## Physics for Scientists and Engineers: A Strategic Approach with Modern Physics (4th Edition)

The total mechanical energy $E$ in the system is constant. At all times during the motion, the sum of the potential energy and the kinetic energy is equal to the total mechanical energy. When all the energy in the system is in the form of elastic potential energy, $U_s = \frac{1}{2}kA^2$. $\frac{1}{2}kA^2 = E$ $A = \sqrt{\frac{2E}{k}}$ Let's suppose the total energy in the system is $2E$. We can find the new value of the amplitude $A'$: $\frac{1}{2}k(A')^2 = 2E$ $A' = \sqrt{\frac{4E}{k}}$ $A' = \sqrt{2}\times \sqrt{\frac{2E}{k}}$ $A' = \sqrt{2}~A$ $A' = (\sqrt{2}) ~(20~cm)$ $A' = 28.3~cm$