Question

The Lagrangian of a simple pendulum, consisting of a bob of mass \( m \) suspended by a string of length \( l \), executing oscillations of amplitude \( \theta \) about the equilibrium position is given by:

• A. \( m \ddot{\theta} - \frac{g}{l} \sin\theta \)
• B. \( \frac{1}{2} m l^2 \dot{\theta}^2 - mg l \cos\theta \)
• C. \( \frac{1}{2} m l^2 \dot{\theta}^2 - mg l (1 + \cos\theta) \)
• D. \( \frac{1}{2} m l^2 \dot{\theta}^2 - mg l (1 - \cos\theta) \)

Hint:

The Lagrangian approach simplifies equations of motion for constrained systems.

Answer 1

The Correct Option is D

The Lagrangian is given by:
\[ L = T - V \] where,
\[ T = \frac{1}{2} m l^2 \dot{\theta}^2 \] \[ V = mg l (1 - \cos\theta) \] Thus,
\[ L = \frac{1}{2} m l^2 \dot{\theta}^2 - mg l (1 - \cos\theta) \]