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Question #43482

Difficulty: 8/10 | 🟢 Free

The Question

A block of mass $m$ is placed on a smooth horizontal surface and attached to a non-linear spring. The spring exerts a restoring force $F = -k_0 x |x|$ , where $k_0$ is a positive constant (in $\text{N}\cdot\text{m}^{-2}$ ) and $x$ is the displacement of the block from its equilibrium position (in $\text{m}$ ). The block is displaced to a distance $x = A$ and released from rest. Which of the following expressions correctly gives the time period $T$ of the resulting oscillatory motion?

Options

A.
$4 \sqrt{\frac{3m}{2k_0 A}} \int_0^1 \frac{du}{\sqrt{1 - u^3}}$
B.
$4 \sqrt{\frac{m}{k_0 A}} \int_0^1 \frac{du}{\sqrt{1 - u^3}}$
C.
$2\pi \sqrt{\frac{m}{k_0 A}}$
D.
$4 \sqrt{\frac{3m}{k_0 A}} \int_0^1 \frac{du}{\sqrt{1 - u^3}}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Non-Uniform Acceleration Constraint|Effective Spring Constant|SHM Definition via Differential Equation

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to analyze anharmonic oscillators using the energy method.

The primary trap lies in reflexively applying the SHM time period formula $T = 2\pi\sqrt{m/k_{eff}}$ by assuming an effective spring constant, which is invalid for non-linear restoring forces.

A secondary, highly lethal trap tests the misconception that potential energy for a spatially varying spring constant $k(x)$ can be written as $U = \frac{1}{2}k(x)x^2$ .

Students must correctly integrate the force $F = -k_0 x^2$ to find $U = \frac{1}{3}k_0 x^3$ , apply conservation of energy to find $v(x)$ , and set up the proper definite integral for the period.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Motion in a Straight Line - Class 11 Part 1
Concept	Non-Uniform Acceleration Constraint
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free