A block of mass m is attached to an ideal horizontal spring with force constant k and allowed to oscillate on a frictionless surface. At t=0 the spring is stretched to its maximum extension (amplitude A) and released from rest.
(a) Derive an expression for the total mechanical energy of the block–spring system in terms of k and A. State any assumptions you make.
(b) Describe qualitatively how the kinetic energy and elastic potential energy vary with time during one complete oscillation. Identify the instants at which each form of energy is at a maximum or minimum.
(c) At an instant when the displacement is x=2A, determine what fraction of the total mechanical energy is kinetic and what fraction is potential. Explain your reasoning without performing an explicit numerical calculation of speed.
(d) Without explicit calculation, compare the speed of the block when its displacement is x=2A to its speed when the displacement is x=43A. Justify your answer using energy considerations.