To solve this problem we will start from the definition of energy of a spring mass system based on the simple harmonic movement. Using the relationship of equality and balance between both systems we will find the relationship of the amplitudes in terms of angular velocities. Using the equivalent expressions of angular velocity we will find the final ratio. This is,
The energy of the system having mass m is,
The energy of the system having mass 2m is,
For the two expressions mentioned above remember that the variables mean
m = mass
Angular velocity
A = Amplitude
The energies of the two system are same then,
Remember that
Replacing this value we have then
But the value of the mass was previously given, then
Therefore the ratio of the oscillation amplitudes it is the same.