Spring Period from Mass

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Step 1

Write an expression for the angular frequency ω of a mass-spring oscillator in terms of k and m.

Hint

Compare x″ + (k/m)x = 0 with the standard form x″ + ω²x = 0. The coefficient of x is ω².

Step 2

Use ω to find the period T. How many radians are in one full cycle?

Solution walkthrough

The equation of motion for a mass on a spring is F = −kx, or equivalently x″ = −(k/m)x. This is exactly x″ + ω²x = 0 with ω² = k/m. The same mathematical structure appears in pendulums (ω² = g/L), LC circuits (ω² = 1/LC), and even molecular bonds — which is why this topic is called 'oscillators everywhere.' Here, ω = √(200/0.5) = √400 = 20 rad/s. The period follows immediately: T = 2π/ω = 2π/20 ≈ 0.314 s. Note that period decreases with stiffer springs (larger k) and increases with heavier masses (larger m) — both intuitive. Heavier mass → more inertia → slower oscillation. Stiffer spring → stronger restoring force → faster oscillation.

Step-by-step solution

Write an expression for the angular frequency ω of a mass-spring oscillator in terms of k and m.

Use ω to find the period T. How many radians are in one full cycle?

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