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Question #43484

Difficulty: 8/10 | 🟢 Free

The Question

The displacement-time ( $x-t$ ) graph of a particle of mass $m = 2.0 \text{ kg}$ moving along the x-axis is described by the function $x(t) = \frac{x_0}{1 + (t/t_0)^2}$ for $t \ge 0$ , where $x_0 = 4.0 \text{ m}$ and $t_0 = 3.0 \text{ s}$ . The particle experiences a position-dependent conservative force $F(x)$ . Let $x_c$ be the position where the $x-t$ graph changes its concavity. What is the effective spring constant $k_{\text{eff}} = -\frac{dF}{dx}$ evaluated at $x = x_c$ ?

Options

A.
$1.0 \text{ N/m}$
B.
$-1.0 \text{ N/m}$
C.
$0.0 \text{ N/m}$
D.
$\frac{8}{3} \text{ N/m}$

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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

Graphical Concavity Significance|SHM Definition via Differential Equation|Effective Spring Constant

Why This Question Is Tricky

Common Mistakes

The question tests the student's ability to link kinematic graphical features (concavity) to dynamic properties (effective stiffness) in a non-linear system.

The primary trap is assuming that at the inflection point where acceleration (and thus net force) is zero, the effective spring constant must also be zero ( $k_{\text{eff}} = -F/x = 0$ ).

This misapplies the linear SHM relation $F = -kx$ to a non-linear regime.

Students must recognize that $k_{\text{eff}}$ is the local gradient of the force curve ( $-\frac{dF}{dx}$ ), requiring them to express acceleration as a function of position $x$ before differentiating, rather than differentiating with respect to time.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Motion in a Straight Line - Class 11 Part 1
Concept	Graphical Concavity Significance
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free