Frequency of Spring-Mass System

A spring-mass system is called a linear simple harmonic oscillator because the spring force is proportional to the first power of displacement, following Hooke's law. On a frictionless surface, weight and normal force cancel, leaving only the spring force as the net force. This linear relationship between force and displacement defines the system's oscillatory behavior.

Angular frequency depends only on the mass and the spring's force constant, not on amplitude or initial conditions. Using Newton's second law with expressions for acceleration and displacement yields the angular frequency equation. A stiffer spring increases angular frequency, while greater mass decreases it, producing faster or slower oscillations respectively.

A stiffer spring produces a shorter period and rapid oscillations, while a more flexible spring produces a longer period. The period depends only on mass and spring force constant. Mathematically, period is inversely related to angular frequency, so increased stiffness reduces the time for one complete oscillation.

Frequency is the inverse of the period of oscillation. If a spring-mass system completes one full oscillation in time T, its frequency is 1/T. Angular frequency relates to period through the equation ω = 2π/T, connecting these fundamental oscillation parameters.

A heavier mass produces sluggish oscillations with a longer period, while a lighter mass produces rapid oscillations with a shorter period. Since period is proportional to the square root of mass, doubling the mass increases the period by a factor of √2, slowing the system's oscillatory motion.

On a frictionless horizontal surface, weight and normal force are equal in magnitude and opposite in direction, so they cancel completely. The net force becomes equal only to the spring force, which acts parallel to the surface. This cancellation simplifies the system to depend solely on spring force and mass.

Start with Newton's second law, F = ma, and substitute Hooke's law (F = -kx) for spring force and the acceleration expression from kinematics. Solving the resulting differential equation yields the angular frequency ω = √(k/m), where k is the force constant and m is mass.