CLASS 11 · CHAPTER 13 · OSCILLATIONS

Oscillations

A mass on a spring, a swinging pendulum, a vibrating string all obey the same restoring-force pattern. The phase space view on the right shows the pattern directly.

01 · See it

Watch it happen

Spring & mass (0.5 kg, frictionless)Phase space: position vs. velocity

Amplitude: 8 cmSpring constant k: 20 N/m

0.99 s

Period, T

1.01 Hz

Frequency

Notice: dragging the amplitude slider changes the ellipse’s size, not the period. That independence is the isochronism this page’s “Break it” section puts to the test.

The dot on the right traces position against velocity. It always draws an ellipse, because in SHM, position and velocity are locked 90° out of phase. That single shape encodes the entire motion.

02 · Derive it

Where the formula comes from

Hooke’s law gives the restoring force on the mass:

F = -kx

Newton’s second law turns this into a differential equation:

m\dfrac{d^2x}{dt^2} = -kx \;\;\Rightarrow\;\; \dfrac{d^2x}{dt^2} + \omega^2 x = 0, \quad \omega^2 = \dfrac{k}{m}

Its general solution is oscillatory:

Position and period

\begin{gathered} x(t) = A\cos(\omega t + \phi) \\[8px] \textcolor{#e08a1e}{T} = \dfrac{2\pi}{\omega} = 2\pi\sqrt{\dfrac{m}{k}} \end{gathered}

Differentiate once for velocity, $v(t) = -A\omega\sin(\omega t + \phi)$ . Notice $T$ depends only on $m$ and $k$ , while amplitude $A$ never appears. That’s isochronism: bigger swings take exactly as long as small ones.

03 · Break it

Where the shortcut stops working

Isochronism is genuinely true for an ideal spring obeying $F = -kx$ exactly, which is what the simulation above shows, and what almost every CBSE and JEE question assumes.

But the same claim gets taught about pendulums as an absolute law, and it isn’t one. A pendulum’s restoring force is $-mg\sin\theta$ , not $-mg\theta$ . The small-angle approximation $\sin\theta \approx \theta$ (the one that makes a pendulum behave like SHM) only holds for a few degrees of swing. Push the amplitude past roughly 20°, and the true period grows with amplitude, described by an elliptic integral, not the simple $T = 2\pi\sqrt{L/g}$ formula.

The lesson isn’t that “the formula is wrong.” It’s that every SHM formula on this page is only exact for the idealised force law it was derived from. Real pendulums, real springs at large stretch, and real systems with friction all drift away from it at the edges.

04 · Master it

Apply it under exam conditions

Q1. A 0.5 kg mass on a spring completes 10 oscillations in 6.28 s. Find the spring constant.

T = 0.628\text{ s} \Rightarrow \omega = \dfrac{2\pi}{T} = 10\text{ rad/s} \Rightarrow k = m\omega^2 = 0.5\times100 = \textcolor{#e08a1e}{50\text{ N/m}}

Q2. Why does doubling the amplitude of a mass-spring system not change its period, even though the mass now travels twice the distance each cycle?

Because it also moves proportionally faster at every point: the restoring force (and hence acceleration) scales linearly with displacement, so a larger swing and a larger speed cancel exactly, leaving the time for one cycle unchanged.

05 · FAQs

Quick answers

What makes a motion "simple harmonic" specifically?+

The restoring force has to be directly proportional to displacement and always point back toward equilibrium, F = −kx exactly. Anything else may be oscillatory, but it isn't SHM.

Does a swinging pendulum ever perform true SHM?+

Only approximately, and only for small angles, where sinθ ≈ θ. Push the amplitude higher and the true motion measurably departs from ideal SHM.

Why is SHM's period independent of amplitude?+

Because the restoring force and the distance to travel scale together: doubling the amplitude doubles the force at every corresponding point too, so the mass covers twice the distance at twice the speed, and the timing comes out unchanged.

What does the phase-space plot actually represent?+

A single ellipse that fully encodes the whole motion at a glance. Its shape shows that position and velocity are always 90° out of phase in ideal SHM.

Is every periodic motion SHM?+

No. Periodic just means it repeats; SHM is a much narrower category defined by that specific linear restoring-force law.

Related concepts

Physics doesn’t stay inside chapter boundaries. Neither should you.

Waves

Tune two close frequencies and hear-see the beat pattern emerge from pure superposition.