Question

Obtain the differential equation of linear simple harmonic motion.

Hint:

The differential equation for SHM relates the displacement to the acceleration, with the restoring force acting opposite to the displacement.

Answer 1

For simple harmonic motion (SHM), the restoring force is proportional to the displacement \( x \) from the equilibrium position:
\[ F = -kx \] By Newton's second law, \( F = ma \), where \( a \) is the acceleration. Therefore, we have:
\[ ma = -kx \] Since acceleration \( a = \frac{d^2x}{dt^2} \), the equation becomes:
\[ m \frac{d^2x}{dt^2} = -kx \] This is the differential equation of SHM, which can be written as:
\[ \frac{d^2x}{dt^2} + \frac{k}{m} x = 0 \]