Two wheels, each of mass \(m\) and radius \(r_{w}\), are connected by an elastic coupler of effective stiffness \(k\) and undeformed length \(L\). The system rolls without slip around a circular track of radius \(R\), as shown. Derive the equations of motion of the wheel system.

In the given problem, two wheels are connected by an elastic coupler and roll on a circular track. We have found the equations of motion for this system by following these steps: 1. Define the forces and position coordinates for each wheel. 2. Apply Newton's Second Law to derive equations for horizontal and vertical forces acting on each wheel. 3. Incorporate kinematic properties of rolling motion in the equations. 4. Apply Hooke's Law to the spring forces between the wheels. 5. Substitute the values of forces and coordinates into the derived equations to find the equations of motion for the entire system. The final equations of motion are given by: $$ k(x_2 - x_1 - L) - N_1\sin\theta = m\ddot{x}_1 $$ $$ -N_2\sin\theta + k(x_1 - x_2 - L) = m\ddot{x}_2 $$ $$ N_1\sin\theta = mg\tan\theta $$ $$ N_2\sin\theta = mg\tan\theta $$ $$ \frac{dx_1}{dt} = r_w\frac{d\theta}{dt} $$ $$ \frac{dx_2}{dt} = r_w\left(\frac{d\theta}{dt} + \frac{d\alpha}{dt}\right) $$