Mechanics Qual Problems

1.) Whan a mass, $M$, 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 $\omega_o$. 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 $M$ to move only in the $x$ direction and the ends of the spring are fixed. Here $x$, the coordinate of $M$, is measured from the equilibrium position of the midpoint of the spring.

a.) Fing, in terms of $\omega_o$, the angular frequency, $\omega_1$, of the oscillations of the mass $M$ 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.