The kinetic energy in terms of the amplitude of oscillation is,
[tex]K=\frac{1}{2}k(A^2-x^2)[/tex]where k is the spring constant and x is the position,
The potential energy in terms of the amplitude of oscillation is,
[tex]U=\frac{1}{2}kx^2[/tex]The value of position when the kinetic energy of the oscillation is 3 times the potential energy is,
[tex]\begin{gathered} K=3U \\ \frac{1}{2}k(A^2-x^2)=\frac{3}{2}kx^2 \\ k(A^2-x^2)=3kx^2 \\ kA^2-kx^2=3kx^2 \end{gathered}[/tex]By simplifying,
[tex]\begin{gathered} 3kx^2+kx^2=kA^2 \\ 4kx^2=kA^2 \\ x^2=\frac{A^2}{4} \\ x=\pm\frac{A}{2} \end{gathered}[/tex]Thus, the position at which the kinetic energy becomes 3 times the potential energy is half of the amplitude.
