Note that the only contributor to the moment is the friction force. This is because the spring force F_k2 passes through the center of mass body 2, so it produces no moment about the center of mass. Recall the constraint equation developed earlier, x₂ = Rθ. Substituting this into Eq. 9↑, the following expression for the friction force is determined:

Equation 10↑ can now be substituted back into Eq. 8↑ and rearranged to give the equation of motion for body 2:

Equations 6↑ and 11↑ are the equations of motion governing the system. These are coupled second-order equations. Not surprisingly, the coupling between the equations of motion is due to k₂, which also couples the two bodies together physically. These equations need to be solved simultaneously to give the response, which can be done analytically using various mathematical methods or numerically. The solution of the equations is left for another topic.

where the summation is evaluated over the number of bodies in the system which is two for the present system. The linear velocity of body 1 is x₁̇î, and body 1 does not rotate, so the kinetic energy of body 1 is

Similarly, the kinetic energy of body 2 is

where ω = θ̇. Substituting the relation from the constraints, the system kinetic energy is

and the potential energy stored in spring 2 is

The system potential energy is just the sum of these potential energies, or