Simple harmonic oscillations

Welcome to the World of Oscillations!

In this chapter, we are going to explore Simple Harmonic Motion (SHM). You’ve seen this everywhere in your daily life without even realizing it: a child on a swing, the swaying of a skyscraper in the wind, or even the atoms vibrating inside your phone.

We’ll learn how to describe these "back-and-forth" movements using math and graphs. Don’t worry if it seems a bit "maths-heavy" at first; once you see the patterns, it becomes much clearer. Let's dive in!

1. The Language of Oscillations

Before we look at the formulas, we need to speak the right language. Here are the key terms you need to know:

Displacement (\(x\)): How far the object is from its middle point (the equilibrium position) at any moment. It has a direction, so it can be positive or negative. Measured in meters (m).
Amplitude (\(A\)): The maximum displacement. This is the furthest the object gets from the center. Measured in meters (m).
Period (\(T\)): The time it takes for one complete "to-and-fro" cycle. Measured in seconds (s).
Frequency (\(f\)): How many full cycles happen every second. Measured in Hertz (Hz).
Phase Difference: A way of describing how "out of sync" two oscillators are. We usually measure this in radians.

Quick Review: The Frequency Formula

Frequency and Period are opposites. If a pendulum takes a long time to swing (\(T\) is big), it doesn't swing very often (\(f\) is small).
\(f = \frac{1}{T}\)

Angular Frequency (\(\omega\))

In SHM, we often use angular frequency instead of regular frequency. Think of it as how fast the "cycle" is moving in terms of angles. Since one full circle is \(2\pi\) radians, the formula is:
\(\omega = 2\pi f\) or \(\omega = \frac{2\pi}{T}\)
Units: radians per second (rad s\(^{-1}\))

Key Takeaway: Amplitude is the max distance, Period is the time for one swing, and Angular Frequency (\(\omega\)) tells us how fast the oscillation is happening.

2. Defining Simple Harmonic Motion (SHM)

Not every back-and-forth motion is "Simple Harmonic." To be SHM, a motion must follow one golden rule.

The Rule: The acceleration of the object must be proportional to its displacement and always directed towards the equilibrium position.

The Defining Equation

\(a = -\omega^2 x\)

What does this mean?
1. The \(\omega^2\) part is a constant. This shows that if you double the displacement (\(x\)), you double the acceleration (\(a\)).
2. The minus sign (\(-\)) is super important! It tells us that the acceleration is always "fighting" the displacement. If you pull a spring to the right (positive \(x\)), the acceleration pulls it back to the left (negative \(a\)).

Analogy: Think of a "Restoring Force." Like a grumpy neighbor who always wants you to get off their lawn and back to your own house, the force in SHM always pushes the object back to the center.

Did you know? Because the period of SHM doesn't change even if the amplitude gets smaller, we call it isochronous. This is why old pendulum clocks keep good time even as the swing fades slightly!

Key Takeaway: SHM is defined by \(a = -\omega^2 x\). Acceleration is always pulling the object back to the middle.

3. Predicting the Position: Sine and Cosine

If we want to know exactly where an object is at a specific time (\(t\)), we use trig functions.

Don't worry if this seems tricky! You just need to choose the right starting point:

Scenario A: Starting at the edge (Max displacement)
If you pull a pendulum back and release it, use the Cosine version:
\(x = A \cos(\omega t)\)

Scenario B: Starting in the middle (Equilibrium)
If you give a resting pendulum a "kick" to start it moving, use the Sine version:
\(x = A \sin(\omega t)\)

IMPORTANT MISTAKE TO AVOID: Always make sure your calculator is in RADIANS mode when doing these calculations. If you use Degrees, your answers will be wrong!

Key Takeaway: Use \(x = A \cos(\omega t)\) if you start at the maximum displacement and \(x = A \sin(\omega t)\) if you start from the center.

4. Velocity in SHM

The speed of the object changes throughout the swing.

- At the center (equilibrium): The object is moving at its maximum velocity.
- At the edges (amplitude): The object momentarily stops to change direction, so velocity is zero.

Maximum Velocity Formula

\(v_{max} = \omega A\)
Memory Aid: "V-Max is WA." (Velocity Max = Omega times Amplitude)

Velocity at any point \(x\)

If you need to find the velocity at a specific position \(x\), use this formula:
\(v = \pm \omega \sqrt{A^2 - x^2}\)

Step-by-Step Check:
1. If \(x = 0\) (the center), the formula becomes \(v = \omega \sqrt{A^2}\), which is just \(\omega A\). This matches our max velocity!
2. If \(x = A\) (the edge), the formula becomes \(v = \omega \sqrt{A^2 - A^2}\), which is \(0\). This matches our "stopping" point!

Key Takeaway: Objects move fastest through the middle and stop at the very edges.

5. Representing SHM with Graphs

In the exam, you are often asked how Displacement (\(x\)), Velocity (\(v\)), and Acceleration (\(a\)) relate to each other on a graph.

1. Displacement vs. Time: A standard Sine or Cosine wave.
2. Velocity vs. Time: This graph is "shifted" by 90 degrees (\(\pi/2\) radians). When displacement is 0, velocity is at its peak.
3. Acceleration vs. Time: This graph is a "mirror image" of displacement (shifted by 180 degrees or \(\pi\) radians). When displacement is at its positive maximum, acceleration is at its negative maximum.

Encouragement: If you find the graphs confusing, just remember the "Defining Rule": acceleration is always opposite to displacement. If the displacement graph goes UP, the acceleration graph must go DOWN at that exact same time!

Key Takeaway: Velocity is 90 degrees out of phase with displacement. Acceleration is 180 degrees out of phase with displacement.

6. Practical Skills (PAG 10)

You need to know how to measure the period/frequency of SHM in the lab. Usually, we use a Mass on a Spring or a Simple Pendulum.

Top Tips for Accuracy:
- Use a Fiducial Marker: Place a pointer (like a pin) at the equilibrium position. It is much easier to time when the object passes the center because it is moving at its fastest there.
- Timing Multiple Swings: Instead of timing 1 swing, time 10 or 20 and then divide by the number of swings. This reduces the effect of your reaction time error.
- Small Angles: For a pendulum, SHM only strictly applies if the angle of the swing is small (less than about 10 degrees).

Key Takeaway: Time from the center (equilibrium) and time multiple oscillations to keep your data accurate.