Mass-Spring Systems in SHM

When a mass attached to a spring is displaced from equilibrium, the spring exerts a restoring force that pushes or pulls the mass back. This force follows Hooke's Law:

$$F_s = -kx$$

where $k$ is the spring constant with units of $\text{N/m}$, measuring the spring's stiffness. The displacement $x$ is measured using a coordinate system with the equilibrium position at the origin. The negative sign ensures the force always points toward equilibrium.

Consider a horizontal mass-spring system. Applying Newton's second law along the $x$-direction:

$$\sum F_x = F_{s}=-kx=ma_x$$

The acceleration is proportional to displacement and directed toward equilibrium.

When stretched ($x > 0$), the force is negative, pointing back toward equilibrium. When compressed ($x < 0$), the force is positive, again pointing toward equilibrium. At equilibrium ($x = 0$), the restoring force vanishes.

This linear relationship between force and displacement is the defining feature connecting Hooke's Law to simple harmonic motion.