Question

A uniform block of mass \(M\) slides on a smooth horizontal bar. Another mass \(m\) is connected to it by an inextensible string of length \(l\) of negligible mass, and is constrained to oscillate in the X-Y plane only. Neglect the sizes of the masses. The number of degrees of freedom of the system is two and the generalized coordinates are chosen as \(x\) and \(\theta\), as shown in the figure. 

If \(p_x\) and \(p_\theta\) are the generalized momenta corresponding to \(x\) and \(\theta\), respectively, then the correct option(s) is(are)

• A. \( p_x = (m + M)\dot{x} + ml \cos\theta \, \dot{\theta} \)
• B. \( p_\theta = ml^2 \dot{\theta} - ml \cos\theta \, \dot{x} \)
• C. \( p_x \) is conserved
• D. \( p_\theta \) is conserved

Hint:

The generalized momenta are obtained by differentiating the Lagrangian with respect to the generalized velocities. Conservation of momentum occurs if no external forces are acting in that direction.

Answer 1

The Correct Option is A, C

The system described consists of two masses and a string that connects them. The generalized coordinates are \(x\) (the displacement of the block \(M\)) and \(\theta\) (the angle of the string relative to the horizontal). For the generalized momenta, we need to compute the partial derivatives of the Lagrangian with respect to the generalized velocities. The Lagrangian is a function of the kinetic and potential energies of the system. The kinetic energy of the system is given by the sum of the energies due to both masses: \[ T = \frac{1}{2}(m + M)\dot{x}^2 + \frac{1}{2} m l^2 \dot{\theta}^2 + m l \dot{x} \dot{\theta} \cos\theta \] The generalized momenta are given by: \[ p_x = \frac{\partial L}{\partial \dot{x}} = (m + M)\dot{x} + ml \cos\theta \, \dot{\theta} \] \[ p_\theta = \frac{\partial L}{\partial \dot{\theta}} = m l^2 \dot{\theta} - m l \cos\theta \, \dot{x} \] Thus, option (A) is correct for \(p_x\), and option (C) is correct as \(p_x\) is conserved (since no external forces are acting in the \(x\)-direction). Therefore, the correct answers are (A) and (C).