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NEET PHYSICSOscillationsEasy

Question

The oscillation of a body on a smooth horizontal surface is represented by the equation, X=Acos⁡(ωt)X = A \cos(\omega t), where X=X = displacement at time tt, ω=\omega = frequency of oscillation. Which one of the following graphs correctly shows the variation of acceleration, aa with time, tt? (T=T = time period)

A

Graph showing positive cosine function

B

Graph showing negative sine function

C

Graph showing negative cosine function

D

Graph showing positive sine function

Step-by-Step Solution

  1. Displacement Equation: The given equation for displacement is X=Acos⁡(ωt)X = A \cos(\omega t).
  2. Velocity: Velocity (vv) is the first derivative of displacement with respect to time: v=dXdt=ddt(Acos⁡ωt)=−Aωsin⁡(ωt)v = \frac{dX}{dt} = \frac{d}{dt} (A \cos \omega t) = -A\omega \sin(\omega t)
  3. Acceleration: Acceleration (aa) is the first derivative of velocity (or second derivative of displacement) with respect to time: a=dvdt=ddt(−Aωsin⁡ωt)=−Aω2cos⁡(ωt)a = \frac{dv}{dt} = \frac{d}{dt} (-A\omega \sin \omega t) = -A\omega^2 \cos(\omega t)
  4. Analysis: The acceleration equation is a=−Aω2cos⁡(ωt)a = -A\omega^2 \cos(\omega t). This is a negative cosine function. At t=0t=0, cos⁡(0)=1\cos(0) = 1, so a=−Aω2a = -A\omega^2 (maximum negative value). At t=T/4t=T/4, cos⁡(π/2)=0\cos(\pi/2) = 0, so a=0a = 0. At t=T/2t=T/2, cos⁡(π)=−1\cos(\pi) = -1, so a=+Aω2a = +A\omega^2 (maximum positive value). At t=Tt=T, cos⁡(2π)=1\cos(2\pi) = 1, so a=−Aω2a = -A\omega^2.
  5. Conclusion: The correct graph must start at a negative maximum, go through zero at T/4T/4, reach a positive maximum at T/2T/2, and return to a negative maximum at TT. This corresponds to a negative cosine curve (often labeled as Option C or 3 in standard exams like AIPMT).

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.

PHYSICSOscillationsoscillationsmoothhorizontalsurfacerepresented

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