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Question #43531

Difficulty: 8/10 | 🟢 Free

The Question

A block of mass $m = 2 \text{ kg}$ is suspended vertically from a specialized non-linear elastic rod. The restoring force exerted by the rod when stretched by a distance $x$ from its natural length is given by $F(x) = -(20x + 80x^3) \text{ N}$ , where $x$ is in meters. The block is allowed to reach its equilibrium position. It is then given an infinitesimally small vertical displacement and released. Assuming the acceleration due to gravity is $g = 10 \text{ m/s}^2$ , what is the time period of the resulting small oscillations?

Options

A.
$\frac{\pi}{\sqrt{10}} \text{ s}$
B.
$\frac{\pi}{\sqrt{5}} \text{ s}$
C.
$\frac{2\pi}{\sqrt{10}} \text{ s}$
D.
$\frac{\pi}{5} \text{ s}$

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

Stress vs. Strain Definition|SHM Definition via Differential Equation|Small Angle Approximation

Why This Question Is Tricky

Common Mistakes

This question tests the student's understanding of Simple Harmonic Motion (SHM) in non-linear systems.

A common misconception is assuming that the effective stiffness of an oscillator is simply the ratio of total force to total extension ( $k_{\text{eff}} = F/x$ ), which is only true for perfectly linear Hookean springs.

In non-linear regimes, students often incorrectly use this secant modulus instead of the tangent modulus, leading to Trap B.

The correct physical principle requires performing a Taylor expansion of the force around the new equilibrium position, defining the effective stiffness as the local derivative $k_{\text{eff}} = dF/dx$ at $x = x_0$ .

Failing to differentiate the cubic term correctly or ignoring it entirely leads to incorrect restoring forces for small perturbations.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Mechanical Properties of Solids - Class 11 Part 2
Concept	Stress vs. Strain Definition
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free