmonic motions of equal frequencies and equal amplitudes are superposed with a phase difference of $\delta = \pi/2$?

Explanation: When $\omega_x = \omega_y$, $a = b$, and $\delta = \pi/2$, the Cartesian equation simplifies to $x^2 + y^2 = a^2$, which represents a circle.

2. Under what condition does the general equation of Lissajous figures for equal frequencies degenerate into a straight line sloping downwards?

Explanation: When $\delta = \pi$ (or integral multiples of $\pi$), the trajectory becomes a straight line with a negative slope, given by $y = -(b/a)x$.

3. If the frequency of the Y-vibration is twice that of the X-vibration ($\omega_y / \omega_x = 2$), how many horizontal loops are typically observed in the closed Lissajous figure?

Explanation: The number of horizontal loops equals the frequency of the Y-vibration, and the number of vertical loops equals the frequency of the X-vibration.

4. Which instrument is commonly used experimentally to display Lissajous figures by applying sinusoidal signals to its X and Y plates?

Explanation: A CRO displays Lissajous figures when sinusoidal signals are simultaneously applied to the horizontal (X) and vertical (Y) deflection plates.

5. What does a stationary, non-rotating Lissajous pattern observed on a CRO indicate about the two applied frequencies?

Explanation: A stationary pattern is formed only when the two frequencies are in a precise rational integer ratio ($n/m$). Otherwise, the pattern drifts continuously.

6. In the tangent counting method for determining frequency ratios, if $N_x$ is the number of tangents touching the top/bottom and $N_y$ is the number touching the sides, what is the frequency ratio $\omega_y / \omega_x$?

Explanation: The frequency ratio is given by $\omega_y / \omega_x = N_x / N_y$, where $N_x$ and $N_y$ correspond to the respective tangent counts.

7. What is the Cartesian equation for Lissajous figures when $\omega_x = \omega_y = \omega$ and phase difference is $\delta$?

Explanation: The general ellipse equation for equal frequencies incorporates the phase difference $\delta$ via the cross-term containing $\cos(\delta)$ and $\sin^2(\delta)$.d

8. If the phase difference $\delta$ between two equal frequency SHMs is $0$, what is the orientation of the straight-line Lissajous figure?

Explanation: When $\delta = 0$, $y = (b/a)x$, which represents a straight line sloping upwards with a positive slope.

9. How can the phase angle $\delta$ for a centered elliptical Lissajous figure with equal frequencies be determined experimentally from its graphical intercepts?

Explanation: The phase angle can be calculated using $\sin(\delta) = y_{(x=0)} / y_{max}$, where $y_{(x=0)}$ is the Y-intercept and $y_{max}$ is the peak Y amplitude.

10. What shape is traced when $\omega_x = \omega_y$, amplitudes $a \neq b$, and the phase difference is $\delta = \pi/2$?

Explanation: When $\delta = \pi/2$, the cross-term vanishes and the equation reduces to $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, which is an ellipse aligned with the X and Y axes.

11. What does the angle of tilt $\theta$ of the principal axes of a general elliptical Lissajous figure depend upon?

Explanation: The orientation angle $\theta$ is given by $\tan(2\theta) = \frac{2ab \cos(\delta)}{a^2 - b^2}$, depending on $a$, $b$, and $\delta$.

12. If two SHMs along X and Y have general displacements $x = a \sin(\omega_x t)$ and $y = b \sin(\omega_y t + \delta)$, what physical quantity does $\delta$ represent?

Explanation: The parameter $\delta$ explicitly denotes the initial phase difference between the Y-motion and the X-motion.

13. Which of the following frequency ratios $\omega_y / \omega_x$ produces a figure with 3 horizontal loops and 2 vertical loops?

Explanation: The ratio of horizontal loops to vertical loops gives the frequency ratio $N_x / N_y = \omega_y / \omega_x$. Here, 3 horizontal loops and 2 vertical loops give $3/2$.

14. Why is a straight line obtained when the phase difference is $\delta = n\pi$?

Explanation: When $\delta = n\pi$, $\sin^2(\delta) = 0$, leading to a degenerate quadratic equation representing a straight line.

15. What is a common misconception regarding straight-line Lissajous figures?

Explanation: A common misconception is that a straight line only means $\delta = 0$. In reality, any $\delta = n\pi$ (including $\pi, 2\pi$, etc.) produces a straight line.

PYQ Patterns

These are pattern-based practice questions. Authentic year labels are only used when verified from a source.

1. A particle is subjected to two mutually perpendicular SHMs given by $x = a \sin(\omega t)$ and $y = a \sin(\omega t + \pi/2)$. The resulting trajectory is:

Explanation: With equal amplitudes ($a = b = a$) and phase difference $\delta = \pi/2$, the equation becomes $x^2 + y^2 = a^2$, which is a circle.

2. If the Lissajous figure formed by two equal frequency perpendicular SHMs is a straight line sloping downwards, the phase difference $\delta$ between them is:

Explanation: A straight line sloping downwards corresponds to $\delta = \pi$ (or an odd multiple of $\pi$), resulting in $y = -(b/a)x$.

3. In a CRO display of Lissajous figures, 4 horizontal tangents and 2 vertical tangents are counted. What is the ratio of the frequency of the Y-vibration to the X-vibration ($\omega_y / \omega_x$)?

Explanation: The frequency ratio $\omega_y / \omega_x = N_x / N_y = 4 / 2 = 2$ (or $2/1$).

4. What is the inclination angle $\theta$ of the principal axes of an elliptical Lissajous figure when the amplitudes along X and Y are equal ($a = b$) and phase difference $\delta = \pi/2$?

Explanation: When $a = b$, the denominator $a^2 - b^2$ in $\tan(2\theta) = \frac{2ab \cos(\delta)}{a^2 - b^2}$ becomes zero, and the figure is a circle where orientation angles are not uniquely defined since it is rotationally symmetric.

5. Which phase difference yields an ellipse whose axes are tilted with respect to the coordinate axes when amplitudes $a$ and $b$ are unequal?

Explanation: When $\delta$ is neither $0$, $\pi/2$, nor $\pi$ (e.g., $\pi/4$), the $\cos(\delta)$ term is non-zero and $a \neq b$, tilting the ellipse.

6. The Cartesian equation for equal frequency Lissajous figures reduces to $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ when the phase difference $\delta$ is:

Explanation: When $\delta = \pi/2$, $\cos(\delta) = 0$ and $\sin^2(\delta) = 1$, giving the standard upright ellipse equation.

Timed Topic Test

Attempt these questions under a ten-minute limit using a separate timer.

1. What is the bounding rectangle dimensions for a Lissajous figure formed by two SHMs with amplitudes $a$ and $b$ along X and Y respectively?

Explanation: The motion is strictly confined within $-a \le x \le a$ and $-b \le y \le b$, making the bounding rectangle of sides $2a$ and $2b$.

2. If a Lissajous figure touches the Y-axis at $y = y_{(x=0)}$ and has a maximum Y amplitude of $y_{max}$, which formula correctly links this to the phase difference for $\omega_x = \omega_y$?

Explanation: At $x = 0$, $x = a \sin(\omega_x t) = 0$, which leads to $\sin(\delta) = y_{(x=0)} / y_{max}$ for centered ellipses.

3. Under what frequency condition does a Lissajous figure form closed loops with multiple intersections rather than a simple single-loop ellipse?

Explanation: When frequencies are in the ratio of small integers ($n/m$), closed open curves and multiple loops are formed.

4. What happens to the Lissajous pattern on a CRO if the unknown frequency drifts slightly from the rational integer ratio relative to the standard oscillator?

Explanation: A slight drift in frequency disrupts the phase locking, causing the Lissajous figure to continuously morph and rotate.

5. For equal frequencies and amplitudes ($a = b$), what is the Lissajous trajectory when $\delta = 3\pi/2$?

Explanation: When $\delta = 3\pi/2$, $\sin(\delta) = -1$ and $\cos(\delta) = 0$, yielding $x^2 + y^2 = a^2$, which is a circle.

6. What is the slope of the straight line obtained when $\omega_x = \omega_y$ and $\delta = 0$?

Explanation: When $\delta = 0$, $y = (b/a)x$, so the slope is $b/a$.

7. In terms of the general equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{2xy}{ab} \cos(\delta) = \sin^2(\delta)$, what is the value of the cross-term coefficient when $\delta = \pi/2$?

Explanation: Since $\cos(\pi/2) = 0$, the entire cross-term vanishes when $\delta = \pi/2$.

8. What is the primary utility of Lissajous figures in electronics and physics laboratories?

Explanation: Lissajous figures are extensively used with CROs to compare unknown frequencies with standard frequencies and to measure phase differences.

9. If $\omega_y / \omega_x = 1/2$, how many vertical loops (tangents touching the sides) are expected in the Lissajous figure?

Explanation: Using $\omega_y / \omega_x = N_x / N_y = 1/2$, the number of vertical loops ($N_y$) corresponding to the X-vibration frequency is 2.

10. Which of the following statements is correct regarding the general parametric equations of Lissajous figures?

Explanation: The parametric equations $x = a \sin(\omega_x t)$ and $y = b \sin(\omega_y t + \delta)$ describe the 2D path resulting from two orthogonal SHMs.

Lissajous Figures

Scope and prerequisites

TGT core

PGT extension

Prerequisites

Definition and Principle of Superposition

Case of Equal Frequencies (&omega;<sub>x</sub> = &omega;<sub>y</sub> = &omega;)

Case of Harmonic Frequencies (&omega;<sub>y</sub> / &omega;<sub>x</sub> = n / m)

Experimental Determination using Cathode-Ray Oscilloscope (CRO)

Derivations

Derivation of the General Cartesian Equation for Equal Frequencies

Determination of Phase Difference from Axial Intercepts

Formula sheet

Graphs and experiments

Lissajous Figure for &delta; = 0 (&omega;<sub>x</sub> = &omega;<sub>y</sub>)

Lissajous Figure for &delta; = &pi;/2, a = b (&omega;<sub>x</sub> = &omega;<sub>y</sub>)

Lissajous Figure for Frequency Ratio 1:2

Solved numericals

Two perpendicular SHMs are given by $x = 3 \sin(\omegat)$ and $y = 3 \sin(\omegat + \pi/2)$. What is the resulting Lissajous figure?

A Lissajous figure formed by two orthogonal SHMs has 3 horizontal tangents and 2 vertical tangents. What is the frequency ratio ωy / ωx?

An elliptical Lissajous figure with equal amplitudes $a = b has a Y-intercept of 2 cm$ and a maximum Y-amplitude of 4 cm. Calculate the phase difference δ.

Assumptions and limitations

Common misconceptions

Quick revision

Must remember

Glossary

Characteristics of Lissajous Figures for Equal Frequencies

Practice Quiz

1. What trajectory is formed when two mutually perpendicular simple harmonic motions of equal frequencies and equal amplitudes are superposed with a phase difference of $\delta = \pi/2$?

2. Under what condition does the general equation of Lissajous figures for equal frequencies degenerate into a straight line sloping downwards?

3. If the frequency of the Y-vibration is twice that of the X-vibration ($\omega_y / \omega_x = 2$), how many horizontal loops are typically observed in the closed Lissajous figure?

4. Which instrument is commonly used experimentally to display Lissajous figures by applying sinusoidal signals to its X and Y plates?

5. What does a stationary, non-rotating Lissajous pattern observed on a CRO indicate about the two applied frequencies?

6. In the tangent counting method for determining frequency ratios, if $N_x$ is the number of tangents touching the top/bottom and $N_y$ is the number touching the sides, what is the frequency ratio $\omega_y / \omega_x$?

7. What is the Cartesian equation for Lissajous figures when $\omega_x = \omega_y = \omega$ and phase difference is $\delta$?

8. If the phase difference $\delta$ between two equal frequency SHMs is $0$, what is the orientation of the straight-line Lissajous figure?

9. How can the phase angle $\delta$ for a centered elliptical Lissajous figure with equal frequencies be determined experimentally from its graphical intercepts?

10. What shape is traced when $\omega_x = \omega_y$, amplitudes $a \neq b$, and the phase difference is $\delta = \pi/2$?

11. What does the angle of tilt $\theta$ of the principal axes of a general elliptical Lissajous figure depend upon?

12. If two SHMs along X and Y have general displacements $x = a \sin(\omega_x t)$ and $y = b \sin(\omega_y t + \delta)$, what physical quantity does $\delta$ represent?

13. Which of the following frequency ratios $\omega_y / \omega_x$ produces a figure with 3 horizontal loops and 2 vertical loops?

14. Why is a straight line obtained when the phase difference is $\delta = n\pi$?

15. What is a common misconception regarding straight-line Lissajous figures?

PYQ Patterns

1. A particle is subjected to two mutually perpendicular SHMs given by $x = a \sin(\omega t)$ and $y = a \sin(\omega t + \pi/2)$. The resulting trajectory is:

2. If the Lissajous figure formed by two equal frequency perpendicular SHMs is a straight line sloping downwards, the phase difference $\delta$ between them is:

3. In a CRO display of Lissajous figures, 4 horizontal tangents and 2 vertical tangents are counted. What is the ratio of the frequency of the Y-vibration to the X-vibration ($\omega_y / \omega_x$)?

4. What is the inclination angle $\theta$ of the principal axes of an elliptical Lissajous figure when the amplitudes along X and Y are equal ($a = b$) and phase difference $\delta = \pi/2$?

5. Which phase difference yields an ellipse whose axes are tilted with respect to the coordinate axes when amplitudes $a$ and $b$ are unequal?

6. The Cartesian equation for equal frequency Lissajous figures reduces to $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ when the phase difference $\delta$ is:

Timed Topic Test

1. What is the bounding rectangle dimensions for a Lissajous figure formed by two SHMs with amplitudes $a$ and $b$ along X and Y respectively?

2. If a Lissajous figure touches the Y-axis at $y = y_{(x=0)}$ and has a maximum Y amplitude of $y_{max}$, which formula correctly links this to the phase difference for $\omega_x = \omega_y$?

3. Under what frequency condition does a Lissajous figure form closed loops with multiple intersections rather than a simple single-loop ellipse?

4. What happens to the Lissajous pattern on a CRO if the unknown frequency drifts slightly from the rational integer ratio relative to the standard oscillator?

5. For equal frequencies and amplitudes ($a = b$), what is the Lissajous trajectory when $\delta = 3\pi/2$?

6. What is the slope of the straight line obtained when $\omega_x = \omega_y$ and $\delta = 0$?

7. In terms of the general equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{2xy}{ab} \cos(\delta) = \sin^2(\delta)$, what is the value of the cross-term coefficient when $\delta = \pi/2$?

8. What is the primary utility of Lissajous figures in electronics and physics laboratories?

9. If $\omega_y / \omega_x = 1/2$, how many vertical loops (tangents touching the sides) are expected in the Lissajous figure?

10. Which of the following statements is correct regarding the general parametric equations of Lissajous figures?

Case of Equal Frequencies (ω<sub>x</sub> = ω<sub>y</sub> = ω)

Case of Harmonic Frequencies (ω<sub>y</sub> / ω<sub>x</sub> = n / m)

Lissajous Figure for δ = 0 (ω<sub>x</sub> = ω<sub>y</sub>)

Lissajous Figure for δ = π/2, a = b (ω<sub>x</sub> = ω<sub>y</sub>)

A Lissajous figure formed by two orthogonal SHMs has 3 horizontal tangents and 2 vertical tangents. What is the frequency ratio ω_y / ω_x?