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Question #43134

Difficulty: 8/10 | 🟢 Free

The Question

A point object $O$ is fixed to the floor of a train car. A convex lens of focal length $f = 20 \text{ cm}$ and mass $m$ is suspended from the roof of the train car by two parallel massless rigid rods of length $l = 50 \text{ cm}$ , forming a parallelogram linkage. This ensures the principal axis of the lens always remains horizontal. When the train is at rest, the lens hangs vertically, and the object $O$ lies exactly on the principal axis at a distance of $u_0 = 60 \text{ cm}$ to the left of the lens. The train then accelerates horizontally to the right with a constant acceleration $a = \frac{3}{4}g$ (where $g = 10 \text{ m/s}^2$ ). Assuming the lens reaches its new equilibrium position without oscillating, what is the magnitude of the displacement of the image of object $O$ relative to the train car, compared to its initial position when the train was at rest?

Options

A.
$30 \text{ cm}$
B.
$20 \text{ cm}$
C.
$10 \text{ cm}$
D.
$0 \text{ cm}$

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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

Newton’s Form of Lens Equation|Frame of Reference|Pseudo Force Directionality

Why This Question Is Tricky

Common Mistakes

This question tests the application of optical principles within a non-inertial reference frame, specifically targeting 'pseudo-force directionality' and coordinate transformations.

Students often fall into multiple traps: ignoring the pseudo-force entirely, applying it in the wrong direction, or failing to account for the vertical displacement of the focal plane caused by the pendulum's circular arc.

A common critical error is calculating the transverse image height relative to the shifted principal axis ( $20\text{ cm}$ ) but forgetting to add the lens's own vertical shift ( $10\text{ cm}$ ) to find the true displacement in the laboratory frame.

The problem elegantly demonstrates the decoupling of the frame's coordinate transformation from Newton's Lens Equation ( $x_1 x_2 = f^2$ ), revealing that the longitudinal image position remains invariant due to optical reversibility, leaving only a pure vertical displacement.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Ray Optics and Optical Instruments - Class 12
Concept	Newton’s Form of Lens Equation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free