Superposition of SHMs and Motion Equations
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Superposition of SHMs and Motion Equations

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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}}$?

  • Ellipse (correct)
  • Circle
  • Parabola
  • Straight line

    • 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?

  • $y = rac{1}{ heta}$
  • $y = rac{1}{2} an( heta) + rac{1}{2} heta$ (correct)
  • $y = an( heta)$
  • $y = e^{ heta} + heta^2$

    • How is the amplitude of a simple harmonic motion represented mathematically?

    <p>$Ampl. = ext{sqrt}(A_{0}^{2}+B_{0}^{2})$</p> Signup and view all the answers

    Which of these equations would not represent simple harmonic motion?

    <p>$y = a + rac{1}{2} heta^2 - heta$</p> Signup and view all the answers

    What is the correct expression for the amplitude of oscillation in SHM?

    <p>$ rac{A^2 + B^2}{2}$</p> Signup and view all the answers

    What type of path does the resultant oscillation describe when two perpendicular SHM waves with different frequencies are combined?

    <p>An ellipse</p> Signup and view all the answers

    Which equation represents an ellipse when combined with SHM equations?

    <p>$ rac{x^2}{A^2} + rac{y^2}{B^2} = 1$</p> Signup and view all the answers

    Which of the following functions does not represent simple harmonic motion?

    <p>$y = A an( heta)$</p> Signup and view all the answers

    Which of the following represents a function that may combine to produce SHM?

    <p>$y = A_1 ext{sin}( heta) + B_2 ext{cos}( heta)$</p> Signup and view all the answers

    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 $y = A_2 \cos(\omega t)$.

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    Description

    Explore the superposition of two mutually perpendicular simple harmonic motions and their resulting elliptical path. Test your understanding of the equations governing simple harmonic motion and how to identify SHM in various functions.

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