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Question #43498

Difficulty: 8/10 | 🟢 Free

The Question

A heavy chain of length $L$ , cross-sectional area $A$ , and Young's modulus $Y$ is suspended vertically from the ceiling of an elevator. The linear mass density of the chain varies as $\lambda(x) = c x$ , where $x$ is the distance from the free bottom end of the chain and $c$ is a positive constant. The total mass of the chain is $M$ . The elevator moves upwards with a time-dependent acceleration $a(t) = kt^2$ , where $k$ is a positive constant and $t$ is the time elapsed. Assuming the deformation is small and elastic, what is the total longitudinal extension of the chain at time $t = t_0$ ? (Let $g$ be the acceleration due to gravity)

Options

A.
$\frac{M L (g + k t_0^2)}{3 A Y}$
B.
$\frac{M L (g + k t_0^2)}{2 A Y}$
C.
$\frac{2 M L (g + k t_0^2)}{3 A Y}$
D.
$\frac{M L (g + k t_0^2)}{A Y}$

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

Tension in Massless vs Massive Strings|Pseudo Force Directionality|Non-Uniform Acceleration Constraint

Why This Question Is Tricky

Common Mistakes

This question tests the application of Newton's laws in a non-inertial frame combined with continuum mechanics and non-uniform mass distributions.

Students often memorize the standard result for a uniform chain ( $\Delta L = \frac{MgL}{2AY}$ ) and incorrectly apply it by simply substituting $g_{\text{eff}} = g + kt^2$ , leading to Option B.

Furthermore, if they misidentify the reference point for integration and calculate the mass from the top instead of the bottom, they will obtain Option C.

The problem forces students to abandon static linear assumptions and derive the spatially dependent tension $T(x)$ from first principles using the correct mass distribution and pseudo-force gradient.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Laws of Motion - Class 11 Part 1
Concept	Tension in Massless vs Massive Strings
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free