Consider a particle of mass m executing simple harmonic motion of amplitude A along a straight line under the action of a linear restoring force provided by an ideal spring.
Answer the following:
(i) Derive expressions for its instantaneous kinetic energy K and potential energy U in terms of its displacement x from the mean position.
(ii) Hence obtain the expression for the total mechanical energy E of the system and show explicitly that E is proportional to the square of the amplitude of oscillation.
(iii) If the amplitude is doubled while the mass and spring constant remain unchanged, by what factor does the total mechanical energy change? Justify your answer.