Physics

The key insight is that a spring's total mechanical energy depends on amplitude squared, not amplitude linearly. Since all the energy in a spring–mass system converts between kinetic and potential at the turning points, the maximum potential energy $\frac{1}{2}kA^2$ equals the total energy. When you double the amplitude, you're quadrupling the energy stored—so the new energy is $4E_0$.

The key insight is that a spring's total mechanical energy depends on amplitude squared, not amplitude linearly. Since all the energy in a spring–mass system converts between kinetic and potential at the turning points, the maximum potential energy $\frac{1}{2}kA^2$ equals the total energy. When you double the amplitude, you're quadrupling the energy stored—so the new energy is $4E_0$.