Mechanics Qual Problems

1.) When a mass, [math]M[/math], is hung from the end of a spring of negligible mass it is found to undergo simple 1-dimensional harmonic motion with an angular frequency [math]\omega_o[/math]. The spring is then cut at its midpoint and the mass is reattached there. The spring is then placed in a horizontal rough that allows [math]M[/math] to move only in the [math]x[/math] direction and the ends of the spring are fixed. Here [math]x[/math], the coordinate of [math]M[/math], is measured from the equilibrium position of the midpoint of the spring.

a.) Find, in terms of [math]\omega_o[/math], the angular frequency, [math]\omega_1[/math], of the oscillations of the mass [math]M[/math] when it is displaced a small distance from equilibrium. You may assume that the contact of the mass and spring with the wall of the trough is frictionless.