The time period of a simple pendulum, given by T = 2π√(l/g), is independent of the mass of the bob. This is because the force causing the pendulum to swing is gravity, which acts equally on all masses. The acceleration due to gravity (g) is constant for a given location, regardless of the mass. When calculating the time period, the mass of the bob cancels out in the equations of motion. Essentially, gravity pulls heavier objects with more force, but these objects also have more inertia, meaning they resist changes in their motion more than lighter objects. This balance results in the time period being the same for different masses. This principle is a fundamental aspect of pendulum motion and demonstrates how gravitational force and inertia interact in harmonic motion.

The stiffness of a spring, characterised by its spring constant 'k', significantly affects the motion of a mass-spring system. The spring constant is a measure of the spring's resistance to being compressed or stretched. A stiffer spring, having a higher 'k' value, exerts a larger force for the same amount of displacement compared to a less stiff spring. In the context of SHM, a stiffer spring leads to a shorter time period of oscillation, as indicated by the formula T = 2π√(m/k). This is because a stiffer spring restores the mass to the equilibrium position more quickly. Consequently, the oscillation frequency increases with the stiffness of the spring. This relationship is fundamental in understanding how physical properties like spring stiffness influence the dynamics of harmonic motion.

A simple pendulum can be considered a simple harmonic oscillator only for small angles of swing, typically less than about 15 degrees. At larger angles, the assumption of simple harmonic motion (SHM) becomes inaccurate. This is because the restoring force (gravity's component acting along the arc) is no longer proportional to the displacement from the equilibrium position. As the angle increases, the difference between the arc length (actual path) and the chord length (straight-line approximation) becomes significant, leading to deviations from the ideal SHM. Consequently, the time period formula T = 2π√(l/g) becomes less accurate, and the motion of the pendulum deviates from the sinusoidal pattern characteristic of SHM.

In the equation x = Acos(ωt + φ), representing the displacement in SHM, the phase constant φ (phi) is significant as it determines the initial phase of the oscillation. This constant represents the displacement of the system at time t = 0. Essentially, it tells us where in its cycle the oscillator starts. For instance, if φ is zero, the motion starts at the maximum displacement (amplitude A). If φ is π/2, the motion starts from the equilibrium position moving in the positive direction. The phase constant is crucial for describing and predicting the motion of the oscillator at any given time, especially when comparing or combining the motions of multiple oscillators. It ensures that the mathematical model of SHM aligns accurately with the physical situation being described.

Damped harmonic motion is crucial in real-world applications because it more accurately represents how oscillatory systems behave in practical situations. In reality, no system is perfectly isolated; they all experience some form of damping due to frictional forces, air resistance, or internal material properties. Understanding damped harmonic motion allows engineers and scientists to predict how systems will behave over time, which is essential for the design and analysis of a wide range of mechanical and structural systems. For instance, in building construction, damping is a critical factor in making structures resilient to vibrations and oscillations, such as those caused by earthquakes or wind. In automotive design, damping in suspension systems is vital for vehicle stability and comfort. The study of damped harmonic motion helps in designing systems that can control or use these energy-dissipating forces effectively.