Phase Space Diagram of Damped Spring Mass System in Water

To determine the correct phase space diagram for the damped spring-mass system, we analyze the physical behavior of the system in terms of the phase space variables: position ($x$) on the horizontal axis and momentum ($p = m \frac{dx}{dt}$) on the vertical axis.

1. Trajectory Direction in Phase Space

The velocity (and hence momentum) is related to the rate of change of position:
$p = m \frac{dx}{dt}$

• Upper half-plane ($p > 0$): Since momentum is positive, $\frac{dx}{dt} > 0$, meaning the position $x$ must increase over time. Therefore, the state point moves from left to right in the upper half-plane.
• Lower half-plane ($p < 0$): Since momentum is negative, $\frac{dx}{dt} < 0$, meaning the position $x$ must decrease over time. Therefore, the state point moves from right to left in the lower half-plane.

Thus, any physically valid phase space trajectory in the $(x, p)$ plane must be traversed in a clockwise direction.

2. Effect of Damping (Fluid Resistance)

When the mass is submerged in water, it experiences a viscous drag force (damping force) proportional to its velocity:
$F_{\text{drag}} = -b \frac{dx}{dt}$

Due to this non-conservative damping force, the mechanical energy $E = \frac{p^2}{2m} + \frac{1}{2}kx^2$ continuously decreases over time:
$\frac{dE}{dt} = -b v^2 < 0$

As a result, the distance of the phase point from the origin in the $(x, p)$ plane decreases with time, causing the trajectory to spiral inwards toward the origin. After one complete cycle, the maximum displacement (amplitude) will be strictly less than the initial displacement:
$x_{\text{final}} < x_{\text{initial}}$

3. Analysis of the Trajectory

Suppose the mass is displaced to a positive position $x_0 > 0$ and released from rest ($p = 0$):

1. Initial state: $(x, p) = (x_0, 0)$ on the positive position axis.
2. First half-cycle: As the spring pulls the mass back toward equilibrium, the mass moves leftwards ($x$ decreases), so momentum becomes negative ($p < 0$). The curve enters the lower half-plane, moving counter-clockwise around the origin towards negative displacement.
3. Second half-cycle: After reaching a negative extreme, the mass moves back toward positive position ($x$ increases), so momentum becomes positive ($p > 0$). The curve enters the upper half-plane, moving rightwards.
4. Completion of one cycle: The mass returns to a positive position, but due to energy loss in water, the new position $x_1$ satisfies $x_1 < x_0$.

This inner-spiraling, clockwise trajectory corresponds precisely to Option (B).