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NEET PHYSICSOscillationsEasy

Question

From the given functions, identify the function which represents a periodic motion:

A

eωte^{\omega t}

B

log⁡e(ωt)\log_e(\omega t)

C

sin⁡ωt+cos⁡ωt\sin \omega t + \cos \omega t

D

e−ωte^{-\omega t}

Step-by-Step Solution

  1. Definition of Periodic Motion: A motion that repeats itself at regular intervals of time is called periodic motion. Mathematically, a function f(t)f(t) is periodic if f(t)=f(t+T)f(t) = f(t + T) for some period TT.
  2. Analysis of Options:
  • eωte^{\omega t}: This is an exponential growth function. It increases monotonically with time and never repeats. Hence, it is non-periodic.
  • log⁡e(ωt)\log_e(\omega t): This is a logarithmic function. It increases monotonically with time and never repeats. Hence, it is non-periodic.
  • e−ωte^{-\omega t}: This is an exponential decay function. It decreases monotonically towards zero but never repeats. Hence, it is non-periodic.
  • sin⁡ωt+cos⁡ωt\sin \omega t + \cos \omega t: Both sine and cosine functions are periodic with period T=2π/ωT = 2\pi/\omega. The sum of two periodic functions with the same frequency is also periodic. We can rewrite this expression: y=sin⁡ωt+cos⁡ωt=2(12sin⁡ωt+12cos⁡ωt)y = \sin \omega t + \cos \omega t = \sqrt{2} \left( \frac{1}{\sqrt{2}}\sin \omega t + \frac{1}{\sqrt{2}}\cos \omega t \right) y=2(sin⁡ωtcos⁡π4+cos⁡ωtsin⁡π4)=2sin⁡(ωt+π4)y = \sqrt{2} \left( \sin \omega t \cos \frac{\pi}{4} + \cos \omega t \sin \frac{\pi}{4} \right) = \sqrt{2} \sin(\omega t + \frac{\pi}{4}) This represents a Simple Harmonic Motion (a specific type of periodic motion) with period T=2π/ωT = 2\pi/\omega. Hence, it is periodic.
  1. Conclusion: Only the function involving sine and cosine represents periodic motion.

Exam Context & Concepts Covered

This question aligns with the NEET PHYSICS syllabus, specifically targeting concepts from Oscillations. Mastering this topic is crucial for scoring well in the upcoming medical entrance examinations. Solving conceptually related problems will help you understand the nuances of these concepts and improve your problem-solving speed.

PHYSICSOscillationsfunctionsidentifyfunctionrepresentsperiodic

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