Paragraph: When a particle of mass $m$ moves on the $x$-axis in a potential of the form $V(x)=k x^2$, it performs simple harmonic motion. The corresponding time period is proportional to $\sqrt{\frac{m}{k}}$, as can be seen easily using dimensional analysis. However, the motion of a particle can…

Question

Paragraph: When a particle of mass $m$ moves on the $x$-axis in a potential of the form $V(x)=k x^2$, it performs simple harmonic motion. The corresponding time period is proportional to $\sqrt{\frac{m}{k}}$, as can be seen easily using dimensional analysis. However, the motion of a particle can be periodic 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 that the particle does not escape to infinity. Consider a particle of mass $\mathrm{m}$ moving on the $x$-axis. Its potential energy is $V(x)=\alpha x^4(\alpha>0)$ for $|x|$ near the origin and becomes a constant equal to $V_0$ for $|x| \geq X_0$ (see figure). Question: If the total energy of the particle is $E$, it will perform periodic motion only if

Aicharya · Accepted Answer

If $E>V_0$, particle will escape. But simultaneously for oscillations, $E>0$. Hence, the correct answer is $V_0>E>0$ or the correct option is (c). ;

Paragraph: When a particle of mass $m$ moves on the $x$-axis in a potential of the form $V(x)=k x^2$, it…

Solution