two equal masses m1 m2 are connected by three identical springs of spring constand k.?

Question

determine the possible frequencies of vibration by assuming a solution fo the form x1=a1coswt x2=a2coswt

Answer 1

Does your question actually mean that you have two fixed points connected together as:spring-mass-spring-mass-spring.

This is a standard Physics Degree Question which is solved in most Mechanics books at that level, in terms of Normal Modes of Oscillation. The concept is then usually extended to a many body situation to model oscillations in materials in Solid State Physics.
It is an important area to become familiar with and to understand.

Answer 2

The work done by the spring is equal to the increase of the potential energy of the spring as it is expanded or compressed. Potential energy = ½ * k * d^2, d is the distance that the spring is elongated or compressed. Since the blocks are moved in opposite directions, the spring is elongated. For Block 1, PE = ½ * k * X1^2 = Work For Block 2, PE = ½ * k * X2^2 = Work