Solve the foregoing problem if the potential energy has the form `U(x)=a//x^(2)-b//x,`, where `a` and `b` are positive constants.

Explanation

If `U(x)=(a)/(x^(2))-(b)/(x)`
then the equilbrium position is `x= x_(0)` when `U^(')(x_(0))=0`
or `-(2a)/(x_(0)^(3))+(b)/(x_(0)^(2))=0implies(2a)/(b)`
Now write `:` `x=x_(0)+y`
Then `U(x)=(a)/(x_(0)^(2))-(b)/(x_(0))+(x-x_(0))+(1)/(2)(x-x_(0))^(2)U^('')(x_(0))`
But `U^('')(x_(0))=(6a)/(x_(0)^(4))-(2b)/(x_(0)^(3))=(2a//b)^(-3)(3b-2b)=b^(4)//8a^(3)`
So finally `:` `U(x)=U(x_(0))+(1)/(2)((b^(4))/(8a^(3)))y^(2)+......`
We neglect remaining terms for small oscillations and compare with the P.E. for a harmonic, oscillator `:`
`(1)/(2)m omega^(2)y^(2)((b^(4))/(8a^(3)))y^(2),` so ` omega=(b^(2))/(sqrt(8a^(3)m))`
Thus `T=2pisqrt(8ma^(3))/(b^(2))`
Note `:` Equilibrium position is generally a minimum of the potential energy. Then `U^(')(x_(0)=0, U^('')(x_(0)) gt 0`) . The equilibrium position can in priciple be a maximum but then `U^('')(x_(0))lt0` and the frequency of oscillation sabout this equilibirium position wil be imaginary.
The answer given in the book in the book s incorrect body numberically and dimensionally.