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Question #61315

Difficulty: 8/10 | 🟢 Free

The Question

Let the position of a particle in the complex plane at time $t \ge 0$ be $z(t) = e^{i \omega t}$ , where $\omega > 0$ is a constant angular velocity. A second particle's position is defined by $w(t) = (z(t))^{1/k}$ , where $k \ge 2$ is an integer and the principal branch of the fractional power is taken such that $\arg(w) \in (-\pi/k, \pi/k]$ . Let $\vec{v}(t) = \frac{dw}{dt}$ denote the velocity vector of the second particle. Determine the magnitude of the jump discontinuity in the velocity of the second particle at $t = \frac{\pi}{\omega}$ , given by $\left| \lim_{t \to (\pi/\omega)^+} \vec{v}(t) - \lim_{t \to (\pi/\omega)^-} \vec{v}(t) \right|$ .

Options

A.
$\frac{2\omega}{k} \sin\left(\frac{\pi}{k}\right)$
B.
0
C.
$\frac{2\omega}{k} \cos\left(\frac{\pi}{k}\right)$
D.
$\frac{\omega}{k}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Physical Interpretation|Euler's Form and Rotations|De Moivre’s Theorem Applications

Why This Question Is Tricky

Examiner Levers

Variable Manipulation

Algebraic solutions only (No numbers); tests pure symbolic logic.

Difficulty Boost : +4

Common Mistakes

This question tests the physical interpretation of complex derivatives and the strict domain constraints of De Moivre's theorem for non-integer powers.

Students commonly fall into the trap of blindly applying De Moivre's theorem as $w(t) = \cos(\omega t / k) + i \sin(\omega t / k)$ for all $t$ , which incorrectly implies the velocity is continuous across the entire domain (yielding a jump of 0, Option B).

By forcing the evaluation at the branch cut $t = \pi/\omega$ , the question requires recognizing that the principal argument of $z(t)$ jumps from $\pi$ to $-\pi$ .

This causes a phase shift in the trajectory of the fractional power, leading to a corresponding discontinuous jump in the velocity vector that must be evaluated using left and right limits.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Limits and Derivatives - Class 11
Concept	Physical Interpretation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free