(a) The " \(6-12\) " potential energy \(U(r)=-2 a / r^{6}+b / r^{12}\), where \(a\) and \(b\) are positive constants, is sometimes used to approximate the potential energy between two atoms in a diatomic molecule, where the atoms are separated by a distance \(r\). Find the effective force constant and the angular frequency of small oscillations of a classical atom of mass \(m\) about the equilibrium point.

(b) Repeat part (a) for the "Morse" potential energy \(U(r)=D_{e}\left(1-e^{a\left(r-r_{0}\right)}\right)^{2}\), where \(D_{e}, a\), and \(r_{0}\) are constants.