7.3 Two identical masses are constrained to move in a horizontal hoop. Two identical springs with spring constant \(k\) connect the masses and wrap around the hoop. Find the normal modes. Do it again for \(N=3\) identical masses and springs. Do it again for general \(N\) identical springs and masses. 

7.4 Consider a double pendulum made of two masses, \(m_1\) and \(m_2\), and two rods of lengths \(l_1\)and \(l_2\). Find the equations of motion. For small oscillations, find the normal modes and their frequencies for the special case \(l_{1\ }=\ l_{2\ }\). Do the same for the case \(m_1\ =\ m_2\).

7.5 A pendulum consists of a mass \(m\) at the end of a massless stick of length \(l\). The other end of the stick is made to oscillate vertically with a position given by \(y\left(t\right)\ =\ A\ \cos\ \left(\omega t\right)\), where \(A<<l\).  It turns out that if \(\omega\) is large enough, and if the pendulum is initially nearly upside-down, then it will surprisingly not fall over as time goes by. Instead, it will (sort of) oscillate back and forth around the vertical position. Find the equation of motion for the angle of the pendulum (measured relative to its upside-down position). And explain why the pendulum doesn’t fall over, and find the frequency of the back and forth motion.