When a particle of mass \( m \) moves on the \( x \)-axis in a potential of the form \( V(x)=k x...
0When 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 \( m, k \) 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 \( m \) moving on the \( x \)-axis. Its potential energy is \( V(x)=\alpha x^{4}(\alpha0) \) for \( |x| \) near the origin and becomes a constant equal to \( V_{0} \) for \( |x| \geq X_{0} \) (see figure).
(IIT-JEE 2010)
The acceleration of this particle for \( |x|X_{0} \) is
(1) proportional to \( V_{0} \)
(2) proportional to \( \frac{V_{0}}{m X_{0}} \)
(3) proportional to \( \sqrt{\frac{V_{0}}{m X_{0}}} \) (4) zero
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