Simple Harmonic Motion

Calculus 2 · Axiom Academy

EXAMPLE Simple Harmonic Motion Solving a mass-spring initial value problem and finding the exact motion A mass on a frictionless spring obeys the differential equation y'' + 4y = 0 , where y(t) is its displacement from rest. Given the starting conditions below, find the function y(t) that describes its exact motion. Nice work — you solved a simple harmonic motion problem end to end. Here's what carried the solution: Standard form: describes simple harmonic motion, where is the angular frequency. General solution: it always takes the form — a blend of cosine and sine. Finding ω: read it straight off the equation. Here y'' + 4y = 0 gives , so . Initial conditions: use y(0) to pin down c_1 and y'(0) to pin down c_2 . Because and , y(0) = c_1 and . Physical meaning: c_1 is the initial displacement and is the initial velocity of the mass. The same recipe handles any undamped, unforced mass-spring system: identify from the coefficient of y , then apply the two initial conditions to lock in the specific solution .

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