OscillationsMedium

Spring and harmonic oscillator

A mass m = 0.5 kg is attached to a spring with k = 200 N/m.\nCalculate: (a) the natural angular frequency, (b) the period, (c) the frequency.

Given data

m = 0.5 kgk = 200 N/m

Steps

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What is the natural angular frequency ω₀ of the oscillator?
Write your working — the numerical result must appear in the text rad/s
What is the oscillation period T?
What is the oscillation frequency f?

Full worked solution

What is the natural angular frequency ω₀ of the oscillator?
$\omega_0 = \sqrt{\dfrac{k}{m}} = \sqrt{\dfrac{200}{0.5}}$
$\omega_0 = \sqrt{k/m} = \sqrt{200/0.5} = \sqrt{400} = \mathbf{20\,rad/s}$ . The natural angular frequency depends on the spring constant and the mass.
What is the oscillation period T?
$T = \dfrac{2\pi}{\omega_0} = \dfrac{2\pi}{20}$
$T = 2\pi/\omega_0 = 2\pi/20 = \pi/10 \approx \mathbf{0.314\,s} \approx \mathbf{314\,ms}$ . The period is the time required for one complete oscillation cycle.
What is the oscillation frequency f?
$f = \dfrac{1}{T} = \dfrac{1}{0.314}$
$f = 1/T = 1/0.314 \approx \mathbf{3.18\,Hz}$ . Alternatively, $f = \omega_0/(2\pi) = 20/(2\pi) \approx 3.18\,\mathrm{Hz}$ . The mass oscillates approximately 3 times per second.

Result:ω₀ = 20 rad/s, T ≈ 0.314 s, f ≈ 3.18 Hz.

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