Question

A particle of mass \( m \) moves in one dimension under the action of a conservative force whose potential energy has the form \( U(x) = \frac{\alpha x}{x^2 + \beta^2} \), where \( \alpha \) and \( \beta \) are dimensional parameters. The angular frequency \( \omega \) of the oscillation is proportional to:

• A. \(\sqrt{\frac{\alpha^3}{m \beta^4}}\)
• B. \(\sqrt{\frac{\alpha}{m \beta^4}}\)
• C. \(\sqrt{\frac{\alpha}{m \beta^3}}\)
• D. \(\sqrt{\frac{\alpha}{m \beta^6}}\)

Hint:

For systems with potential energy containing higher-order terms, use small displacement approximation and equate the resulting equation of motion to find the angular frequency.