A physics demonstration has two metal spheres of mass m hanging from (nearly) frictional pivots on light (nearly massless) rigid rods of length L. The pivots are separated by a distance d. The two masses are linked by a (nearly massless) spring of rest length = d. We describe the motion of the system by giving the horizontal displacements of each mass from its equilibrium position, x₁, and x₂, as shown. ("Nearly" = can be ignored.)

(a) Discuss what you think the normal modes of this system look like and explain why you think so.  

(b) For small displacements from equilibrium, derive the equations of motion ( i.e., Newton's second law of motion) for the two masses and write them as coupled ordinary differential equations that you could solve used standard ODE techniques.