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https://texercises.com/exercise/velocity-for-simple-harmonic-oscillation/
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Exercise:
A simple harmonic motion is given by yt A cosomega t Using the expression cosomega t frace^iomega t+e^-iomega t derive the expreesion for the velocity and verify that it has the well-known form.

Solution:
The velocity is the derivative of the displacement. For the derivative of cosx we find fractextrmdtextrmdtleftfrace^iomega t+e^-iomega tright frace^iomega t iomega+e^-iomega t-iomega iomega frace^iomega t-e^-iomega t i^omegafrace^iomega t-e^-iomega ti -omegasinomega t As expected it follows for the velocity that vt dot yt Afractextrmdtextrmdtcosomega t -Aomegasinomega t
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Exercise:
A simple harmonic motion is given by yt A cosomega t Using the expression cosomega t frace^iomega t+e^-iomega t derive the expreesion for the velocity and verify that it has the well-known form.

Solution:
The velocity is the derivative of the displacement. For the derivative of cosx we find fractextrmdtextrmdtleftfrace^iomega t+e^-iomega tright frace^iomega t iomega+e^-iomega t-iomega iomega frace^iomega t-e^-iomega t i^omegafrace^iomega t-e^-iomega ti -omegasinomega t As expected it follows for the velocity that vt dot yt Afractextrmdtextrmdtcosomega t -Aomegasinomega t
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euler formula, simple harmonic motion
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Calculative / Quantity
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