Question

A bead is constrained to move along a long, massless, frictionless horizontal rod parallel to the \( x \)-axis. The rod itself is moving vertically upward along the \( z \)-direction against gravity with a constant speed, starting from \( z = 0 \) at \( t = 0 \), and remains horizontal. The conjugate momenta are denoted by \( p_x, p_y, p_z \) and the Hamiltonian by \( H \). Which of the following option(s) is/are correct?

• A. \( H \) is the total energy of the system and is conserved
• B. \( H \) is the total energy of the system and is not conserved
• C. \( H \) is not the total energy of the system, but it is conserved
• D. \( H \) is not the total energy of the system and is not conserved

Hint:

In systems with external motion, such as a moving rod, the total energy described by the Hamiltonian may not be conserved. The motion of the external system (the rod in this case) affects the energy of the system.

Answer 1

The Correct Option is D

1. Total Energy:
The system involves a bead constrained to move along a horizontal rod, and the rod itself moves vertically upward with a constant speed. The bead experiences kinetic energy due to its motion along the rod, and gravitational potential energy due to its height relative to the rod. The total energy of the system would typically include both the kinetic energy of the bead and the potential energy due to gravity. 2. Conservation of the Hamiltonian:
The system’s Hamiltonian \( H \) includes contributions from both the bead’s motion and the motion of the rod. However, since the rod is moving vertically with a constant velocity, the height of the bead relative to the rod changes with time. This implies that the total energy of the system is not conserved due to the constant upward motion of the rod. 3. Interpretation of the Hamiltonian:
The Hamiltonian in this case does not represent the total energy of the system, and because of the upward motion of the rod, the total energy of the system is not conserved. Therefore, the correct answer is (D), as the Hamiltonian is not the total energy and it is not conserved.