Solved: PHYSICS Question for NEET

NEET PHYSICSMedium

A particle of mass $m$ 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 sine wave starting from origin ( $x=0$ at $t=0$ )

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

A straight line indicating constant velocity

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

Step-by-Step Solution

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

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