Answer
$t = 0.096~s$
Work Step by Step
We can write an expression for the total energy in the system as:
$E = \frac{1}{2}kA^2$
If the kinetic energy is twice the potential energy, then $K = \frac{2E}{3}$ and $U_s = \frac{E}{3}$.
We then find and expression for $x$ when $U_s = \frac{E}{3}$;
$U_s = \frac{E}{3}$
$\frac{1}{2}kx^2 = \frac{\frac{1}{2}kA^2}{3}$
$x^2 = \frac{A^2}{3}$
$x = \sqrt{\frac{1}{3}}~A$
We then find the time $t$ when $x = \sqrt{\frac{1}{3}}~A$:
$x(t) = A~cos(10t)$
$\sqrt{\frac{1}{3}}~A = A~cos(10t)$
$cos(10t) = \sqrt{\frac{1}{3}}$
$t = \frac{arccos(\sqrt{\frac{1}{3}})}{10}$
$t = 0.096~s$
