Question

Superposition of two perpendicular harmonic oscillation by graphical and analytical methods

Answer 1

Answer:

When a particle is subject to more than one harmonic force, each trying to move the particle in is own direction with SHM, we say that there is an interference of SHMs.

Explanation:

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Answer 2

The superposition of two perpendicular harmonic oscillation by graphical and analytical methods is given below.

Given:

The number of perpendicular harmonic oscillators = 2.

To Find:

We have to find the superposition of two perpendicular harmonic oscillation by graphical and analytical methods.

Solution:

• Simple harmonic motion (SHM) is a periodic motion in which the restoring force on the moving body acts towards its own equilibrium position and is directly proportional to the magnitude of the displacement of the object.

Consider two perpendicular harmonic oscillators having same frequency. The SHM produced by the first oscillator in x-direction is given by,

x = $A_1$ sin ωt

The SHM produced by the second oscillator in y-direction is given by,

y = $A_2$ sin (ωt + δ).

δ is the phase difference between the two perpendicular oscillators. The resultant motion of the two oscillators is a combination of the above two SHM's.

The resultant motion of two oscillators follows a two-dimensional elliptical path. The equation for this path is obtained by eliminating the term "t" from x and y.

From the equation of SHM produced by the first oscillator, we get,

sin ωt = $\frac{x}{A_1}$

Or, cos ωt = $\sqrt{1- (\frac{x}{A_1})^2}$.

Consider the SHM, y = $A_2$ sin (ωt + δ).

This can be written as,

y = $A_2$ [sin ωt × cos δ + cos ωt × sin δ]

Substituting the values of sin ωt and cos ωt in the above equation, we get,

y = $A_2$  $[\frac{x}{A_1}$ × cos δ + $\sqrt{1- (\frac{x}{A_1})^{2} }$ × sin δ]

Or, $(\frac{y}{A_2} - \frac{x}{A_1}$  cos δ$)^{2}$ = $(1- (\frac{x}{A_1})^2)$ sin² δ)

$\frac{x^{2} }{(A_1)^2} + \frac{y^{2} }{(A_2)^2} - \frac{2xy.cosδ}{A_1 . A_2}$ = sin² δ.

This is in the form of the equation for an ellipse. The motion of the two oscillators always remains inside the rectangle defined by $x =$ ± $A_1$ and $y =$ ± $A_2$.

The graphical representation of the two-dimensional elliptical path followed by the oscillators is given below.

Hence, the superposition of two perpendicular harmonic oscillation by graphical and analytical methods is represented as above.

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