A physics student observes four different oscillating systems, each initially displaced from its equilibrium position and released from rest.
System I: A 0.50kg mass attached to an ideal horizontal spring on a frictionless track.
System II: A small magnet oscillates between two identical repelling magnets so that the net force on the central magnet varies with displacement according to F(x)=Cx−2, where C is a positive constant and x is the position measured from the midpoint.
System III: A lightweight pendulum 1.0m long released from an initial angle of 35∘.
System IV: A glider of mass 0.40kg moves back and forth in a shallow, frictionless U-shaped track whose height above the lowest point varies with horizontal position according to h(x)=2Rx2, where R=1.2m.
Assume air resistance is negligible unless stated otherwise.
(a) For each system, state whether the resulting motion meets the criteria for simple harmonic motion (SHM). Provide a clear justification that refers specifically to the form of the restoring force or the energy–displacement relationship for that system.
(b) Choose one system you concluded is not SHM. Propose a realistic modification that would cause its motion to become SHM, and explain why your modification succeeds.
(c) Select one system you classified as SHM and one you classified as non-SHM. For each, discuss qualitatively how (or whether) the period of oscillation depends on the amplitude of motion. Justify your reasoning without performing numerical calculations.