Question

setup the second order differential equation of simple harmonic oscillator and solve it to find the position and velocity of oscillator.​

Answer 1

Solution :

From Newton's Second Law of Motion,

$\sf{f \propto \: - x} \\ \\ \longmapsto \: \sf{ma \propto - x} \\ \\ \longmapsto \: \sf{ma = - kx} \\ \\ \longmapsto \: \sf{ma + kx = 0} \\ \\ \longmapsto \: \boxed{ \boxed{\sf{a + \dfrac{k}{m}x = 0 }}}$

We know that,

$\displaystyle \sf{{ \omega = \sqrt{ \dfrac{k}{m} } }}$

and,

$\sf{a = \dfrac{d {}^{2}x }{ {dt}^{2} } }$

Now,

$\longmapsto \: \sf{ \dfrac{ {d}^{2}x }{ {dt}^{2} } + { \omega}^{2} x = 0 }$