Q.6 Obtain the differential equation of linear simple harmonic motion.
Best answer
Answer: \(\dfrac{d^2x}{dt^2} + \omega^2 x = 0\), where \(\omega^2 = k/m\).
In linear S.H.M., the restoring force is directly proportional to the displacement from the mean position and is directed towards the mean position:
\(F = -kx\) … (1) where \(k\) is the force constant.
By Newton's second law, \(F = ma = m\dfrac{d^2x}{dt^2}\) … (2)
From (1) and (2): \(m\dfrac{d^2x}{dt^2} = -kx\), i.e. \(m\dfrac{d^2x}{dt^2} + kx = 0\), or
\(\dfrac{d^2x}{dt^2} + \dfrac{k}{m}x = 0\). Putting \(\dfrac{k}{m} = \omega^2\) (\(\omega\) = angular frequency):
\(\dfrac{d^2x}{dt^2} + \omega^2 x = 0\). This is the differential equation of linear S.H.M.
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