Simple Harmonic Motion (SHM) - The Physics of Repetition
Welcome to one of the most fundamental and fascinating topics in A-Level Physics: Simple Harmonic Motion (SHM)!
Don't worry if this sounds intimidating. SHM is just the physics of things that wobble, swing, or vibrate in the simplest possible way. From the ticking of a grandfather clock to the microscopic vibrations of atoms, SHM governs much of the world around us. Mastering this chapter provides the foundation for understanding waves, sound, and even complex electrical circuits.
Let’s break down the rules that govern this simple, repetitive motion!
1. The Defining Condition of Simple Harmonic Motion
SHM is a very specific type of oscillation. Not all oscillations are SHM, but all SHM is an oscillation. To qualify, the motion must obey one critical rule.
The Characteristic Features
There are two key features that define SHM:
- The acceleration (\(a\)) of the object is always directly proportional to its displacement (\(x\)) from the equilibrium position.
- The acceleration (\(a\)) is always directed towards the equilibrium position (the centre point).
This second point is crucial. It means that when the object is moving away from the centre, the acceleration is pulling it back, and vice versa. This requires a Restoring Force.
The Defining Equation (The SHM Condition)
We combine these two features into a single mathematical expression:
$$a = -\omega^2 x$$
Where:
- \(a\) is the acceleration (in \(\text{m s}^{-2}\)).
- \(x\) is the displacement from the equilibrium position (in \(\text{m}\)).
- \(\omega\) (omega) is the Angular Frequency (in \(\text{rad s}^{-1}\)). It's a constant for a given SHM system.
- The Negative sign (\(-\)) indicates that the acceleration is always in the opposite direction to the displacement (i.e., it is a restoring acceleration).
Analogy: Imagine holding a ball attached to a rubber band. If you pull it \(1 \text{ cm}\) away from the centre, the restoring force (and thus acceleration) is small. If you pull it \(10 \text{ cm}\) away, the restoring force is \(10\) times larger, meaning the acceleration is \(10\) times larger! This direct proportionality is the heart of SHM.
If a system obeys \(a \propto -x\), it is undergoing SHM.
2. Key Parameters of SHM
A. Amplitude (\(A\))
The Amplitude (\(A\)) is the maximum displacement from the equilibrium position (the centre). This is the furthest the object gets before the restoring force manages to turn it around.
- Units: metres (\(\text{m}\)).
B. Period (\(T\)) and Frequency (\(f\))
The Period (\(T\)) is the time taken for one complete oscillation (cycle). The Frequency (\(f\)) is the number of oscillations per unit time (\(f = 1/T\)).
C. Angular Frequency (\(\omega\))
Angular frequency is perhaps the most important concept in SHM calculations. It links the period/frequency directly to the acceleration equation.
The relationship is:
$$\omega = \frac{2\pi}{T} = 2\pi f$$
- Units: radians per second (\(\text{rad s}^{-1}\)).
Did you know? We use radians because SHM is fundamentally linked to circular motion. Imagine an object moving in a circle; its shadow projected onto a wall moves exactly in SHM. The \(\omega\) here is the angular speed of the object moving in the circle.
3. The Kinematics of SHM: Displacement, Velocity, and Acceleration
Since the acceleration is constantly changing (it's non-uniform), we cannot use the standard kinematic equations (\(v=u+at\)). Instead, we use equations derived from calculus, which rely on the angular frequency \(\omega\).
3.1 Displacement (\(x\))
The displacement varies sinusoidally (using sine or cosine). Assuming the oscillation starts at maximum amplitude (\(x=A\) when \(t=0\)), the equation is:
$$x = A \cos \omega t$$
If the oscillation started at the equilibrium position (\(x=0\) when \(t=0\)), we would use \(x = A \sin \omega t\).
3.2 Velocity (\(v\))
Velocity is the rate of change of displacement, meaning it is the gradient of the displacement-time graph. Velocity is zero when the displacement is maximum (\(x = \pm A\)), and maximum when the displacement is zero (\(x = 0\)).
The maximum speed is given by:
$$v_{max} = \omega A$$
The speed at any displacement \(x\) is given by:
$$v = \pm \omega \sqrt{A^2 - x^2}$$
Tip for Calculations: When calculating \(v\), make sure \(x\) and \(A\) are in the same units, and \(\omega\) is in \(\text{rad s}^{-1}\).
3.3 Acceleration (\(a\))
Acceleration is the rate of change of velocity, meaning it is the gradient of the velocity-time graph. Maximum acceleration occurs at maximum displacement (where velocity is momentarily zero).
The maximum acceleration is given by:
$$a_{max} = \omega^2 A$$
Note that we drop the negative sign here as we are calculating the magnitude of the maximum acceleration.
The maximum acceleration occurs when the speed is zero (at the limits of the oscillation, \(x=\pm A\)). The acceleration is zero when the speed is maximum (at the equilibrium, \(x=0\)). They are always out of phase!
4. Energy in Simple Harmonic Motion
Since the system is typically assumed to be isolated (no friction/damping), the Total Energy of the oscillator remains constant. Energy simply transforms back and forth between Kinetic Energy (\(E_k\)) and Potential Energy (\(E_p\)).
4.1 Kinetic Energy (\(E_k\))
Kinetic energy depends on the object's speed: \(E_k = \frac{1}{2} m v^2\).
- \(E_k\) is maximum at the equilibrium position (\(x=0\)) because speed is maximum (\(v = v_{max}\)).
- \(E_k\) is zero at the maximum displacement (\(x = \pm A\)) because speed is zero.
4.2 Potential Energy (\(E_p\))
This is the stored energy due to the object's position (e.g., elastic potential energy in a spring). \(E_p\) is proportional to \(x^2\).
- \(E_p\) is maximum at the maximum displacement (\(x = \pm A\)).
- \(E_p\) is zero at the equilibrium position (\(x=0\)).
4.3 Total Energy (\(E_{Total}\))
The total energy is always constant and is equal to the maximum kinetic energy (which occurs at \(x=0\)) or the maximum potential energy (which occurs at \(x=\pm A\)).
$$E_{Total} = E_k + E_p = \text{constant}$$
Since \(E_{Total} = E_{k, max}\) and \(v_{max} = \omega A\), we can substitute this into the kinetic energy equation:
$$E_{Total} = \frac{1}{2} m v_{max}^2$$
$$E_{Total} = \frac{1}{2} m (\omega A)^2$$
$$E_{Total} = \frac{1}{2} m \omega^2 A^2$$
Key Takeaway: The total energy depends on mass (\(m\)), angular frequency (\(\omega\)), and the square of the amplitude (\(A^2\)). If you double the amplitude, the energy increases four-fold!
5. Systems Undergoing SHM: Period Formulas
The period \(T\) for an SHM system depends only on the inherent properties of that system (mass, stiffness, length, gravity), not on the amplitude \(A\).
5.1 The Mass-Spring System
Consider a mass \(m\) oscillating horizontally on a spring with stiffness constant \(k\). The period \(T\) is:
$$T = 2\pi \sqrt{\frac{m}{k}}$$
Mnemonic: Think of a mass oscillating on a spring: Timmy (T) is a small man (m) under a little K (k).
This formula requires the system to obey Hooke's Law, meaning the restoring force \(F = -kx\), which leads directly to the SHM condition \(a = -\omega^2 x\) (since \(a=F/m\)).
5.2 The Simple Pendulum
For a simple pendulum (a point mass at the end of a light string of length \(l\)), the period \(T\) is:
$$T = 2\pi \sqrt{\frac{l}{g}}$$
Where \(l\) is the length of the string and \(g\) is the acceleration due to gravity.
Crucial Restriction: This formula is only valid if the angle of swing is small (usually less than about \(10^\circ\) or \(15^\circ\)). If the amplitude is large, the motion is still oscillatory, but it is no longer truly Simple Harmonic.
You can experimentally determine \(g\) by measuring the period \(T\) for different lengths \(l\) of a simple pendulum. Graphing \(T^2\) against \(l\) yields a straight line where the gradient relates to \(g\).
6. Real Oscillations: The Effects of Damping
In the real world, no oscillation lasts forever. Energy is always lost to the surroundings, usually as heat, due to resistive forces like air resistance or friction. This process is called Damping.
What is Damping?
Damping is the loss of mechanical energy from an oscillating system due to resistive forces, causing the amplitude of the oscillations to gradually decrease over time.
Types of Damping (Qualitative Treatment)
The effect of damping depends on the magnitude of the resistive force:
- Light Damping (Underdamped): The system oscillates for many cycles, but the amplitude slowly decreases exponentially over time. (Example: A swing gradually slowing down in the air.)
- Critical Damping: The system returns to the equilibrium position in the quickest possible time without oscillating at all. This is the ideal for many systems. (Example: Car shock absorbers or door closing mechanisms.)
- Heavy Damping (Overdamped): The system returns slowly to equilibrium without oscillating. It takes longer than critical damping because the resistive force is too strong. (Example: A pendulum swinging in very thick oil.)
Key Takeaway: Damping causes the amplitude to decrease, but it generally has little or no effect on the period \(T\) or frequency \(f\) of the oscillation, provided the damping is light.