Answer:
Explanation:
Given
Potential Energy is given by
[tex]U(x)=ax^2+\frac{b}{x^2}[/tex]
And Force is given by
[tex]F=-\frac{\mathrm{d} U}{\mathrm{d} x}[/tex]
Particle will be at equilibrium when Potential Energy is either minimum or maximum
[tex]F=-\left ( 2ax-\frac{2b}{x^3}\right )[/tex]
i.e.[tex]ax=\frac{b}{x^3}[/tex]
[tex]x_0=(\frac{b}{a})^{0.25}[/tex]
So angular Frequency of small oscillation is given by
[tex]\omega =\sqrt{\frac{U''(x)}{m}}[/tex]
for [tex]m=1[/tex]
we get [tex]\omega =\sqrt{\frac{U''(x_0)}{1}}[/tex]
[tex]U''(x_0)=2a+6a= 8a[/tex]
[tex]\omega =\sqrt{8a}[/tex]
