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Question #42021

Difficulty: 8/10 | 🟢 Free

The Question

A particle of mass $m = 1 \text{ kg}$ is subjected to two orthogonal restoring forces such that its position coordinates as a function of time $t$ are given by $x(t) = A \sin(\omega t)$ and $y(t) = A \sin\left(\omega t + \frac{\pi}{3}\right)$ , where $A = 2 \text{ m}$ and $\omega = \pi \text{ rad/s}$ . What is the radius of curvature of the particle's trajectory at the point where its speed is maximum?

Options

A.
$3\sqrt{2} \text{ m}$
B.
$\sqrt{6} \text{ m}$
C.
$2\sqrt{3} \text{ m}$
D.
$6 \text{ m}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Dimensional Independence vs. Scalar/Vector Nature|Superposition of SHMs|Vector Cross-Products

Why This Question Is Tricky

Common Mistakes

This question tests the geometric and kinematic interpretation of orthogonal coupled SHMs (Lissajous figures).

A common trap is assuming that the maximum speed of the particle occurs at the origin, as it does in 1D SHM.

However, due to the phase difference, the particle follows an elliptical trajectory and never passes through the origin; its maximum speed occurs at the ends of the semi-minor axis.

Furthermore, students often fail to recognize that dimensional independence implies the net acceleration is a central force ( $\vec{a} = -\omega^2\vec{r}$ ), allowing the radius of curvature to be elegantly found using $a_n = v^2/\rho$ without computing complex parametric derivatives.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Units and Measurements - Class 11 Part 1
Concept	Dimensional Independence vs. Scalar/Vector Nature
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free