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Question #42063

Difficulty: 8/10 | 🟢 Free

The Question

A long taut string is aligned along the negative $x$ -axis ( $x < 0$ ). At $x=0$ , the string is attached to a massless ring that is constrained to slide without friction along a rigid, straight rod. The rod lies in the $y$ - $z$ plane and makes an angle $\theta$ with the positive $y$ -axis. An incident transverse wave, polarized purely along the $y$ -axis, is given by $\vec{D}_i(x,t) = A \sin(kx - \omega t) \hat{j}$ . Which of the following represents the reflected wave $\vec{D}_r(x,t)$ ?

Options

A.
$-A \cos(2\theta) \sin(kx + \omega t) \hat{j} - A \sin(2\theta) \sin(kx + \omega t) \hat{k}$
B.
$-A \cos^2\theta \sin(kx + \omega t) \hat{j} - A \sin\theta \cos\theta \sin(kx + \omega t) \hat{k}$
C.
$A \cos(2\theta) \sin(kx + \omega t) \hat{j} + A \sin(2\theta) \sin(kx + \omega t) \hat{k}$
D.
$-A \sin(kx + \omega t) \hat{j}$

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

Particle Velocity vs. Wave Velocity|Boundary Condition Reflection|Vector Decomposition Ambiguity

Why This Question Is Tricky

Common Mistakes

This problem tests the student's ability to apply vector boundary conditions to transverse waves at a constrained interface.

A common trap is assuming the ring acts as a simple free boundary because it can slide, leading students to incorrectly conclude the reflected wave is just $-A \sin(kx + \omega t) \hat{j}$ .

Another mistake is projecting the incident wave onto the rod but forgetting that the perpendicular component experiences a fixed-boundary reflection.

By forcing the decomposition of the particle displacement vector $\vec{D}$ into an orthogonal basis defined by the boundary constraints ( $\hat{u}$ and $\hat{v}$ ), the question highlights the ambiguity that arises when the wavefront normal is not aligned with the physical constraints of the medium.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Motion in a Plane - Class 11 Part 1
Concept	Vector Decomposition Ambiguity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free