A thin uniform rigid bar of length L and M is hinged at

A thin uniform rigid bar of length L and M is hinged at point O, located at a distance of L/3 from one of its ends. The bar is further supported using springs, each of stiffness k located at the two ends. A particle of mass m = M/4 is fixed at one end of the bar, as shown in the figure. For small rotations of the bar about O, the natural frequency of the system is

I = (I_{about Ⅰ})_rod + (I_{about Ⅰ})_{point mass}

I =		ml²	+ m		2l	-	l		²			m		2l		²
		12			3		2					4		3

I =	ml²	+	ml²	=	2ml²
	9		9		9

∑M_I = 0

I θ̇̇ + Kx₂ +	2l	+ kx₁ ×	l	= 0
	3		3

2ml²	+ k		5	l²		θ = 0
9			9