7.7 A mass \(M\) is fixed at the right-angled vertex where a massless rod of length
\(l\) is connected to a very long massless rod (see Figure). A mass \(m\) is free
to move frictionlessly along the long rod. The rod of length \(l\) is hinged at a
support, and the whole system is free to rotate, in the plane of the rods, about
the support.
Let \(\theta\) be the angle of rotation of the system, and let \(x\) be the distance between
\(m\) and \(M\). Find the equations of motion. Find the normal modes when \(\mu\) and \(x\) are both very small.