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Question #42200

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The Question

A particle of mass $m$ is constrained to move along the x-axis. It is subjected to a force that consists of a static destabilizing component and a viscoelastic stabilizing memory component, given by: $F(t) = k x(t) - \int_0^t \beta e^{-\gamma(t-\tau)} x(\tau) d\tau$ where $k$ , $\beta$ , and $\gamma$ are strictly positive constants, and $x(t)$ is the displacement from the equilibrium position $x=0$ . The particle is released from rest at $t=0$ with a small displacement $x_0$ . Which of the following conditions on $\beta$ ensures that the equilibrium position $x=0$ is dynamically stable?

Options

A.
$\beta > k \gamma$
B.
$\beta < k \gamma$
C.
$\beta > k \gamma + m \gamma^3$
D.
$\text{The equilibrium is always unstable for any } \beta > 0$

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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

Elastic Hysteresis Loop|Equilibrium Stability|Differential Equations

Why This Question Is Tricky

Common Mistakes

This question tests the critical distinction between static effective stiffness and dynamic stability in systems with time-varying relaxation (elastic hysteresis).

A common trap is to assume a static displacement $x(t) = x_0$ , which yields an effective static force $F_{eff} = (k - \beta/\gamma)x_0$ , leading students to incorrectly conclude that $\beta > k\gamma$ provides stability.

However, applying the Boltzmann Superposition Principle and analyzing the integro-differential equation via Laplace transforms reveals that the characteristic equation always contains a negative coefficient ( $-k$ ).

Thus, the memory effect cannot dynamically stabilize a statically unstable stiffness, regardless of its magnitude.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Mechanical Properties of Solids - Class 11 Part 2
Concept	Elastic Hysteresis Loop
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free