Welcome to the World of Simple Harmonic Motion!
Hello there! Today we are diving into one of the most rhythmic and repetitive topics in Further Mathematics: Simple Harmonic Motion (SHM). Whether it's the swing of a grandfather clock, the bounce of a car's suspension, or even the vibration of a guitar string, SHM is everywhere.
Don't worry if the math looks a bit "wavy" at first. By the end of these notes, you’ll understand how objects move back and forth and how we use equations to predict exactly where they will be at any given time. Let's get moving!
1. What Exactly is SHM?
At its heart, Simple Harmonic Motion is a specific type of periodic motion. But not every back-and-forth movement is SHM. To be "Simple Harmonic," a motion must follow one golden rule:
The acceleration of the object must be directly proportional to its displacement from a fixed point and always directed toward that point.
The "Rubber Band" Analogy
Imagine you are holding a ball attached to a rubber band. If you pull the ball to the right (displacement), the rubber band pulls it back to the left (acceleration). The further you pull it, the harder it snaps back. That "pulling back to the center" is what makes it SHM.
The Defining Equation
In your exam, you will define SHM using this formula:
\( a = -\omega^2 x \)
- \( a \): Acceleration (in \(ms^{-2}\)).
- \( x \): Displacement from the center (equilibrium) position (in \(m\)).
- \( \omega \): The angular frequency (in \(rad \ s^{-1}\)).
- The Minus Sign (\(-\)): This is vital! It shows that acceleration is always in the opposite direction to displacement. If you pull it right, it pulls back left.
Quick Review: For SHM to happen, acceleration must look towards the center and get stronger the further away you go!
2. Key Terms You Need to Know
Before we look at the graphs, let's get our vocabulary straight. These terms will appear in almost every exam question:
- Equilibrium Position: The "rest" position where the net force is zero (usually \( x = 0 \)).
- Amplitude (\(A\)): The maximum displacement from equilibrium. It’s the furthest the object ever gets from the center.
- Period (\(T\)): The time taken for one full oscillation (e.g., from far right, to far left, and back to far right).
- Frequency (\(f\)): The number of full oscillations per second. Measured in Hertz (\(Hz\)).
- Angular Frequency (\(\omega\)): A measure of how fast the oscillation is happening in radians.
The "Math Connection" Box
You can jump between these terms using these handy formulas:
\( T = \frac{2\pi}{\omega} \)
\( f = \frac{1}{T} \)
\( \omega = 2\pi f \)
Did you know? In SHM, the Period (\(T\)) is independent of the Amplitude (\(A\)). This means a pendulum takes the same time to swing back and forth whether you give it a tiny nudge or a big push!
3. The Equations of Motion
How do we find out where an object is at a specific time? We use trigonometry! Because SHM is repetitive, sine and cosine curves are perfect for describing it.
Displacement (\(x\))
If we start timing when the object is at its maximum displacement (the end of the swing):
\( x = A \cos(\omega t) \)
If we start timing when the object is passing through the equilibrium (the center):
\( x = A \sin(\omega t) \)
Velocity (\(v\))
Sometimes you need to know how fast the object is going without knowing the time. Use this "magic" formula:
\( v^2 = \omega^2(A^2 - x^2) \)
Step-by-Step Logic:
1. At the center (\(x = 0\)), velocity is at its maximum: \( v_{max} = \omega A \).
2. At the edges (\(x = A\)), the object momentarily stops to change direction, so velocity is zero.
Common Mistake to Avoid: Always ensure your calculator is in RADIANS mode when calculating \( \cos(\omega t) \) or \( \sin(\omega t) \). Using degrees will give you a very wrong answer!
4. Simple Pendulums and Springs
The Oxford AQA syllabus often asks you to apply SHM to two specific real-world systems: mass on a spring and the simple pendulum.
Mass on a Spring
For a mass \(m\) on a spring with stiffness \(k\):
\( T = 2\pi \sqrt{\frac{m}{k}} \)
Mnemonic: Think of "My Keys" (\( m / k \)) to remember the order of the fraction!
Simple Pendulum
For a string of length \(l\) in gravity \(g\):
\( T = 2\pi \sqrt{\frac{l}{g}} \)
Analogy: A Long pendulum takes a Long time to swing (increasing \(l\) increases \(T\)).
Key Takeaway: For a pendulum, the mass of the bob doesn't matter! A heavy person and a light person on the same length swing will have the same period.
5. Summary and Exam Tips
Quick Checklist for Problems:
- Check if the question gives you Frequency or Period; immediately calculate \( \omega \).
- Identify the Amplitude (\(A\)) — usually the maximum distance mentioned.
- If the question asks for "maximum speed," use \( v = \omega A \).
- If it asks for "maximum acceleration," use \( a = \omega^2 A \).
Don't worry if this seems tricky at first! SHM is just circular motion viewed from the side. Once you get used to the link between \(\omega\), \(T\), and \(A\), the rest is just plugging numbers into the formulas.
Final Summary Key Point:
SHM is defined by \( a = -\omega^2 x \). Everything else—the sine waves, the pendulum swings, and the spring bounces—comes from this one simple relationship between acceleration and position.