Phase space diagrams are useful tools in analyzing all kinds of dynamical problems. They are especially useful in studying the changes in motion as initial position and momentum are changed. Here we consider some simple dynamical systems in one-dimension. For such systems, phase space is a plane in which position is plotted along horizontal axis and momentum is plotted along vertical axis. The phase space diagram is x(t) vs. p(t) curve in this plane. The arrow on the curve indicates the time flow. For example, the phase space diagram for a particle moving with constant velocity is a straight line as shown in the figure. We use the sign convention in which position or momentum upwards (or to right) is positive and downwards (or to left) is negative.
Consider the spring-mass system, with the mass submerged in water, as shown in the figure. The phase space diagram for one cycle of this system is
(A)(see structure)
(B)(see structure)✓ Correct
(C)(see structure)
(D)(see structure)
Answer: (B)
Solution
A spring-mass system submerged in water experiences damping due to viscous forces. This is a damped harmonic oscillator.
In phase space, damped oscillations appear as spirals that converge toward the origin (equilibrium position with zero momentum). The amplitude decreases with each cycle due to energy dissipation.
For one complete cycle, the trajectory forms a spiral loop that doesn't close on itself but moves inward. Option (B) shows this characteristic inward spiral pattern of a damped oscillator.
JEE Advanced · 2011 · 10 Apr 2011 · Paper 1 · Single correct option
