IIT-JEE 2010 · Paper 1, Physics question 72

Paragraph for Questions 70 to 72

When a particle of mass m moves on the x-axis in a potential of the form $V(x)=kx^2$, it performs simple harmonic motion. The corresponding time period is proportional to $\sqrt{\dfrac mk}$, 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 $kx^2$ and its total energy is such that the particle does not escape to infinity. Consider a particle of mass 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 $|x|\ge X_0$ (see figure).