'perpetual motion' is ruled out of the realm of possibility because of which law of thermodynamics?

That depends on the design of the machine. Any design will violate at least one law of thermodynamics (1st, 2nd, or 3rd). Having said that, most people think that it is the 1st or 2nd law that would have to be violated in designing a perpetual motion machine. The first law states that the energy output of a machine, system or process cannot exceed the energy input. The 2nd Law states that entropy increases in all processes so the work output is always less than the energy input. Some of the energy that enters a system is wasted increasing its entropy (disorder).

The notion of perpetual motion is ruled out by the first and second laws of thermodynamics.

The first law of thermodynamics, also known as the law of conservation of energy, states that energy cannot be created or destroyed within an isolated system. It can only be converted from one form to another or transferred between systems. In other words, the total amount of energy in a closed system remains constant.

The second law of thermodynamics is the one directly relevant to perpetual motion. It states that, in any isolated system, the entropy (a measure of energy dispersal or disorder) tends to increase over time. This means that energy becomes less available for useful work, and more dispersed or dissipated as heat. In simple terms, it is not possible to convert all the input energy into useful work without losing some energy as waste heat.

Perpetual motion machines claim to provide unlimited energy output without consuming any external energy source. However, they violate these laws because they either require input energy greater than the energy output (violating the conservation of energy) or they contradict the tendency of energy to disperse and eventually be converted entirely into heat (violating the second law of thermodynamics).

To reach this conclusion, one can study and understand the laws of thermodynamics, which are fundamental principles in physics. These laws have been extensively tested and proven through numerous experiments and observations. By comprehending these laws, one can grasp the reasons why perpetual motion machines are not feasible based on the principles they establish.