Test Academy | 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 1 2 m x ˙ 2 + 1 2 m b θ ˙ x ˙ cos ⁡ θ + 1 6 m b 2 θ ˙ 2 − 1 2 k x 2 , 1 2 m x ˙ 2 + 1 2 m b θ ˙ x ˙ cos ⁡ θ + 1 6 m b 2 θ ˙ 2 − 1 2 k x 2 + m g b 2 cos ⁡ θ + f b sin ⁡ θ , 1 2 ( M + m ) x ˙ 2 + 1 2 m b θ ˙ x ˙ cos ⁡ θ + 1 6 m b 2 θ ˙ 2 − 1 2 k x 2 + m g b 2 cos ⁡ θ , 1 2 ( M + m ) x ˙ 2 + 1 2 m b 2 θ ˙ 2 − 1 2 k x 2 + m g b 2 cos ⁡ θ

Question

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 :

$\frac{1}{2} m {\dot{x}}^{2} + \frac{1}{2} m b \dot{θ} \dot{x} \cos θ + \frac{1}{6} m b^{2} {\dot{θ}}^{2} - \frac{1}{2} k x^{2}$
$\frac{1}{2} m {\dot{x}}^{2} + \frac{1}{2} m b \dot{θ} \dot{x} \cos θ + \frac{1}{6} m b^{2} {\dot{θ}}^{2} - \frac{1}{2} k x^{2} + m g \frac{b}{2} \cos θ + f b \sin θ$
$\frac{1}{2} (M + m) {\dot{x}}^{2} + \frac{1}{2} m b \dot{θ} \dot{x} \cos θ + \frac{1}{6} m b^{2} {\dot{θ}}^{2} - \frac{1}{2} k x^{2} + m g \frac{b}{2} \cos θ$
$\frac{1}{2} (M + m) {\dot{x}}^{2} + \frac{1}{2} m b^{2} {\dot{θ}}^{2} - \frac{1}{2} k x^{2} + m g \frac{b}{2} \cos θ$

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