Question

Derive differential equation for simple harmonic motion

Answer 1

⭐《ANSWER》

$\huge\mathfrak{\underline {\underline\pink {ANSWER}}}$

↪Actually welcome to the concept of the SIMPLE HARMONIC MOTION ,

↪Basically we know that , in the Simple harmonic motion , the Restoring force is always directed towards the displacement of the particle from the mean postion in the opposite direction,

↪So we mathematically get as ,

↪F is directly proportional to the x

↪here , F = Force , x = displacemet from the mean position , k = force constant ,

↪so removing the proportional Sign we get as,

〽F = - k x

↪now , we aslo know that , according to the NEWTONS SECOND LAW OF MOTION ,

↪F = m a ___( where , a = Acceleration of the particle and m is the mass)

↪so we equate anf get as ,

↪ma = -k x

==》 ma + kx = 0

↪ since in the calculus form , a = d^2x / dt^2

==》 m d^2x/dt^2 + k x = 0

↪Dividing throughout by 'm' ,

〽d^2x/dt^2 + kx/m = 0

↪now here , we know that ,

↪w = underoot k/m , so, w^2 = k/m

↪now ,

↪d^2x/dt^2 + w^2 x = 0

↪here , w = Angular frequency ,

↪so the differential equation of SHM is ,

⭐d^2x/dt^2 + w^2 x =0