A concentrated mass m is attached at the centre of a rod

A concentrated mass m is attached at the centre of a rod of Iength 2L as shown in the figure. The rod is kept in a horizontal equilibrium position by a spring of stiffness k. For very small amplitude of vibration, neglecting the weights of the rod and spring, the undamped natural frequency of the system is

Equation of motion, is
m( Lθ̇ ).Lθ + kx × 2 L = 0
⇒ m( Lθ̇ ).Lθ + k(2L.θ) × 2 L = 0
⇒ mL²θ̇̇ + 4kL² θ = 0
⇒ θ̇̇ + √4k / mθ = 0

∴ ω_n = √	4k
	m