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

• A. \( \frac{1}{2}\left( {M + m} \right){\dot x^2} + \frac{1}{2}mb\dot \theta \dot x\cos \theta + \frac{1}{6}m{b^2}{\dot \theta ^2} - \frac{1}{2}k{x^2} + mg\frac{b}{2}\cos \theta \)
• B. \(\frac{1}{2}m{\dot x^2} + \frac{1}{2}mb\dot \theta \dot x\cos \theta + \frac{1}{6}m{b^2}{\dot \theta ^2} - \frac{1}{2}k{x^2} + mg\frac{b}{2}\cos \theta + fb\sin \theta \)
• C. \(\frac{1}{2}m{\dot x^2} + \frac{1}{2}mb\dot \theta \dot x\cos \theta + \frac{1}{6}m{b^2}{\dot \theta ^2} - \frac{1}{2}k{x^2} \)
• D. \( \frac{1}{2}\left( {M + m} \right){\dot x^2} + \frac{1}{2}mb^2\dot \theta^2 - \frac{1}{2}k{x^2} + mg\frac{b}{2}\cos \theta \)