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NEET PHYSICSOscillationsEasy

Question

Which one of the following equations of motion represents simple harmonic motion where kk, k0k_0, k1k_1, and aa are all positive?

A

Acceleration=−k0x+k1x2\text{Acceleration} = -k_0x + k_1x^2

B

Acceleration=−k(x+a)\text{Acceleration} = -k(x+a)

C

Acceleration=k(x+a)\text{Acceleration} = k(x+a)

D

Acceleration=kx\text{Acceleration} = kx

Step-by-Step Solution

  1. Identify the Condition for SHM: For a particle to execute Simple Harmonic Motion, its acceleration aa must be directly proportional to its displacement XX from the mean (equilibrium) position and directed towards it. This is mathematically expressed as a=−ω2Xa = -\omega^2 X, where ω2\omega^2 is a positive constant.
  2. Analyze the Given Options:
  • Option A (a=−k0x+k1x2a = -k_0x + k_1x^2): The acceleration depends on x2x^2, meaning it is not linearly proportional to the displacement. Thus, it does not represent SHM.
  • Option B (a=−k(x+a)a = -k(x+a)): If we consider the mean position to be at x=−ax = -a, the displacement from the mean position is X=x−(−a)=x+aX = x - (-a) = x + a. Substituting this into the equation gives a=−kXa = -kX. Since kk is given as positive, this matches the standard SHM equation a=−ω2Xa = -\omega^2 X with ω=k\omega = \sqrt{k}. This represents SHM.
  • Option C (a=k(x+a)a = k(x+a)): The positive sign indicates that the acceleration is in the same direction as the displacement (the force is repulsive, not restoring). Thus, it is not SHM.
  • Option D (a=kxa = kx): Similar to Option C, the lack of a negative sign means the force is not a restoring force. It does not represent SHM.
  1. Conclusion: Only the equation Acceleration=−k(x+a)\text{Acceleration} = -k(x+a) satisfies all conditions for Simple Harmonic 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.

PHYSICSOscillationsfollowingequationsmotionrepresentssimple

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