Final answer:
Yes, the sum of kinetic and potential energy can change without work being done by an external force. This can occur in a closed system where internal forces are at play, such as gravity in free fall, consistent with the Law of Conservation of Mechanical Energy and the work-energy theorem.
Step-by-step explanation:
Is it possible for the sum of kinetic energy and potential energy of an object to change without work having been done on the object? The answer to this question is yes, if internal forces are involved. According to the Law of Conservation of Mechanical Energy, in a closed system where no dissipative forces like friction are acting, the mechanical energy, which is the sum of gravitational potential energy and kinetic energy, remains constant. However, this does not mean that the kinetic and potential energies cannot change; they can transform into one another.
For instance, when an object falls, its potential energy is converted into kinetic energy. This transformation does not require work by an external force, because gravity, an internal force in a closed system, is doing the work. The object's potential energy decreases while its kinetic energy increases, resulting in no net change to the total mechanical energy of the system. The work-energy theorem supports this by stating that work done by gravity is the change in kinetic energy.
Similarly, in systems with internal structures such as mass-spring oscillators or pendulums, changes in the system's configuration can lead to changes in internal energy, which can alter the balance of kinetic and potential energy without external work.