Answer
The ratio of the amplitudes is 1.73
Work Step by Step
$f = \frac{1}{2\pi}~\sqrt{\frac{k}{m}}$
If the mass and frequency are equal, then $k$ must also be equal for both oscillations.
We can write an expression for the energy of each oscillation as:
$E_1 = \frac{1}{2}kA_1^2$
$E_2 = \frac{1}{2}kA_2^2$
We can divide the first equation by the second equation. We know that $E_1 = 3.0~E_2$. Therefore;
$\frac{E_1}{E_2} = \frac{\frac{1}{2}kA_1^2}{\frac{1}{2}kA_2^2}$
$\frac{3.0~E_2}{E_2} = \frac{A_1^2}{A_2^2}$
$\frac{A_1}{A_2} = \sqrt{3.0} = 1.73$
The ratio of the amplitudes is thus 1.73.
