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Question #42630

Difficulty: 8/10 | 🟢 Free

The Question

A block of mass $m$ is placed on a smooth horizontal floor of a carriage and attached to a vertical wall of the carriage via a spring of force constant $k$ and a dashpot of damping coefficient $b$ . The carriage moves along the horizontal x-axis with an acceleration $a(t) = \alpha t$ for $t \ge 0$ , where $\alpha$ is a positive constant. Let $x(t)$ be the displacement of the block relative to the carriage from the unstretched position of the spring. As $t \to \infty$ , the phase space trajectory $(x(t), v(t))$ of the block relative to the carriage converges to a shifting attractor. Identify the coordinates of this attractor $(x_{ss}(t), v_{ss}(t))$ .

Options

A.
$\left( -\frac{m\alpha}{k}t + \frac{bm\alpha}{k^2}, -\frac{m\alpha}{k} \right)$
B.
$\left( -\frac{m\alpha}{k}t, -\frac{m\alpha}{k} \right)$
C.
$\left( -\frac{m\alpha}{k}t - \frac{bm\alpha}{k^2}, -\frac{m\alpha}{k} \right)$
D.
$\left( \frac{bm\alpha}{k^2}, 0 \right)$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Damped Oscillation Equation|Phase Space Trajectory|Variable Force Integration

Why This Question Is Tricky

Common Mistakes

This question tests the dynamics of a damped oscillator in a non-inertial reference frame with a time-varying acceleration.

The primary trap lies in the assumption that the steady-state equilibrium position is simply determined by balancing the instantaneous pseudo-force with the spring force ( $kx = -ma(t)$ ).

Students falling for this trap ignore the velocity-dependent damping force's contribution to the steady-state spatial shift.

Because the attractor itself is moving at a constant velocity in phase space, the damper exerts a constant drag force that must be counteracted by an additional constant stretch in the spring, leading to the $\frac{bm\alpha}{k^2}$ offset.

This forces a rigorous application of non-homogeneous ODE particular solutions rather than static equilibrium intuition.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Oscillations - Class 11 Part 2
Concept	Damped Oscillation Equation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free