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NEET PHYSICSOscillationsMedium

Question

A particle of mass mm is released from rest and follows a parabolic path as shown. Assuming that the displacement of the mass from the origin is small, which graph correctly depicts the position of the particle as a function of time?

A

A sine wave starting from origin (x=0x=0 at t=0t=0)

B

A negative cosine wave starting from negative extreme (x=−Ax=-A at t=0t=0)

C

A straight line indicating constant velocity

D

A cosine wave starting from positive extreme (x=+Ax=+A at t=0t=0)

Step-by-Step Solution

  1. Identify the Type of Motion: The particle is constrained to move on a parabolic path, so its height yy is proportional to x2x^2 (i.e., y=cx2y = cx^2). For small displacements, its potential energy is U=mgy=mgcx2U = mgy = mgcx^2. Since U∝x2U \propto x^2, the restoring force F=−dU/dx∝−xF = -dU/dx \propto -x, which is the definitive condition for Simple Harmonic Motion (SHM).
  2. Determine Initial Conditions: The particle is 'released from rest', which means at time t=0t=0, its velocity is 00 and its displacement is at a maximum amplitude (x=+Ax = +A or x=−Ax = -A).
  3. Select the Correct Graph: The displacement equation for SHM starting from a positive extreme position is given by x(t)=Acos⁡(ωt)x(t) = A \cos(\omega t). The graph of this function is a typical cosine wave that starts at a maximum positive value on the y-axis when t=0t=0. Therefore, the correct graph must be a cosine curve.

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.

PHYSICSOscillationsparticlereleasedfollowsparabolicassuming

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