Three frictionless spring–block oscillators, X, Y, and Z, each have a block of mass 0.25kg. The spring constants and amplitudes are shown.
System k (N/m) A (m)
X 80 0.10
Y 130 0.08
Z 200 0.07
All three blocks are released from rest at their maximum displacements and allowed to oscillate along the same horizontal line.
(a) Derive an algebraic expression for the total mechanical energy of a spring–block oscillator in terms of its spring constant and amplitude.
(b) Using your result from part (a) and the values in the table, rank the three systems from greatest to least total mechanical energy. Support your ranking with calculations.
(c) Consider the instant when each block is 0.060m from equilibrium and moving toward equilibrium. Rank the three systems by potential energy at that instant, justifying your answer analytically.
(d) For the same instant described in part (c), rank the systems by kinetic energy. Explain how your ranking follows from energy conservation principles.