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 \( m \). \[ \text{Assume that the spring deflects only horizontally and force } F \text{ is applied horizontally at Q. For this system, the Lagrangian, } L \text{ is} \]

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

Hint:

When deriving the Lagrangian for a system, include the kinetic and potential energy terms for both translational and rotational motion, and consider the work done by any external forces.

Answer 1

The Correct Option is B

The Lagrangian \( L \) is given by the difference between the kinetic and potential energies of the system.
- The kinetic energy consists of:
1. The translational kinetic energy of the mass \( M + m \) moving horizontally with velocity \( \dot{x} \).
2. The rotational kinetic energy of the rod, which involves the angular velocity \( \dot{\theta} \) and the moment of inertia of the rod.
- The potential energy consists of:
1. The spring potential energy, \( \frac{1}{2} k x^2 \). 2. The gravitational potential energy, \( mg \cos \theta \), where \( g \) is the gravitational constant.
Additionally, there is an applied horizontal force \( F \) at point Q, and the Lagrangian includes the work done by this force, \( F b \sin \theta \).
Thus, the Lagrangian for this system is given by: \[ L = \frac{1}{2} (M + m) \dot{x}^2 + \frac{1}{2} m b \dot{x} \cos \theta + \frac{1}{6} m b^2 \dot{\theta}^2 - \frac{1}{2} k x^2 + mg \cos \theta + F b \sin \theta \] Therefore, the correct answer is (B).