Final answer:
The change in gravitational potential energy of the box must be considered when it compresses a vertical spring, as energy conservation principles dictate that this energy is converted into the spring's elastic potential energy.
Step-by-step explanation:
Yes, the change in the gravitational potential energy of the box while the spring is compressing does need to be considered in this problem. When a box is dropped onto a spring, the gravitational potential energy of the box is converted into the elastic potential energy of the spring as it compresses. The energy conservation principle states that energy within a closed system must be conserved unless external work is done or non-conservative forces, like friction, are present.
The gravitational potential energy change is calculated based on the height the box descends from the equilibrium position until the spring is fully compressed. This energy transition is an integral part of the energy transformations that occur within the system. Therefore, to analyze the system's dynamics and calculate various energy quantities accurately, the change in gravitational potential energy must be included.