A cart of mass 0.40kg rests on a horizontal, frictionless track and is attached to a non-ideal spring fixed at one end. The spring’s restoring force F as a function of cart displacement x from the spring’s natural (unstretched) length is described by the following verbal "graph":
• From x=−0.10m to x=0m, F varies linearly from +4.0N to 0N.
• From x=0m to x=+0.10m, F varies linearly from 0N to −6.0N.
Assume the spring exerts no other forces outside this displacement range and that the cart remains on the track throughout the motion.
(a) Identify all positions within the interval −0.10m≤x≤+0.10m where the cart could be in translational equilibrium.
(b) For each equilibrium position you listed in part (a), state whether it is stable or unstable. Justify your classification using features of the force–displacement relationship.
(c) The cart is released from rest at x=+0.040m. Describe qualitatively the initial direction of acceleration and the subsequent motion for small oscillations, citing evidence from the force information provided.