A particle of mass \(m\) is attached to the end of a light spring of equilibrium length \(a\), whose other end is fixed, so that the spring is free to rotate in a horizontal plane. The tension in the spring is \(k\) times its extension. Initially the system is at rest and the particle is given an impulse that starts it moving at right angles to the spring with velocity \(v\). Write down the equations of motion in polar co-ordinates. Given that the maximum radial distance attained is \(2 a\), use the energy and angular momentum conservation laws to determine the velocity at that point, and to find \(v\) in terms of the various parameters of the system. Find also the values of \(\ddot{r}\) when \(r=a\) and when \(r=2 a\).

In physics, the conservation of energy principle is pivotal. It states that energy cannot be created or destroyed, only transformed from one form to another. When dealing with the mechanics of particle motion, particularly in systems with no external forces or friction, such as a particle attached to a spring, this principle becomes a crucial tool for analysis.

For the particle in question, its mechanical energy is a combination of its kinetic energy, due to its motion, and potential energy, associated with the spring's extension. At different points in the particle's motion, though these energy forms interchange, their total remains constant. This allows us to equate the total mechanical energy at any two points in the motion to find unknown quantities like the velocity at the maximum radial distance (see Step 3 in the solution).

Understanding and applying the conservation of energy concept not only simplifies problem-solving but also anchors our grasp of fundamental physics that governs systems ranging from simple springs to complex planetary orbits.

Angular momentum conservation is another foundational principle in the study of dynamics, especially within the realm of circular motion and central forces. For a system with no external torques, such as our particle-spring setup in a horizontal plane, angular momentum remains unchanged throughout the motion.

In the provided exercise, angular momentum is conserved because the spring force, acting as a central force, induces no torque. The particle's initial angular momentum, imparted by the impulse, equates to its angular momentum at any later stage of the motion (refer to Step 4). When applying this principle, it is often helpful to select moments where calculations are simplified, such as when the particle is at maximum extension and its radial velocity is zero. By understanding and using the conservation of angular momentum, we can reveal relationships between velocities at different positions without detailed knowledge of intervening forces.

In polar coordinates, forces are often decomposed into radial and tangential components to analyze the motion more effectively. The radial component acts along the line joining the particle to the origin, while the tangential component is perpendicular to this line, influencing the rotational aspect.

Our system's initial motion is purely tangential, given by the initial impulse perpendicular to the spring. Hence, the initial radial force is zero. As the motion continues, the spring's tension impacts both radial and tangential acceleration (see Step 2). This decomposition simplifies complex motions into two mutually perpendicular motions, each with its separate equations, thereby enabling us to tackle intricate dynamics problems one dimension at a time.

The mechanics of particle motion is the study of how particles move under various forces. Newton's laws serve as the bedrock of this field. They help in setting up equations of motion that describe how forces affect the velocity and acceleration of a particle.

For our setup with the particle attached to the spring, examining its motion in polar coordinates reveals the underlying mechanics (see Step 1). We account for the forces (both radial and tangential), understand their origins, and use them to express the particle's motion equations. These equations are crucial as they enable us to predict future motion states and reverse-engineer the forces from observed motions.

A harmonic oscillator is a system where the force acting on a particle is proportional to its displacement from equilibrium, often resulting in sinusoidal motion. In the classical sense, a mass attached to a spring exemplifies a harmonic oscillator, provided the forces remain conservative and the system is undamped.

Our exercise, while also involving a spring and mass, adds an interesting twist with rotational motion in a horizontal plane. Nevertheless, the radial behavior of the mass tied to the extension of the spring offers harmonic qualities. The system's behavior embraces harmonicity in terms of spring force being proportional to the distance from equilibrium length, albeit with a clear distinction due to the additional rotational dynamics (related to the angular momentum conservation as seen in Step 4). Such systems demand an intricate understanding of both linear and circular motion concepts, underscoring the rich phenomenology often encountered in real-world applications of harmonic oscillators.