InterviewSolution
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InterviewSolution
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Identify which of the following function represent simple harmonic motion. (i) `Y = Ae^(I omega t)` (ii) `Y = a e^(- omega t)` (iii) `y = a sin^(2) omega t` (iv) `y = a sin omega t + b cos omega t` (v) `y = sin omega t + b cos 2 omega t` |
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Answer» (i) According to given equation in problem differentiating with respect to time, we get `(dy)/(dt)` `= I A omega e^(i omega t)` differentiating again with respect to time , we get `(d^(2)y)/(dt^(2)) + - omega^(2) A e^(i omega t) = - omega^(2) y [as y = A e^(i omega t)]` Thus we have `(d^(2)y)/(dt^(2)) + omega^(2) y = 0` This is the basic differential equation of SHM. (ii) The function `y = ae^(-omega t)` is not harmonic as it is not expressed in terms of sine and cosine functions, So, it cannot simple harmonic. Moreover, this function is not periodie. (iii) `y = a sin^(2) omega t` The function `y = a sin^(2) omega t` is harmonic. To become simple harmonic `(d^(2)y)/(dt^(2)) prop y` Here, `(dy)/(dt) = 2 a omega sin ^(2) omega t cos omega t` `(d^(2)y)/(dt^(2)) = 2 a omega^(2) [cos^(2) omega t - sin ^(2) omega t] = 2 a omega^(2) [1 - 2 sin^(2) omega t]` `= 2a omega^(2) [1 - (2y)/(a)]` The function is not simple harmonic. (iv) `y = a sin^(2) omega t + b cos omega t` The function `y = a sin omega t` is simple harmonic Because `(dy)/(dt) = omega a cos omega t - omega b sin omega t` `(d^(2)y)/(dt^(2)) = - omega^(2) a sin omega t - omega^(2) b cos omega t implies (d^(2)y)/(dt^(2)) = - omega^(2) y` This is the basic different equation of SHM. ` y = sin omega t + cos 2 omega t` The function `y = sin omega t + cos 2 omega t` is not simple harmonic `(d^(2)y)/(dt^(2)) = - omega^(2) sin omega t - 4 omega^(2) cos 2 omega t = - omega^(2) [sin omega t + 4 cos 2 omega t]` `(d^(2)y)/(dt^(2)) "ne" - omega^(2) y` The function is not simple harmonic. |
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