Passage
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 \( \sqrt{m / k} \), as can be seen easily using dimensional analysis. However, the motion of
P a particle can be periodic even when its potential energy
W 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).
(2010)
For periodic motion of small amplitude \( A \), the time period \( T \) of this particle is proportional to
(a) \( A \sqrt{\frac{m}{\alpha}} \)
(b) \( \frac{1}{A} \sqrt{\frac{m}{\alpha}} \)
(c) \( A \sqrt{\frac{\alpha}{m}} \)
(d) \( \frac{1}{A} \sqrt{\frac{\alpha}{m}} \)
📲PW App Link - https://bit.ly/YTAI_PWAP
🌐PW Website - https://www.pw.live
0