Revise Simple harmonic motion for IB Physics (C.1)

For an object to undergo SHM, two things must be true. First, it oscillates about a fixed equilibrium position — the point where it would rest undisturbed, because the net force there is zero. Second, whenever it is displaced, it feels a restoring force that is directly proportional to the displacement and always directed back towards equilibrium. The further you pull it away, the harder it is pulled back. Together these give the single defining condition of SHM: the acceleration is proportional to the displacement and points in the opposite direction.

$a = -\omega^2 x$

This is why a constant force (like a steady push) or a force that does not point back to a centre cannot produce SHM. It is the proportional, restoring nature of the force — captured entirely by $a = -\omega^2 x$ — that makes the motion repeat with a fixed rhythm.

Three quantities describe the timing of an oscillation, and they are all connected. The period $T$ is the time for one complete oscillation (in seconds). The frequency $f$ is the number of oscillations per second (in hertz, Hz), so $f = 1/T$. The angular frequency $\omega$ measures the rate of oscillation in radians per second and ties directly to the defining equation. All three are linked by a single relationship.

$\omega = \frac{2\pi}{T} = 2\pi f$

Because $\omega$ appears in $a = -\omega^2 x$, knowing the period of an oscillator immediately tells you its acceleration at any displacement, and vice versa. Everything about the timing of the motion is carried by this one constant.

Two systems dominate SHM problems, and each has a period formula you must be able to use. A simple pendulum — a small bob on a light string — swings with SHM as long as the angle stays small. A mass on a spring oscillates with SHM because the spring supplies a restoring force proportional to extension (Hooke's law). Notice what each period depends on, and — just as importantly — what it does not.

$\text{Simple pendulum:} \quad T = 2\pi\sqrt{\dfrac{l}{g}}$ $\text{Mass–spring:} \quad T = 2\pi\sqrt{\dfrac{m}{k}}$

In SHM the displacement traces out a sinusoidal curve in time — a sine or cosine, depending only on where you start the clock. If timing begins when the object is at maximum displacement (the amplitude $x_0$), the displacement follows a cosine, the velocity follows a negative sine, and the acceleration follows a negative cosine. You do not need to derive these at SL, but you must know how the three quantities relate in size and in phase.

$x = x_0\cos(\omega t), \quad v = -\omega x_0\sin(\omega t), \quad a = -\omega^2 x_0\cos(\omega t) = -\omega^2 x$

In an ideal, undamped oscillator the total mechanical energy is constant; it only shuttles between kinetic energy $E_k$ and potential energy $E_p$. At the extremes the object is momentarily at rest, so all the energy is potential. As it accelerates towards the centre that potential energy converts into kinetic energy, reaching all-kinetic at the equilibrium position, where the speed is greatest. The cycle then reverses. Because the total is fixed by the amplitude, $E_{total} \propto x_0^2$: doubling the amplitude quadruples the energy.

$E_{total} = E_k + E_p = \tfrac{1}{2}mv^2 + \tfrac{1}{2}kx^2 = \text{constant} = \tfrac{1}{2}kx_0^2 = \tfrac{1}{2}m\omega^2 x_0^2$

In Paper 2 the marks are analytic: each is tied to a specific line of working — a method mark (M) or an answer mark (A) — and error-carried-forward (ECF) means a wrong number early on does not have to cost you the marks that follow. But that protection only exists if your method is written down. Study how each mark below is earned by a specific line in a two-part SHM calculation.

SHM is the seed of wave physics. The sinusoidal displacement of a single oscillator, repeated across many connected particles, is exactly what a travelling wave is — so the definitions of amplitude, period, frequency and phase carry straight into the wave topics that follow in unit C. The energy picture reappears too: standing waves, resonance and damping all build on the idea of energy stored and exchanged in an oscillator. Master the habit here — check the defining condition, read the timing from $\omega$, keep speed and acceleration at the right places, and follow the energy — and wave behaviour becomes variations on a motion you already own.