Simple harmonic motion can in some cases be considered to be the one-dimensional projection of uniform circular motion . If an object...
Simple harmonic motion can in some cases be considered to be the one-dimensional projection of uniform circular motion. If an object moves with angular speed ω around a circle of radius A centered at the origin of the x−y plane, then its motion along each coordinate is simple harmonic motion with amplitude A and angular frequency ω.
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| Understanding Phase Difference |
Q1: given that, a circular motion can be described by x = A cos(ω t) and y = A sin(ω t) what is the y-component model-equation that can describe the motion of a uniform circular motion?
A1: \(y = A\sin \omega t\)
Q2: When the x-component of the circular motion is modelled by x = A cos(ω t) and y = A sin(ω t) suggest an model-equation for y.
A2: \(y = A\cos \omega t\) for top position or \(y = - A\cos \omega t\) for bottom position
Q3: Explain why are the models for both x and y projection of a uniform circular motion, a simple harmonic motion?
A3: both \(x = A\cos \omega t\) and \(y = a\sin \omega t\) each follow the defining relationship for SHM as ordinary differential equations of \(\frac{{{d^2}x}}{{d{t^2}}} = - {\omega ^2}x\) and \(\frac{{{d^2}y}}{{d{t^2}}} = - {\omega ^2}y\) respectively.
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