# Simple Harmonic Motion

> IB Physics SL · IB Physics SL 2025
> Source: https://www.owlsprep.com/study/ib-physics-sl-u3-simple-harmonic-motion/

This module covers the core definition, kinematic equations, energy transformations, and common examples of simple harmonic motion (SHM), a foundational concept for all IB Physics SL wave topics.

**Prerequisites:** [Basic kinematics and energy conservation](https://www.owlsprep.com/study/ib-physics-sl-mechanics-energy-conservation/); [Angular quantities from circular motion](https://www.owlsprep.com/study/ib-physics-sl-kinematics-circular-motion/)

## Learning objectives

- Define simple harmonic motion using its core acceleration-displacement relationship
- Use kinematic equations for SHM to solve for displacement, velocity and acceleration
- Analyze energy transformations in undamped SHM systems
- Calculate periods for common SHM systems (mass-spring, simple pendulum)

## Core Definition and SHM Condition

**Simple Harmonic Motion** — Periodic oscillatory motion that satisfies the core relationship $a = -\omega^2 x$, where $a$ is acceleration, $x$ is displacement from equilibrium, and $\omega$ is constant angular frequency.

*Notation:* SHM

This defining condition separates SHM from other periodic motions (like bouncing balls). A key property of SHM is isochronism: period is independent of amplitude, so changing how far you pull a mass does not change how fast it oscillates.

**Worked example:** Displacement of an oscillator is $x = 0.5 \sin(3t)$. Show that the motion is SHM.

1. To confirm SHM, prove the motion satisfies $a = -\omega^2 x$:
2. Calculate velocity (first derivative of displacement):
3. $$v = \frac{dx}{dt} = 1.5 \cos(3t)$$
4. Calculate acceleration (second derivative of displacement):
5. $$a = \frac{d^2x}{dt^2} = -4.5 \sin(3t)$$
6. Substitute $x = 0.5 \sin(3t) \implies \sin(3t) = 2x$ into acceleration:
7. $$a = -4.5 (2x) = -9x$$
8. This matches the SHM condition with $\omega^2 = 9$, so motion is SHM.

> **tip**
>
> The negative sign is non-negotiable: it tells you acceleration always points back towards equilibrium. Examiners regularly penalize missing negative signs in SHM answers.

## Equations of Motion for SHM

Starting from the core condition $a = -\omega^2 x$, we get general displacement equations that depend on initial conditions. Angular frequency is always defined as:

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

- If $x = A$ (max displacement) at $t=0$: $x = A \cos(\omega t)$
- If $x = 0$ (equilibrium) at $t=0$: $x = A \sin(\omega t)$
- Velocity for any displacement $x$: $v = \pm \omega \sqrt{A^2 - x^2}$

**Worked example:** A SHM oscillator has amplitude 2 cm, period $4\pi$ s. At $t=0$, $x = 2$ cm. Find velocity at $x = 1$ cm.

1. First calculate angular frequency:
2. $$\omega = \frac{2\pi}{T} = \frac{2\pi}{4\pi} = 0.5 \text{ rad s}^{-1}$$
3. Use the velocity-displacement relation:
4. $$v = \pm \omega \sqrt{A^2 - x^2}$$
5. Substitute values $\omega = 0.5$, $A = 2$, $x = 1$:
6. $$v = \pm 0.5 \sqrt{2^2 - 1^2} = \pm 0.5\sqrt{3} \approx \pm 0.87 \text{ cm s}^{-1}$$
7. Unless direction is specified, both positive and negative values are acceptable.

**Check your understanding**

Test your understanding:

1. What is acceleration when displacement $x = A$ (amplitude)?

   - $a = 0$
   - $a = -\omega^2 A$
   - $a = \omega A^2$
   - Cannot be determined

   *Answer:* $a = -\omega^2 A$

   *Why:* When $x = A$, substitute directly into $a = -\omega^2 x$ to get maximum acceleration opposite displacement.

## Energy in Undamped SHM

In undamped SHM (no energy loss to friction), total mechanical energy is conserved. Energy continuously swaps between kinetic energy ($E_k$) and potential energy ($E_p$):

- At $x = \pm A$ (max displacement): $E_k = 0$, $E_p = E_{\text{total}}$
- At $x = 0$ (equilibrium): $E_p = 0$, $E_k = E_{\text{total}}$

**Total Energy of Undamped SHM** — Total energy is constant and equal to $E_{\text{total}} = \frac{1}{2} m \omega^2 A^2 = \frac{1}{2} k A^2$

**Worked example:** A 0.2 kg mass undergoes SHM with amplitude 0.1 m and angular frequency 4 rad s⁻¹. Calculate total energy.

1. Maximum speed in SHM is $v_{\text{max}} = \omega A$:
2. $$v_{\text{max}} = 4 \times 0.1 = 0.4 \text{ m s}^{-1}$$
3. Total energy equals maximum kinetic energy, since potential is zero at equilibrium:
4. $$E_{\text{total}} = \frac{1}{2} m v_{\text{max}}^2 = 0.5 \times 0.2 \times 0.4^2 = 0.016 \text{ J}$$

## Common SHM Systems

IB Physics SL regularly tests two standard SHM systems: mass-spring systems and small-angle simple pendulums. Their period formulas are summarized below:

| System | Period Formula | SHM Conditions |
| --- | --- | --- |
| Horizontal mass-spring | $T = 2\pi \sqrt{\frac{m}{k}}$ | Spring obeys Hooke's law |
| Vertical mass-spring | $T = 2\pi \sqrt{\frac{m}{k}}$ | Spring obeys Hooke's law (gravity shifts equilibrium only) |
| Simple pendulum | $T = 2\pi \sqrt{\frac{l}{g}}$ | Displacement angle < 10° (small angle approximation) |

**Worked example:** A simple pendulum has period 2 s on Earth ($g = 9.8$ m s⁻²). Calculate its length.

1. Rearrange the pendulum period formula to solve for $l$:
2. $$l = \frac{g T^2}{4 \pi^2}$$
3. Substitute values $T = 2$ s, $g = 9.8$ m s⁻²:
4. $$l = \frac{9.8 \times (2)^2}{4 \pi^2} = \frac{9.8}{\pi^2} \approx 1.0 \text{ m}$$

> **warning**
>
> Vertical mass-springs have the same period as horizontal ones! Gravity only changes where equilibrium is, not how fast it oscillates.

## Common pitfalls

- **Wrong:** Forgetting the negative sign in $a = -\omega^2 x$
  - Why it fails: The negative sign is a core part of the SHM definition, showing acceleration opposes displacement.
  - Correct: Always include the negative sign when stating the SHM condition or writing acceleration equations.
- **Wrong:** Claiming period of SHM depends on amplitude
  - Why it fails: A defining property of SHM is isochronism: period is independent of amplitude for valid SHM systems.
  - Correct: Use the system-specific period formula, which never includes amplitude as a variable.
- **Wrong:** Thinking vertical mass-springs have different periods than horizontal ones
  - Why it fails: Gravity only shifts the equilibrium position, it does not change the restoring force or period.
  - Correct: Use $T = 2\pi \sqrt{m/k}$ for all mass-spring systems, regardless of orientation.
- **Wrong:** Using the simple pendulum period formula for large displacements
  - Why it fails: The formula only works for SHM, which requires the small angle approximation (< 10°).
  - Correct: Recognize that for angles larger than 10°, the pendulum does not obey SHM and the formula is invalid.
- **Wrong:** Confusing total energy with kinetic energy at non-zero displacement
  - Why it fails: Total energy is only equal to maximum kinetic energy at equilibrium, not at any other displacement.
  - Correct: Calculate potential energy for any non-zero displacement and add it to kinetic energy to get total energy.

## Cheatsheet

| Concept | Formula |
| --- | --- |
| SHM Core Condition | $a = -\omega^2 x$ |
| Angular Frequency | $\omega = 2\pi f = \frac{2\pi}{T}$ |
| Displacement ($t=0, x=A$) | $x = A \cos(\omega t)$ |
| Displacement ($t=0, x=0$) | $x = A \sin(\omega t)$ |
| Velocity | $v = \pm \omega \sqrt{A^2 - x^2}$ |
| Total SHM Energy | $E_{\text{total}} = \frac{1}{2} m \omega^2 A^2$ |
| Period: Mass-Spring | $T = 2\pi \sqrt{\frac{m}{k}}$ |
| Period: Simple Pendulum | $T = 2\pi \sqrt{\frac{l}{g}}$ |

## What's next

Simple harmonic motion is the foundational concept for all oscillation and wave topics in IB Physics SL. All wave motion can be modeled as a collection of coupled SHM oscillators, so mastering SHM makes understanding more advanced topics much easier. Next, you will build on this foundation to learn what happens when SHM systems lose energy or are driven by external forces, leading to the important phenomenon of resonance, which is a common exam question in Paper 1 and Paper 2. Explore the following topics to continue building your knowledge of wave behaviour.

- [Wave characteristics](https://www.owlsprep.com/study/ib-physics-sl-u3-wave-characteristics/)
- [Wave interference](https://www.owlsprep.com/study/ib-physics-sl-u3-wave-interference/)
- [Standing waves](https://www.owlsprep.com/study/ib-physics-sl-u3-standing-waves/)

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