Answer
$k=833 N/m$
Work Step by Step
Since $K_s=4.0J$ and it is two-thirds of the maximum kinetic energy, $K_{max}=6.0J$. Since energy is conserved, the maximum kinetic energy is equal to the maximum potential energy. Therefore, $$U_{max}=\frac{1}{2}kA^2$$ $$6.0J=\frac{1}{2}kA^2$$ Solving for $k$ yields $$k=\frac{2(6.0J)}{A^2}$$ According to the graph, the point at which there is no kinetic energy, the amplitude, is 12cm (0.12m). Therefore, by substituting the known value of $A=0.12m$ yields a spring constant of $$k=\frac{2(6.0J)}{(0.12m)^2}=833N/m$$
