A uniform rigid rod of mass m = 1 kg and length L = 1 m

A uniform rigid rod of mass m = 1 kg and length L = 1 m is hinged at its centre & laterally supported at one end by a spring of spring constant k = 300 N/m. The natural frequency ω_n in rad/s is

	1	Iθ̇ ² +	1	kx² = Constant
	2		2

	1	Iθ̇ ² +	1	k		L	θ		²	= Constant
	2		2			2

Differentiating above equation w.r.t. t, we get

	1	Iθ̇̇ +	kL²θ	= 0
	2		4

I =	mL²
	12

⇒	mL²	θ̇̇ +	kL²	θ = 0
	12		4

θ̇̇ +	3k	θ = 0
	m

ω_n = √3k / m = √(3 × 300) / 1 = 30 rad/s