Simple Harmonic Oscillations (SHM) - Comprehensive Study Notes (9702 A Level Physics)

Welcome to the fascinating world of Oscillations! This chapter explores how things move back and forth, from the swing of a pendulum to the vibrations that produce sound. Understanding Simple Harmonic Motion (SHM) is fundamental because many real-world processes—like clocks, musical instruments, and even atoms in a solid—can be modelled using these principles. Don't worry if the math looks intimidating at first; we will break down the concepts step-by-step using clear analogies!

17.1 Defining Simple Harmonic Motion (SHM)

Basic Terminology of Oscillations

Before diving into SHM, we need to establish the vocabulary used to describe repetitive motion. Imagine a mass oscillating on a spring.

  • Displacement (\(x\)): This is the distance of the oscillating object from its equilibrium position (the central resting point). Displacement is a vector quantity, so it includes direction (positive or negative).
  • Amplitude (\(x_0\)): This is the maximum displacement from the equilibrium position. It measures the size of the oscillation. \(x_0\) is always positive.
  • Period (\(T\)): The time taken for one complete oscillation or cycle (e.g., from maximum positive displacement back to maximum positive displacement). Measured in seconds (s).
  • Frequency (\(f\)): The number of complete oscillations occurring per unit time. Measured in Hertz (Hz) or \(s^{-1}\).

    • \(T = \frac{1}{f}\)

Understanding Angular Frequency (\(\omega\))

Although the object is moving linearly (back and forth), it is often mathematically easier to relate its motion to circular motion. This leads to the concept of angular frequency.

  • Angular Frequency (\(\omega\)): The rate of change of phase, measured in radians per second (\(rad \ s^{-1}\)).

    • Think of SHM as the projection of uniform circular motion onto a diameter. If an object takes time \(T\) to complete one cycle, it travels \(2\pi\) radians in that time.

    Formula relating period and frequency to angular frequency:

    \(\omega = \frac{2\pi}{T} = 2\pi f\)

Phase Difference (\(\phi\))

The phase difference describes how 'out of step' two or more oscillations are.

If two masses are oscillating with the same frequency, but one reaches maximum amplitude at time \(t=0\) and the other reaches maximum amplitude at \(t = T/4\), they have a phase difference.

  • If \(\phi = 0\) (or \(360^\circ / 2\pi \ rad\)): The oscillations are in phase (they move together).
  • If \(\phi = 180^\circ / \pi \ rad\): The oscillations are in anti-phase (one is at max positive displacement while the other is at max negative).
Quick Review: Core Terms

Amplitude is Maximum displacement.
Time for one cycle is Period.
Frequency is \(1/T\).
Angular frequency (\(\omega\)) uses Radians (\(2\pi/T\)).

17.1 The Defining Condition of SHM

SHM is not just any oscillation; it is a very specific type of oscillation where the restoring force (and thus the acceleration) behaves in a very predictable way.

The Defining Law

Simple Harmonic Motion (SHM) occurs when the acceleration (\(a\)) of the oscillating object is:

  1. Proportional to its displacement (\(x\)) from the fixed equilibrium point.
  2. Always directed towards the fixed equilibrium point (in the opposite direction to the displacement).

Analogy: The Restoring Force Cop
Imagine a police officer (the Restoring Force) standing at the equilibrium point. If the object moves to the right (+x), the police officer accelerates it strongly back to the left (–a). If the object moves a small amount, the force is small. If it moves a large amount, the force is large. The force always tries to restore the object to the center.

The SHM Equation

Mathematically, the defining condition is written as:

\[a = -\omega^2 x\]

  • \(a\) is the acceleration.
  • \(x\) is the displacement.
  • \(\omega\) is the angular frequency (which is constant for a given SHM system).
  • The minus sign (\(-\)) shows that the acceleration is always in the opposite direction to the displacement (the restoring condition).

Important Link to Force: Since $F = ma$, the force acting on the mass must also be proportional to displacement and directed towards equilibrium: \(F = m(-\omega^2 x)\).

Common Mistake Alert!

Students often forget the negative sign! Without the negative sign, the object would accelerate away from the center, leading to unstable, exponential motion, not oscillation.

17.1 Kinematics of SHM: Displacement, Velocity, and Acceleration

Since the acceleration in SHM is not constant (it changes with \(x\)), we must use specific equations derived using calculus (though you don't need to derive them, you must use them!).

Displacement Equation

If an oscillation starts at the equilibrium position (\(x=0\) at \(t=0\)), the displacement is given by:

\[x = x_0 \sin (\omega t)\]

If it starts at maximum amplitude (\(x=x_0\) at \(t=0\)), a cosine function is used: \(x = x_0 \cos (\omega t)\).

Velocity Equations

Velocity is maximum when displacement is zero (at the equilibrium point) and zero when displacement is maximum (at the amplitudes).

1. Velocity as a function of time: (Derived from the sine displacement equation)

\[v = v_0 \cos (\omega t)\]

Here, \(v_0\) is the maximum speed, found at the equilibrium position ($x=0$).

2. Velocity as a function of displacement (The 'quick' equation):

This is often the most useful equation in problem-solving:

\[v = \pm \omega \sqrt{x_0^2 - x^2}\]

Note: The maximum velocity \(v_0\) occurs when \(x=0\), so \(v_{max} = \omega \sqrt{x_0^2 - 0} = \omega x_0\).

Graphical Representation Analysis

It is essential to analyze and interpret the graphs showing how \(x\), \(v\), and \(a\) vary with time. Since they are all based on sine/cosine functions, they look like waves, but they are shifted relative to each other.

  1. Displacement (\(x\)) vs. \(t\): Follows a sine or cosine wave.
  2. Velocity (\(v\)) vs. \(t\): The velocity curve is a quarter cycle ahead (or 90° / \(\pi/2\) rad ahead) of the displacement curve. When displacement is maximum (a turning point), velocity is momentarily zero.
  3. Acceleration (\(a\)) vs. \(t\): The acceleration curve is a half cycle ahead (or 180° / \(\pi\) rad ahead) of the displacement curve. When displacement is max positive, acceleration is max negative.
Did You Know? The Pendulum

A simple pendulum only undergoes true SHM if the angle of swing is small (usually less than about 10 degrees). This is because the restoring force is only proportional to displacement for small angles.

17.2 Energy in Simple Harmonic Motion

In any ideal (undamped) SHM system, total mechanical energy is conserved. The energy continuously converts between Kinetic Energy (KE) and Potential Energy (PE).

Interchange of Energy

Analogy: The Trampoline
When you bounce on a trampoline:

  • At the top/bottom (Maximum Amplitude, \(x = \pm x_0\)): You stop momentarily. KE is zero, and all energy is stored as maximum Potential Energy (PE). (Elastic PE for a spring, Gravitational PE for a pendulum).
  • At the middle (Equilibrium Position, \(x = 0\)): You are moving fastest. PE is zero (or minimum), and all energy is converted to maximum Kinetic Energy (KE).

Maximum Potential Energy (\(PE_{max}\))

The total energy stored in the oscillation system is equal to the maximum Potential Energy (which occurs at \(x=x_0\)).

For a mass on a spring, the potential energy is often called the elastic potential energy, \(PE = \frac{1}{2} k x^2\). Since \(k\) is related to $\omega$ (specifically \(k=m\omega^2\)), we can express the total energy \(E\) using the terms we know for SHM.

Total Energy in SHM (\(E\))

The total energy (\(E\)) of the system, which is constant, is calculated using the maximum amplitude and angular frequency:

\[E = \frac{1}{2} m \omega^2 x_0^2\]

Where:

  • \(m\) is the mass of the oscillating object.
  • \(\omega\) is the angular frequency.
  • \(x_0\) is the amplitude.

Key Takeaway: The total energy of an SHM system is proportional to the square of the amplitude (\(x_0^2\)). If you double the amplitude, you quadruple the energy!

17.3 Damped and Forced Oscillations, Resonance

Ideal SHM (where energy is conserved) rarely happens in the real world. Real systems always lose energy, leading to damping.

Damping

Damping is the reduction in the amplitude of an oscillation over time due to a resistive force (like air resistance or internal friction) acting on the system.

This resistive force always opposes the motion, converting the system’s mechanical energy (KE and PE) into heat energy (internal energy). This process causes the amplitude ($x_0$) to decrease exponentially over time.

We classify damping into three main types, based on the magnitude of the resistive force:

  1. Light Damping: The resistive forces are small. The system oscillates many times, but the amplitude gradually decreases over several periods. (Example: A car with worn-out shocks, or a grandfather clock pendulum).
  2. Heavy Damping: The resistive forces are large. The object returns slowly to the equilibrium position without oscillating. The motion takes a long time. (Example: A pendulum swinging in very thick oil).
  3. Critical Damping: The resistive force is exactly the right size to bring the oscillating body back to the equilibrium position in the shortest possible time without overshooting or oscillating. This is the ideal behaviour for many applications. (Example: Car shock absorbers (dampers) are designed to be critically damped).

Sketching Displacement-Time Graphs:
You must be able to sketch the displacement-time graphs for these types of damping. The key is how quickly the oscillation stops or how slowly it returns to zero.

Forced Oscillations and Natural Frequency

If a system is left alone to oscillate (after an initial push), it oscillates at its own specific frequency, called the natural frequency (\(f_0\)).

A forced oscillation occurs when an external periodic force (the driving force) is applied to the system, causing it to oscillate at the frequency of the external force (the driving frequency, \(f_{driver}\)).

Resonance

Resonance is an extremely important concept in physics and engineering.

  • Definition: Resonance occurs when an oscillating system is forced to oscillate at its natural frequency (\(f_{driver} = f_0\)).
  • Effect: This causes the system to oscillate with a maximum amplitude.

The extent of this amplitude increase depends strongly on damping:

  • Low Damping: Leads to a very sharp, tall resonance peak (large maximum amplitude).
  • High Damping: Leads to a lower, broader resonance peak (small maximum amplitude).

Real-World Application:

Resonance can be useful (like tuning a radio to a specific frequency) or disastrous (like the famous Tacoma Narrows Bridge collapse in 1940, where wind forced the bridge to oscillate at its natural frequency). Engineers must design structures to have natural frequencies far away from any common driving frequencies they might encounter.

Chapter Summary: Key Takeaways

1. Definition: SHM means acceleration is proportional to displacement ($a \propto -x$), defined by \(a = -\omega^2 x\).

2. Kinematics: Maximum velocity is $v_0 = \omega x_0$. Velocity at any point $x$ is $v = \pm \omega \sqrt{x_0^2 - x^2}$.

3. Energy: Total energy is conserved and given by \(E = \frac{1}{2} m \omega^2 x_0^2\). Energy swaps between KE (max at equilibrium) and PE (max at amplitude).

4. Damping: Resistive forces reduce amplitude (exponential decay). Critical damping returns the system to equilibrium fastest without oscillation.

5. Resonance: Maximum amplitude occurs when the driving frequency matches the natural frequency ($\omega_{driver} = \omega_0$). Low damping yields a sharper, higher peak.