Consider a two degree of freedom system as shown in the figure, where PQ is a rigid uniform rod of length, 𝒃 and mass, 𝒎.

Assume that the spring deflects only horizontally and force F is applied horizontally at Q. For this system, the Lagrangian, L is

Options :

1. $\frac{1}{2}m{\dot{x}}^2+\frac{1}{2}mb\dot{\theta }\dot{x}\cos \theta +\frac{1}{6}m{b}^2{\dot{\theta }}^2-\frac{1}{2}k{x}^2$

2. $\frac{1}{2}m{\dot{x}}^2+\frac{1}{2}mb\dot{\theta }\dot{x}\cos \theta +\frac{1}{6}m{b}^2{\dot{\theta }}^2-\frac{1}{2}k{x}^2+mg\frac{b}{2}\cos \theta +fb\sin \theta$

3. $\frac{1}{2}\left(M+m\right){\dot{x}}^2+\frac{1}{2}mb\dot{\theta }\dot{x}\cos \theta +\frac{1}{6}m{b}^2{\dot{\theta }}^2-\frac{1}{2}k{x}^2+mg\frac{b}{2}\cos \theta$

4. $\frac{1}{2}\left(M+m\right){\dot{x}}^2+\frac{1}{2}m{b}^2{\dot{\theta }}^2-\frac{1}{2}k{x}^2+mg\frac{b}{2}\cos \theta$