Consider a mass m suspended from a massless spring that obeys Hooke’s Law (i.e. the force…

Consider a mass m suspended from a massless spring that obeys Hooke’s Law (i.e. the force required to stretch or compress it is proportional to the distance stretched/compressed). The kinetic energy T of the system is mv2/2, where v is the velocity of the mass, and the potential energy V of the system is kr-/2, where k is the spring constant and x is the displacement of the mass from its gravitational equilibrium position. Using Lagrange’s equations for mechanics (with x as the only generalized coordinate), derive the differential equation that describes the motion of the mass. (Do NOT solve this differential equation: its general solution is just an arbitrary linear combination of sin ax and cos ax, where a is the square root of k/m.)