Q.At what distance from the mean position is the kinetic energy of a particle performing S.H.M. of amplitude 8 cm, three times its potential energy?

A particle in S.H.M. possesses kinetic energy everywhere except at the two extreme positions, and potential energy everywhere except at the mean position; the two interconvert continuously as the particle oscillates, but their SUM -- the total energy -- stays constant throughout the motion. Kinetic energy at displacement x is E_k = (1/2)momega^2*(A^2 - x^2) = (1/2)k(A^2 - x^2), which as a function of time varies as cos^2 of the phase. Potential energy, found as the work done against the restoring force to reach displacement x, is E_p = (1/2)kx^2 = (1/2)momega^2*x^2, varying with time as sin^2 of the phase. …