Consider a system of two particles with Lagrangian L = 52ẋ2 + l2θ̇2 + 2lẋθ̇ cos(θ) + 2gl cos(θ), where x, θ are generalized coordinates, and g, l are positive constants. The non-zero frequency of the normal mode of the system for small oscillations (|θ| << 1) is:

A diatomic molecule is modeled as two atoms of equal mass m connected by a spring of force constant k. The molecule moves in one dimension under the influence of an external potential: V(x1, x2) = 12mω02(x12 + x22) If one normal mode frequency of the molecule is ω0, then the other normal mode frequency is:

The potential energy between two particles is given by V(r) = -a/r4 + b/r10, where 'a' and 'b' are positive constants. The potential energy at equilibrium separation is: