Time Period, Frequency and Angular Frequency

Three quantities describe the rate of oscillation. They are all equivalent — knowing one gives you all three.

Time period $T$

Time for one complete oscillation. Unit: seconds.

$T = 2\pi\sqrt{\frac{m}{k}}$

SpringMass amplitude 60 k 100 m 1
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Count one full cycle on the x(t) graph. The readout value of T matches exactly.

Frequency $f$

Oscillations per second. Unit: Hz.

$f = \frac{1}{T} = \frac{1}{2\pi}\sqrt{\frac{k}{m}}$

Angular frequency $\omega$

Radians per second. The most natural unit for SHM mathematics — it appears directly in $x = A\cos(\omega t)$.

$\omega = 2\pi f = \frac{2\pi}{T} = \sqrt{\frac{k}{m}}$

Effect of $k$: stiffer → faster

$f \propto \sqrt{k}$. Quadrupling $k$ doubles $\omega$, halves $T$.

Soft spring ($k = 25$):

SpringMass amplitude 60 k 25 m 1
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Stiff spring ($k = 400$):

SpringMass amplitude 60 k 400 m 1
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The x(t) graphs have the same amplitude but very different cycle widths. $k$ increased 16× → $\omega$ increased 4× → 4 times as many cycles in the same window.

Effect of $m$: heavier → slower

$f \propto 1/\sqrt{m}$.

Light ($m = 0.5$ kg):

SpringMass amplitude 60 k 100 m 0.5
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Heavy ($m = 2$ kg):

SpringMass amplitude 60 k 100 m 2
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JEE caution: units of $\omega$

$\omega$ is in rad/s, not Hz, not degrees/s. When a problem says "angular frequency 10 rad/s":

$\omega = 10 \implies T = \frac{2\pi}{10} \approx 0.63 \text{ s}, \quad f = \frac{10}{2\pi} \approx 1.6 \text{ Hz}$

Never use $\omega$ directly as $f$ — they differ by a factor of $2\pi$.