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Question #43889

Difficulty: 8/10 | 🟢 Free

The Question

A plano-convex lens of refractive index $\mu = 1.5$ and radius of curvature $R = 20 \text{ cm}$ has its flat surface silvered. The lens is suspended from a vertical spring and constrained to move only along its principal axis. The vertical displacement $y$ (in cm) of the lens from its equilibrium position satisfies the differential equation $\frac{d^2y}{dt^2} + 100y = 0$ . The amplitude of oscillation is $A = 2 \text{ cm}$ . A luminous point object is fixed on the ground, exactly on the principal axis, such that the equilibrium position of the lens is at a height $H = 30 \text{ cm}$ above the object. What is the speed of the image formed by the silvered lens at the instant the lens passes through its equilibrium position?

Options

A.
$0.2 \text{ m/s}$
B.
$0.6 \text{ m/s}$
C.
$0.8 \text{ m/s}$
D.
$1.0 \text{ m/s}$

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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

Lens Silvering Logic|SHM Definition via Differential Equation|Vertical Oscillations

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of SHM kinematics with the optics of moving mirrors.

The primary trap lies in the relative velocity calculation.

Students often memorize the image velocity formula as $v_I = -m^2 v_O$ assuming a stationary mirror.

When the silvered lens (acting as a mirror) moves, the correct Galilean transformation is $\vec{v}_I - \vec{v}_L = -m^2 (\vec{v}_O - \vec{v}_L)$ .

Failing to account for the frame of the moving lens leads to the incorrect answer of $0.8 \text{ m/s}$ (calculating only $m^2 v_L$ ), while a sign error in the relative velocity expansion leads to $0.6 \text{ m/s}$ (calculating $(m^2-1)v_L$ ).

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Ray Optics and Optical Instruments - Class 12
Concept	Lens Silvering Logic
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free