A horizontal piston-cylinder system containing \(n\) mol of ideal gas is surrounded by air at temperature \(T_{0}\) and pressure \(p_{0}\). If the piston is displaced slightly from equilibrium, show that it executes simple harmonic motion with angular frequency \(\omega=A p_{0} / \sqrt{M n R T_{0}}\) where \(A\) and \(M\) are the piston area and mass, respectively. Assume the gas temperature remains constant.

The ideal gas law is a fundamental equation in thermodynamics and physical chemistry, relating the pressure, volume, temperature, and amount of an ideal gas. This equation is represented as \( PV = nRT \), where \(P\) stands for the pressure of the gas, \(V\) represents the volume, \(n\) the number of moles, \(R\) is the ideal gas constant, and \(T\) is the absolute temperature of the gas. In the context of our piston-cylinder system, the ideal gas law provides insights into the relationship between pressure exerted by the gas and the volume it occupies, as well as how the gas pushes against the piston.
An understanding of the ideal gas law can help predict how changes in the system, such as the movement of the piston, alter the internal pressure of the gas. When the piston moves slightly, it changes the volume of the gas. Since the temperature remains constant (isothermal process), according to the law, the pressure must adjust to maintain the relationship defined by the equation. This change in pressure is what causes the restoring force that leads to simple harmonic motion of the piston.

Simple harmonic motion (SHM) is a type of periodic motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement. It is a simplified model that is foundational to understanding more complex vibratory systems. An object in SHM will oscillate back and forth over a central point, known as the equilibrium position, in a sinusoidal pattern.
In the piston-cylinder problem, when the piston is displaced, the ideal gas inside acts to restore it to its equilibrium position. As the gas expands or compresses due to the piston's displacement, it exerts a force on the piston to return it to where the gas pressure balances the external atmospheric pressure. This restoring force results in SHM of the piston as long as the displacement from the equilibirum is small and the force can be linearized. The back and forth movement of the piston can be highly predictable, and typical examples of SHM in other contexts include the oscillations of a spring-mass system or a swinging pendulum.

Newton's second law of motion is one of the fundamental principles in classical mechanics, depicted by the equation \( F = ma \), where \(F\) is the net force applied to an object, \(m\) is the mass of the object, and \(a\) is its acceleration. This law describes how the velocity of an object changes when it is subjected to external forces.
In the case of our piston-cylinder system, this law is applied to relate the net force coming from the pressure difference between the inside and outside of the cylinder to the acceleration of the piston. As the piston experiences a displacement, there's a change in net force, leading therefore to an acceleration. The law provides the crucial link between the macroscopic motion of the piston and the microscopic forces exerted by the gas molecules. By expressing the force balance in these terms, we are able to use this physical principle to connect the movement of the piston with the ideal gas law and characterize the system's motion as simple harmonic.

Angular frequency \(\omega\) is a measure of how quickly something undergoes one complete cycle of oscillation and is related to the concept of frequency \(f\) by the relation \(\omega = 2 \pi f\). The angular frequency is particularly important when dealing with systems in simple harmonic motion, as it can describe not only the rate of oscillation but also the energy involved in the motion.
For the piston-cylinder system, the angular frequency determines how rapidly the piston oscillates around its equilibrium position. The formula \(\omega = A p_{0} / \sqrt{M n R T_{0}}\) derived from the problem connects various physical quantities of our system—the atmospheric pressure, the area and mass of the piston, the amount of gas in moles, the ideal gas constant, and the absolute temperature—and encapsulates how these properties influence the rate of harmonic motion. Recognizing how to calculate angular frequency and how it is affected by these parameters is crucial for understanding the dynamics of this mechanical system and similar oscillatory systems in physics.