Solution Found!
Consider a two-dimensional isotropic oscillator moving
Chapter 5, Problem 5.16(choose chapter or problem)
Consider a two-dimensional isotropic oscillator moving according to Equation (5.20). Show that if the relative phase is \(\delta=\pi / 2\), the particle moves in an ellipse with semimajor and semiminor axes \(A_{x}\) and \(A_{y}\).
Questions & Answers
QUESTION:
Consider a two-dimensional isotropic oscillator moving according to Equation (5.20). Show that if the relative phase is \(\delta=\pi / 2\), the particle moves in an ellipse with semimajor and semiminor axes \(A_{x}\) and \(A_{y}\).
ANSWER:
Step 1 of 2
The displacement along x direction is \(x(t)\)
The displacement along y direction is \(y(t)\)
The angular speed is \(\omega\)
The amplitude along x direction is \(A_{x}\)
The amplitude along y direction is \(A_{y}\)
The phase along x direction is \(\delta_{x}\)
The phase along y direction is \(\delta_{y}\)
The time is t
The displacement along x direction is,
\(x(t)=A_{x} \cos \omega t\)
Squaring the above,
\(\begin{array}{l} x^{2}=A_{x}^{2} \cos ^{2} \omega t \\ \frac{x^{2}}{A_{x}^{2}}=\cos ^{2} \omega t \end{array}\)
The displacement along y-direction is,
\(y(t)=A_{y} \cos (\omega t-\delta)\)