Welcome to Simple Harmonic Motion (SHM)
Have you ever watched a playground swing move back and forth, plucked a guitar string, or seen a pendulum clock tick? All of these are examples of oscillations. In this chapter, we will explore a special and very important type of oscillation called Simple Harmonic Motion (SHM).
Don't worry if the mathematics looks intimidating at first glance! We will break everything down step-by-step, connect equations to real-life movements, and give you memory tips to master your exam questions.
1. Defining Simple Harmonic Motion
An oscillation is a back-and-forth repetitive motion about a central balance point, known as the equilibrium position.
The Golden Definition of SHM
In physics, a body undergoes Simple Harmonic Motion if and only if:
1. Its acceleration is directly proportional to its displacement from the equilibrium position.
2. Its acceleration is always directed towards the equilibrium position.
Mathematically, we write this fundamental relationship as:
\(a \propto -x\)
Which leads to the definitive equation:
\(a = -\omega^2 x\)
Where:
• \(a\) = acceleration of the oscillator in metres per second squared (\(\text{m s}^{-2}\))
• \(x\) = displacement from the equilibrium position in metres (\(\text{m}\))
• \(\omega\) = angular frequency in radians per second (\(\text{rad s}^{-1}\))
• The minus sign (\(-\)) shows that the acceleration is always in the opposite direction to the displacement (it always pulls or pushes back towards the centre).
Key Terms You Need to Know
• Displacement (\(x\)): The distance and direction of the oscillating object from its equilibrium position (measured in \(\text{m}\)).
• Amplitude (\(A\) or \(x_0\)): The maximum displacement from the equilibrium position (measured in \(\text{m}\)).
• Period (\(T\)): The time taken to complete one full oscillation (measured in seconds, \(\text{s}\)).
• Frequency (\(f\)): The number of complete oscillations per unit time (measured in hertz, \(\text{Hz}\) or \(\text{s}^{-1}\)). Recall that \(f = \frac{1}{T}\).
• Angular Frequency (\(\omega\)): The rate of change of phase angle, defined as:
\(\omega = 2\pi f = \frac{2\pi}{T}\)
Memory Aid: An oscillation is like a rubber band stretched away from your hand: the further away you pull it (large displacement \(x\)), the harder it snaps back towards the centre (large acceleration \(a\)).
Key Takeaway: Whenever you are asked to prove a system is performing SHM, you must show that its acceleration matches \(a = -\text{constant} \times x\).
2. Kinematics of SHM: Displacement, Velocity, and Acceleration
How Position Changes with Time
If an object begins oscillating from its maximum displacement at time \(t = 0\), its displacement is modelled by:
\(x = A \cos(\omega t)\)
If the object starts at the equilibrium position at \(t = 0\), its displacement is modelled by:
\(x = A \sin(\omega t)\)
Velocity in SHM
As the object swings back and forth, its velocity changes continuously:
• Velocity is maximum when passing through the equilibrium position (\(x = 0\)).
• Velocity is zero at the maximum displacement (amplitude) (\(x = \pm A\)), because the object momentarily stops to change direction.
To find the velocity at any specific displacement \(x\), we use:
\(v = \pm \omega \sqrt{A^2 - x^2}\)
The maximum velocity (\(v_{\text{max}}\)) occurs when \(x = 0\):
\(v_{\text{max}} = \omega A\)
Acceleration in SHM
• Acceleration is zero at the equilibrium position (\(x = 0\)).
• Acceleration is maximum at the extremes (\(x = \pm A\)).
The maximum acceleration (\(a_{\text{max}}\)) is given by:
\(a_{\text{max}} = \omega^2 A\)
Phase Relationships
Understanding how displacement, velocity, and acceleration graphs relate to one another is a classic exam question:
• Velocity leads displacement by \(\frac{\pi}{2}\) radians (\(90^\circ\)) or a quarter of a cycle.
• Acceleration leads velocity by \(\frac{\pi}{2}\) radians (\(90^\circ\)).
• Acceleration is \(\pi\) radians (\(180^\circ\)) out of phase with displacement (they are in exact antiphase, explaining the minus sign in \(a = -\omega^2 x\)).
Summary Table: At a Glance
• At Equilibrium Position (\(x = 0\)): Displacement = \(0\), Velocity = \(\text{Maximum}\) (\(\pm \omega A\)), Acceleration = \(0\).
• At Maximum Amplitude (\(x = \pm A\)): Displacement = \(\pm A\), Velocity = \(0\), Acceleration = \(\text{Maximum}\) (\(\mp \omega^2 A\)).
Common Mistake to Avoid: Students often think velocity is maximum at the extremes because that is where force is greatest. Remember: at the turning points, the object stops completely for a split second (\(v = 0\))!
3. Classic Simple Harmonic Oscillators
1. The Simple Pendulum
A simple pendulum consists of a small mass (bob) suspended from a light string. For small angles of displacement (typically \(\theta < 10^\circ\)), its motion is simple harmonic.
The periodic time \(T\) is given by:
\(T = 2\pi \sqrt{\frac{l}{g}}\)
Where:
• \(l\) = length of the pendulum in metres (\(\text{m}\))
• \(g\) = acceleration due to gravity (\(9.81\text{ m s}^{-2}\))
Did you know? The mass of the bob and the amplitude (for small angles) do not affect the period of a simple pendulum at all!
2. Mass on a Helical Spring
A mass attached to a helical spring will oscillate when displaced vertically or horizontally.
The periodic time \(T\) is given by:
\(T = 2\pi \sqrt{\frac{m}{k}}\)
Where:
• \(m\) = mass attached to the spring in kilograms (\(\text{kg}\))
• \(k\) = spring constant in newtons per metre (\(\text{N m}^{-1}\))
Key Takeaway: Both oscillators are isochronous, meaning the period \(T\) is independent of the amplitude \(A\).
4. Energy in Simple Harmonic Motion
During undamped SHM, energy continuously interchanges between Kinetic Energy (\(E_k\)) and Potential Energy (\(E_p\)), while the Total Energy (\(E_{\text{total}}\)) remains constant (assuming no energy is lost to resistive forces).
Energy Equations
• Kinetic Energy:
\(E_k = \frac{1}{2} m v^2 = \frac{1}{2} m \omega^2 (A^2 - x^2)\)
• Potential Energy:
\(E_p = \frac{1}{2} m \omega^2 x^2\)
• Total Mechanical Energy:
\(E_{\text{total}} = E_k + E_p = \frac{1}{2} m \omega^2 A^2\)
Energy vs Displacement Graphs
• The \(E_p\) curve is an upright parabola with a minimum at \(x = 0\) and peaks at \(x = \pm A\).
• The \(E_k\) curve is an inverted parabola with a maximum at \(x = 0\) and zeros at \(x = \pm A\).
• The \(E_{\text{total}}\) line is a horizontal straight line across the top at value \(\frac{1}{2}m\omega^2 A^2\).
• At any displacement \(x\), the sum of \(E_k\) and \(E_p\) equals \(E_{\text{total}}\).
Quick Check: When \(x = \frac{A}{2}\), what fraction of the total energy is potential energy?
Using \(E_p \propto x^2\), when \(x = \frac{A}{2}\), \(E_p = \frac{1}{4} E_{\text{total}}\). The remaining \(\frac{3}{4}\) is kinetic energy!
5. Damping in Oscillations
In the real world, oscillating systems experience resistive forces (such as friction and air resistance). Damping is the process by which an oscillating system loses energy to its surroundings, causing the amplitude of oscillation to decrease over time.
Degrees of Damping
1. Light Damping (Underdamping):
• The amplitude decays exponentially over time.
• The period \(T\) and frequency \(f\) remain almost unchanged.
• Example: A pendulum swinging in air.
2. Critical Damping:
• The system returns to the equilibrium position in the shortest possible time without oscillating.
• Example: Car suspension systems (shock absorbers) and needle indicators on analogue meters.
3. Heavy Damping (Overdamping):
• The resistive forces are very large. The system returns towards the equilibrium position very slowly without oscillating.
• Example: Automatic closing mechanisms on heavy fire doors.
Key Takeaway: Critical damping is ideal when you need rapid stabilization without overshoot or unwanted oscillations.
6. Forced Oscillations and Resonance
Free vs. Forced Oscillations
• Free Oscillations: An oscillation that occurs when a system is displaced and released without any external periodic driving force. The system oscillates at its natural frequency (\(f_0\)).
• Forced Oscillations: An oscillation driven by a continuous periodic external force. The system is forced to oscillate at the driving frequency (\(f\)) of the external driver.
The Phenomenon of Resonance
Resonance occurs when the periodic driving frequency matches the natural frequency of the vibrating system (\(f = f_0\)).
When resonance happens:
• There is a maximum transfer of energy from the periodic driver to the oscillating system.
• The amplitude of the oscillations reaches a dramatic maximum.
The Effect of Damping on the Resonance Curve
When plotting amplitude against driving frequency:
• As damping increases, the maximum amplitude of the peak decreases.
• As damping increases, the resonance curve becomes broader and flatter.
• With heavy damping, the peak frequency shifts slightly to a value just below the natural frequency \(f_0\).
Real-World Examples of Resonance
• Useful Applications:
– Radio receivers: Tuning circuits resonate at a specific radio station frequency to amplify its signal.
– Microwave ovens: Microwaves cause water molecules in food to oscillate, heating the food.
– Musical instruments: Acoustic bodies (e.g., of violins and guitars) resonate to amplify sound waves.
• Destructive Effects:
– Bridges and Buildings: Wind or marching troops can match the natural frequency of a suspension bridge, causing large, destructive oscillations (e.g., the Tacoma Narrows Bridge collapse). Engineers add heavy dampers to prevent this!
Quick Chapter Summary
• SHM Condition: \(a = -\omega^2 x\).
• Velocity: \(v = \pm \omega \sqrt{A^2 - x^2}\), with \(v_{\text{max}} = \omega A\) at equilibrium.
• Period equations: Pendulum: \(T = 2\pi \sqrt{\frac{l}{g}}\); Mass-Spring: \(T = 2\pi \sqrt{\frac{m}{k}}\).
• Energy: Total energy is conserved: \(E_{\text{total}} = \frac{1}{2} m \omega^2 A^2\).
• Damping: Dissipates energy; critical damping stops oscillation in the minimum time.
• Resonance: Happens when driving frequency \(f = \text{natural frequency } f_0\), resulting in maximum energy transfer and maximum amplitude.