Why Everything Oscillates
~6 min read · Tier toggle changes depth (T1 plain · T2 standard · T3 rigour)
You should be comfortable with basic calculus, specifically differentiation and integration, and have a general intuition for Newton’s laws of motion. No prior knowledge of differential equations is required, as this lecture introduces the concepts from first principles.
Why this matters
Oscillation is the fundamental language of nature, appearing in everything from atomic bonds to planetary orbits.
Before we can analyse complex wave phenomena, we must understand the simplest possible system: a mass attached to a spring. This lecture establishes the physical intuition behind periodic motion. We will explore why systems naturally return to equilibrium, how energy exchanges between kinetic and potential forms, and why this specific type of motion is so prevalent in physics. By the end of this session, you will see that the simple harmonic oscillator is not just a toy model, but the building block for understanding sound, light, and quantum fields.
The Intuition of Equilibrium
Imagine a ball resting at the bottom of a smooth, curved bowl. If you nudge it slightly to the left, gravity pulls it back towards the centre. As it moves towards the centre, it speeds up. However, due to its inertia, it does not stop exactly at the bottom; it overshoots to the right. Now, gravity pulls it back to the left. This back-and-forth motion is an oscillation.
The key feature here is the restoring force. Whenever the object is displaced from its equilibrium position, a force acts to bring it back. In the case of the bowl, this force is a component of gravity. In the case of a spring, it is the elastic tension. Crucially, for small displacements, the strength of this restoring force is directly proportional to the displacement. This linear relationship is what gives rise to the simple, predictable rhythm of harmonic motion. If the force were not proportional to displacement, the motion would be more complex and irregular. This linear, proportional relationship between force and position is what produces the sinusoidal ("harmonic") motion.
Newton’s Second Law and the Equation of Motion
To describe this motion mathematically, we apply Newton’s second law, \( F = ma \). For a mass \( m \) attached to a spring with spring constant \( k \), Hooke’s Law tells us that the restoring force is \( F = -kx \), where \( x \) is the displacement from equilibrium. The negative sign indicates that the force always acts in the direction opposite to the displacement.
Substituting this into Newton’s second law gives us the fundamental differential equation for simple harmonic motion:
Rearranging this equation, we get:
This equation states that the acceleration of the mass is proportional to its displacement but in the opposite direction. This is the defining characteristic of simple harmonic motion. The term \( \frac{k}{m} \) is a constant that determines how quickly the system oscillates. We define the angular frequency \( \omega \) such that \( \omega^2 = \frac{k}{m} \). Thus, the equation becomes \( \frac{\mathrm{d}^2 x}{\mathrm{d}t^2} + \omega^2 x = 0 \). The solution to this equation is a sine or cosine function, which describes the periodic back-and-forth motion we observed intuitively.
Energy Exchange in the Oscillator
Another way to understand oscillation is through energy. In a frictionless system, the total mechanical energy is conserved. The energy exists in two forms: kinetic energy, associated with the motion of the mass, and potential energy, stored in the stretched or compressed spring.
The kinetic energy is given by \( K = \frac{1}{2}mv^2 \), and the potential energy is \( U = \frac{1}{2}kx^2 \). As the mass moves from the maximum displacement (amplitude) towards the equilibrium position, the potential energy decreases and the kinetic energy increases. At the equilibrium point, the displacement is zero, so the potential energy is zero, and the kinetic energy is at its maximum. As the mass moves past equilibrium and slows down, the kinetic energy is converted back into potential energy.
This continuous exchange between kinetic and potential energy is the engine of the oscillation. The total energy \( E = K + U \) remains constant throughout the cycle. The amplitude of the oscillation is determined by the total energy input into the system. A larger amplitude means a greater maximum displacement, which corresponds to a higher total energy stored in the system.
Why "Simple" Harmonic Motion is Special
Not all oscillations are simple harmonic. If you stretch a rubber band too far, the force is no longer proportional to the displacement, and the motion becomes anharmonic. However, for small displacements around any stable equilibrium point, the potential energy curve can be approximated by a parabola. This is a consequence of the Taylor series expansion of the potential energy function.
Because the potential energy near a minimum is approximately quadratic, the restoring force is approximately linear. This means that almost any stable system, when perturbed slightly, will exhibit simple harmonic motion. This universality is why the simple harmonic oscillator is so important. It provides a baseline model for understanding vibrations in molecules, bridges, electrical circuits, and even quantum fields. By mastering this simple case, we gain the tools to analyse more complex systems by breaking them down into their fundamental oscillatory components.
A common misconception is that the frequency of oscillation depends on the amplitude. In reality, for an ideal simple harmonic oscillator, the frequency \( \omega = \sqrt{k/m} \) is independent of the amplitude. Whether you pull the mass a little or a lot, it takes the same amount of time to complete one cycle. This property, known as isochronism, is a direct result of the linearity of the restoring force. In real-world systems with damping or non-linear forces, the frequency may shift slightly with amplitude, but in the ideal model, it remains constant. C
Worked example
Problem. A mass of \( 0.50\ \mathrm{kg} \) is attached to a spring with a spring constant of \( 20\ \mathrm{N/m} \). The mass is displaced \( 0.10\ \mathrm{m} \) from its equilibrium position and released from rest. Calculate the angular frequency \( \omega \) and the maximum speed \( v_{\mathrm{max}} \) of the mass.
- Identify the given values: \( m = 0.50\ \mathrm{kg} \), \( k = 20\ \mathrm{N/m} \), amplitude \( A = 0.10\ \mathrm{m} \).
- Calculate the angular frequency using the formula \( \omega = \sqrt{\frac{k}{m}} \). Substituting the values: \( \omega = \sqrt{\frac{20}{0.50}} = \sqrt{40} \approx 6.32\ \mathrm{rad/s} \).
- Determine the maximum speed. The maximum speed occurs at the equilibrium position where all energy is kinetic. Using conservation of energy: \( \frac{1}{2}kA^2 = \frac{1}{2}mv_{\mathrm{max}}^2 \). Solving for \( v_{\mathrm{max}} \): \( v_{\mathrm{max}} = A\sqrt{\frac{k}{m}} = A\omega \).
- Substitute the values: \( v_{\mathrm{max}} = 0.10 \times 6.32 = 0.632\ \mathrm{m/s} \).
Units check. For \( \omega \): \( \sqrt{\frac{\mathrm{N/m}}{\mathrm{kg}}} = \sqrt{\frac{\mathrm{kg \cdot m/s^2 / m}}{\mathrm{kg}}} = \sqrt{\frac{1}{\mathrm{s^2}}} = \frac{1}{\mathrm{s}} = \mathrm{rad/s} \). For \( v_{\mathrm{max}} \): \( \mathrm{m} \times \mathrm{s^{-1}} = \mathrm{m/s} \). The units are consistent.
Three quick tests
- If the mass of the oscillator is doubled, what happens to the angular frequency?
Answer
It decreases by a factor of \( \sqrt{2} \), since \( \omega \propto 1/\sqrt{m} \). - At which position is the kinetic energy of the oscillator maximum?
Answer
At the equilibrium position, where displacement is zero and potential energy is minimum. - Does the period of oscillation depend on the amplitude in an ideal simple harmonic oscillator?
Answer
No, the period is independent of amplitude, a property known as isochronism.
Problem set for PU-103 →
Each derivation in this unit ends with five graded problems with full solutions.
Lecture notes drafted with a local model and checked line by line by the editor; every number in the worked example was recomputed.