When a particle of mass m moves on the x-axis in a potential form $V (x) = k x^{2}$ , it performs simple harmonic motion. The corresponding time period is proportional to $\frac{m}{K}$ , as can be seen easily using dimensional analysis. However, the motion of a particle can be preiodic even when its potential energy increases on both sides of x=0 in a way different from $k x^{2}$ and its total energy is such taht the particle does not escape to infinity. consider a particle of mass m moving on the x -axis. its potential energy is $V (x) = α x^{4} (α > 0)$ for |x| near the origin and becomes a constant equal to $V_{o}$ for |x| $\geq X_{0}$

For periodic motion of small amplitude A, the time period T of this particle is proportional to

Question

When a particle of mass m moves on the x-axis in a potential form

V (x) = k x^{2}

, it performs simple harmonic motion. The corresponding time period is proportional to

\frac{m}{K}

, as can be seen easily using dimensional analysis. However, the motion of a particle can be preiodic even when its potential energy increases on both sides of x=0 in a way different from

k x^{2}

and its total energy is such taht the particle does not escape to infinity. consider a particle of mass m moving on the x -axis. its potential energy is

V (x) = α x^{4} (α > 0)

for |x| near the origin and becomes a constant equal to

V_{o}

for |x|

\geq X_{0}

For periodic motion of small amplitude A, the time period T of this particle is proportional to

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