A particle of mass m executes simple harmonic motion (SHM) about the origin under the action of an ideal spring of force constant k. Its total mechanical energy at any time is E.
(i) Starting from the definition of total mechanical energy in SHM, derive an expression for the amplitude A in terms of E and k.
(ii) If the amplitude of oscillation is doubled while k remains unchanged, by what factor does the total mechanical energy change? Give reasons for your answer.
(iii) The kinetic and potential energies of the particle vary continuously during the motion, yet the total mechanical energy is constant. Explain why, citing the relevant physical principle.