A particle of mass $m$ is released from rest and follows a parabolic path as shown. Assuming that the displace

Question

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?

Accepted Answer

1. **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).
2. **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 $x = -A$).
3. **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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