Podcast
Questions and Answers
What is the shape of the resultant path when two mutually perpendicular SHMs are superimposed with a phase difference of $ rac{ rac{ heta}{2}}$?
What is the shape of the resultant path when two mutually perpendicular SHMs are superimposed with a phase difference of $ rac{ rac{ heta}{2}}$?
- Ellipse (correct)
- Circle
- Parabola
- Straight line
Which equation correctly describes the motion of a particle in simple harmonic motion?
Which equation correctly describes the motion of a particle in simple harmonic motion?
- $x = A an( heta) + B heta$
- $x = A rac{1}{ heta} + B heta^2$
- $x = A_{0} rac{1}{ heta} + B_{0} heta$
- $x = A_{0} an( heta) + B_{0} an( heta)$ (correct)
Which of the following functions represents a particle exhibiting SHM?
Which of the following functions represents a particle exhibiting SHM?
- $y = rac{1}{ heta}$
- $y = rac{1}{2} an( heta) + rac{1}{2} heta$ (correct)
- $y = an( heta)$
- $y = e^{ heta} + heta^2$
How is the amplitude of a simple harmonic motion represented mathematically?
How is the amplitude of a simple harmonic motion represented mathematically?
Which of these equations would not represent simple harmonic motion?
Which of these equations would not represent simple harmonic motion?
What is the correct expression for the amplitude of oscillation in SHM?
What is the correct expression for the amplitude of oscillation in SHM?
What type of path does the resultant oscillation describe when two perpendicular SHM waves with different frequencies are combined?
What type of path does the resultant oscillation describe when two perpendicular SHM waves with different frequencies are combined?
Which equation represents an ellipse when combined with SHM equations?
Which equation represents an ellipse when combined with SHM equations?
Which of the following functions does not represent simple harmonic motion?
Which of the following functions does not represent simple harmonic motion?
Which of the following represents a function that may combine to produce SHM?
Which of the following represents a function that may combine to produce SHM?
Flashcards are hidden until you start studying
Study Notes
Superposition of Two Mutually Perpendicular SHMs
- If two perpendicular simple harmonic motions (SHMs) with the same frequency and a phase difference of $\frac{\pi}{2}$ are superimposed, the resulting path of the particle is an ellipse.
- This can be represented by the equation: $\frac{x^{2}}{A_{1}^{2}} + \frac{y^{2}}{A_{2}^{2}} = 1$ where $x = A_{1} \sin(\omega t)$ and $y = A_{2} \cos(\omega t)$
Equation of Simple Harmonic Motion
- The equation of SHM for a particle is given by: $x = A_{0} \sin(\omega t) + B_{0} \cos(\omega t)$.
- The amplitude of this SHM is: $Ampl.= \sqrt{A_{0}^{2}+B_{0}^{2}}$.
Identifying SHM
- A function represents SHM if it can be written in the form: $y = A_{0} + A \sin(\omega t) + B \cos(\omega t)$.
- The amplitude of such a function is given by: $Y = A_{0} + A \sin(\omega t) + B \cos(\omega t)$ and Amplitude $= \sqrt{A^2 + B^2}$
- The following functions represent SHM:
- $y = \sin(\omega t) - \cos(\omega t)$
- $y = \cos(\frac{\pi}{2} - \omega t)$
- $y = a \sin(\omega t) + b \cos(\omega t)$
- The following does not represent SHM:
- $y = a + \omega t + \omega^{2} t^{2} /2$
Resultant Path of Superimposed SHMs
- The resultant path of two superimposed SHMs will be an ellipse if they are perpendicular and have different frequencies.
- This can be represented by the equation $\frac{x^2}{A_1^2} + \frac{y^2}{A_2^2} = 1$, where $x = A_1 \sin(\omega t)$ and $y = A_2 \cos(\omega t)$.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
