A platform capable of moving horizontally is outfitted with a motion sensor that records its position relative to a laboratory reference frame. A lightweight hanger of mass m is suspended from a vertical spring of constant k that is rigidly attached to the platform. While the platform executes small-amplitude sinusoidal oscillations in the horizontal direction, the hanger is free to move only vertically and is observed to perform independent vertical simple harmonic motion about a point that is itself shifting horizontally with the platform.
For analysis, a student chooses a coordinate system whose origin is fixed to the laboratory, with upward y positive and rightward x positive.
(a) Explain why the vertical motion of the hanger can be treated as simple harmonic even though the platform is simultaneously moving horizontally.
(b) At a moment when the hanger is displaced downward by 0.015m from its instantaneous equilibrium position and is instantaneously at rest relative to the platform, draw and label a free-body diagram for the hanger in the laboratory frame.
(c) Using your diagram, determine the direction (upward or downward) of the net force acting on the hanger at that instant and justify your conclusion based solely on the sign of the displacement.
(d) Discuss how the student could redefine the coordinate system to simplify the analysis of the hanger’s vertical motion, and explain the advantages of the new choice in terms of force-displacement relationships.