Introduction to Simple Harmonic Systems

Welcome! In the previous chapter, we explored the basic rules of Simple Harmonic Motion (SHM). Now, we are going to look at how these rules apply to real-world systems like springs and pendulums. We will also investigate why things eventually stop moving (damping) and how we can use external forces to make things vibrate with huge amplitudes (resonance).

Don't worry if this seems a bit "maths-heavy" at first. We will break down every formula and use everyday examples to make sense of the physics!

1. Simple Harmonic Systems

There are two classic systems you need to know inside out for your AQA exams: the mass-spring system and the simple pendulum.

The Mass-Spring System

Imagine a mass \(m\) attached to a spring with a spring constant \(k\). If you pull it and let go, it oscillates. The time it takes for one full oscillation (the period, \(T\)) depends on how heavy the mass is and how stiff the spring is.

The formula for the period is:
\(T = 2\pi\sqrt{\frac{m}{k}}\)

  • Increase the mass \(m\): The period \(T\) increases (it moves slower because it has more inertia).
  • Increase the spring constant \(k\): The period \(T\) decreases (it moves faster because the restoring force is stronger).

The Simple Pendulum

A simple pendulum consists of a small mass (a bob) on a string of length \(l\). For small angles (less than about \(10^{\circ}\)), it performs SHM.

The formula for the period is:
\(T = 2\pi\sqrt{\frac{l}{g}}\)

Important Note: Notice that the mass of the bob \(m\) is not in the formula! This means a heavy bob and a light bob will have the same period if the strings are the same length. This is a common "trick" question in multiple-choice papers.

Quick Review: To find the frequency \(f\), just remember that \(f = \frac{1}{T}\). So, for a pendulum, \(f = \frac{1}{2\pi}\sqrt{\frac{g}{l}}\).

2. Energy in SHM

In a perfect SHM system with no friction, the total energy remains constant. However, the energy constantly swaps between two forms:

  • Potential Energy (\(E_p\)): This is maximum when the displacement is at its peak (the "turning points").
  • Kinetic Energy (\(E_k\)): This is maximum when the object is rushing through the center (the equilibrium position).

Key Takeaway:
At maximum displacement (\(x = A\)): \(E_p\) is Max, \(E_k\) is Zero.
At equilibrium (\(x = 0\)): \(E_p\) is Zero, \(E_k\) is Max.

Did you know? The graph of energy against displacement is a parabola. The total energy line is a horizontal flat line above them, showing that \(E_k + E_p = \text{constant}\).

3. Damping: Slowing Things Down

In the real world, oscillations don't last forever. Friction and air resistance take energy away from the system and turn it into heat. This is called damping.

Types of Damping

You need to be able to identify three main types of damping:

  1. Light Damping: The amplitude gradually decreases over time, but the period remains almost unchanged. Example: A pendulum swinging in air.
  2. Critical Damping: This is the "just right" amount of damping. It returns the object to equilibrium in the shortest time possible without overshooting. Example: Car suspension systems or high-quality door closers.
  3. Heavy Damping (Overdamping): The damping is so strong that the object takes a long time to return to equilibrium and never oscillates. Example: Trying to swing a pendulum in thick treacle.

Common Mistake: Students often think critical damping means the object stops instantly. It doesn't! It just means it gets back to the "zero" position as fast as physically possible without "wobbling" past it.

4. Forced Vibrations and Resonance

Now, let's look at what happens when we apply an external rhythmic force to an oscillator.

Key Definitions

  • Natural Frequency (\(f_0\)): The frequency an object vibrates at if you just give it one tap and let it go (a "free vibration").
  • Driving Frequency (\(f\)): The frequency of the external force being applied to the system (a "forced vibration").

What is Resonance?

Resonance occurs when the driving frequency is equal to the natural frequency of the system (\(f = f_0\)).

When this happens, the energy transfer is at its most efficient, and the amplitude of the vibrations becomes very large. Think of pushing a friend on a swing: if you push at exactly the same time the swing starts to move away from you, they go higher and higher. If you push at the wrong time, you actually slow them down!

Resonance and Damping

Damping has a big effect on the "resonance curve" (a graph of Amplitude vs. Driving Frequency):

  • Lightly damped systems: Have a very sharp, tall peak at the natural frequency.
  • Heavily damped systems: Have a flatter, broader peak, and the maximum amplitude occurs at a frequency slightly lower than the natural frequency.

Key Takeaway: As damping increases, the resonance peak becomes lower and flatter (less "sharp").

5. Required Practical 7: Investigating SHM

For your exam, you must be familiar with how to investigate these systems experimentally. Here are the top tips for success:

  • Use a fiducial marker: Place a pointer (like a needle or a mark on a card) at the equilibrium position. It is much easier to time when the mass passes the center than when it reaches its highest point because it is moving fastest at the center.
  • Timing multiple oscillations: Never time just one swing! Time 10 or 20 full oscillations and divide the total time by the number of swings. This reduces the percentage uncertainty caused by your reaction time.
  • Small angles: For the pendulum, ensure the angle of swing is less than \(10^{\circ}\) to keep the SHM approximation valid.

Summary Table:

Mass-Spring Period: \(T = 2\pi\sqrt{\frac{m}{k}}\)
Pendulum Period: \(T = 2\pi\sqrt{\frac{l}{g}}\)
Resonance Condition: \(f_{\text{driving}} = f_{\text{natural}}\)
Damping Effect: Reduces amplitude and "flattens" the resonance peak.

Final Tip: When drawing resonance graphs, always label your axes! Amplitude goes on the y-axis, and Driving Frequency goes on the x-axis.