Simple Harmonic Motion: MDCAT Physics notes
Simple Harmonic Motion notes for MDCAT: a = −ω²x, mass–spring and pendulum periods, energy in SHM, phase relations, resonance and common traps.
What is SHM?
A body performs simple harmonic motion when its acceleration is directly proportional to its displacement from the mean (equilibrium) position and always directed towards it:
$$a = -\omega^2 x$$
The restoring force obeys Hooke's law, $F = -kx$. Without resistive forces (no damping), the oscillations never stop; total energy stays constant.
Position-dependent quantities
| Quantity | Mean position ($x = 0$) | Extreme position ($x = \pm x_0$) |
|---|---|---|
| Speed | Maximum, $\omega x_0$ | Zero |
| Acceleration | Zero | Maximum, $\omega^2 x_0$ |
| Kinetic energy | Maximum | Zero |
| Potential energy | Zero (spring unstretched) | Maximum |
Speed at displacement $x$: $v = \omega\sqrt{x_0^2 - x^2}$. From mean to extreme, KE decreases continuously to zero while PE increases.
Energy
$$E = \tfrac12 k x_0^2 = \tfrac12 m\omega^2 x_0^2$$
The total energy depends on the square of the amplitude. KE and PE each go through a maximum twice per cycle, so they vary with frequency $2f$.
Phase relations
If $x = x_0\sin\omega t$, then $v = \omega x_0\cos\omega t$ and $a = -\omega^2 x_0\sin\omega t$. Velocity leads displacement by $\pi/2$; acceleration leads velocity by $\pi/2$; acceleration and displacement are out of phase by $\pi$.
Mass–spring system
$$T = 2\pi\sqrt{\frac{m}{k}}, \qquad f = \frac{1}{2\pi}\sqrt{\frac{k}{m}}$$
- $T \propto \sqrt m$: a 16 times heavier mass gives 4 times the period (1 s becomes 4 s).
- $f \propto 1/\sqrt m$: reducing mass to one quarter doubles the frequency.
- For a vertical spring with static extension $e$: $k = mg/e$, so $T = 2\pi\sqrt{e/g}$. With $e = 9.8\ \text{cm}$: $T = 2\pi\sqrt{0.01} = 2\pi/10\ \text{s}$.
- Same force on springs with $k_1 : k_2 = 3 : 4$: $x \propto 1/k$, so $x_2 = \tfrac34 x_1$.
- Maximum acceleration $= kx_0/m$: $k = 20\ \text{N m}^{-1}$, $x_0 = 0.2\ \text{m}$, $m = 10\ \text{kg}$ gives $0.4\ \text{m s}^{-2}$.
Simple pendulum
For small amplitudes,
$$T = 2\pi\sqrt{\frac{l}{g}}$$
- Independent of the mass of the bob and (for small swings) of amplitude.
- $T \propto \sqrt l$: doubling $l$ multiplies $T$ by $\sqrt2$; four times $l$ doubles $T$.
- A second's pendulum has period 2 s (length about 1 m).
- A brass pendulum expands when heated; longer $l$ means a longer period, so the clock runs slow.
Resonance
When a periodic driving force has a frequency equal to a natural frequency of the body, the body oscillates with maximum amplitude. This is resonance. Examples: tuning a radio, a child on a swing pushed in step, microwave heating of water.
Key formulas
- $a = -\omega^2x$, $\omega = 2\pi f = 2\pi/T$
- $T = 2\pi\sqrt{m/k}$; $T = 2\pi\sqrt{l/g}$
- $E = \tfrac12 kx_0^2$; $v = \omega\sqrt{x_0^2 - x^2}$
Common MDCAT traps
- Acceleration is maximum at the extremes, zero at the mean position.
- Energy depends on amplitude, not frequency alone.
- Quarter the mass: frequency doubles, not quadruples.
- KE varies at $2f$, not $f$.
- Pendulum period depends on length and $g$, not mass.
Quick revision
- SHM: $a \propto -x$.
- Undamped oscillations never stop.
- Spring PE is zero at equilibrium.
- Heating a pendulum: period increases.
- Resonance: driving frequency = natural frequency.