Oscillations are all around us, from the macroscopic world of pendulums and the vibration of strings to the microscopic world of the motion of electrons in atoms and electromagnetic radiation.
Motion like this that undergoes a predictable repeating pattern is known as periodic motion or oscillatory motion, and learning about the quantities that allow you to describe any type of oscillatory motion is a key step in learning the physics of these systems.
One particular type of periodic motion that’s easy to describe mathematically is simple harmonic motion, but once you’ve understood the key concepts, it’s easy to generalize to more complex systems.
Periodic motion, or simply repeated motion, is defined by three key quantities: amplitude, period and frequency. The amplitude A of any periodic motion is the maximum displacement from the equilibrium position (which you can think of as the “rest” position, such as the stationary position of a string or the lowest point on a pendulum’s path).
The period T of any oscillatory motion is the time it takes for the object to complete one “cycle” of motion. For example, a pendulum on a clock might complete one complete cycle every two seconds, and so it would have T = 2 s.
The frequency f is the inverse of the period, or in other words, the number of cycles completed per second (or unit of time, t). For the pendulum on a clock, it completes half a cycle per second, and so it has f = 0.5 Hz, where 1 hertz (Hz) means one oscillation per second.
Simple Harmonic Motion (SHM)
Simple harmonic motion (SHM) is a special case of periodic motion, where the only force is a restorative force and the motion is a simple oscillation. One of the basic properties of SHM is that the restoring force is directly proportional to the displacement from the equilibrium position.
Returning to the example of a string being plucked, the farther you pull it from the resting position, the faster it will move back towards it. The other major property of simple harmonic motion is that the amplitude is independent of the frequency and period of the motion.
The simplest case of simple harmonic motion is when the oscillatory motion is only in one direction (i.e., movement back and forth), but you can model other types of motion (e.g., circular motion) as a combination of multiple cases of simple harmonic motion in different directions, too.
Some examples of simple harmonic motion include a mass on a spring bobbing up and down as a result of an extension or compression of the spring, a small angle pendulum rocking backwards and forwards under the influence of gravity and even two-dimensional examples of circular motion like a child riding around on a carousel or merry-go-round.
Equations of Motion for Simple Harmonic Oscillators
As pointed out in the previous section, there is an interesting relationship between uniform circular motion and simple harmonic motion. Imagine a point on a circle rotating at a constant rate on a fixed axis, and that you were tracking the x-coordinate of this point throughout its circular motion.
The equations that describe the x position, x velocity and x acceleration of this point describe the motion of a simple harmonic oscillator. Using x(t) for position as a function of time, v(t) for velocity as a function of time and a(t) for acceleration as a function of time, the equations are:
Where ω is the angular frequency (related to ordinary frequency by ω = 2πf) in units of radians per second, and we use time t like in most equations. As stated in the first section, A is the amplitude of the motion.
From these definitions, you can characterize simple harmonic motion and oscillatory motion in general. For example, you can see from the sine function in both the position and acceleration equations that these two vary together, and so maximum acceleration occurs at the maximum displacement. The velocity equation depends on cosine, which takes its maximum (absolute) value exactly half way between the maximum acceleration (or displacement) in the x or -x direction, or in other words, at the equilibrium position.
Mass on a Spring
Hooke’s law describes a form of simple harmonic motion for a spring and states that the restoring force for the spring is proportional to the displacement from equilibrium (∆x, i.e., change in x), and has a “constant of proportionality” called the spring constant, k. In symbols, the equation states:
The negative sign here tells you that the force is a restoring force, which acts in the opposite direction to the displacement and is measured in the SI unit of force, the newton (N).
For a mass m on a spring, the maximum displacement (amplitude) is again called A, and ω is defined as:
This equation can be used with the position equation for simple harmonic motion (to find the position of the mass at any time), and then substituted into the place of the ∆x in Hooke’s law to determine the size of the restoring force at any time t. The complete relation for the restoring force would be:
Small Angle Pendulum
For a small angle pendulum, the restoring force is proportional to the maximum angular displacement (i.e., the change from the equilibrium position expressed as an angle). Here the amplitude A is the maximum angle of the pendulum and ω is defined as:
Where g = 9.81 m/s2 and L is the length of the pendulum. Again, this can be substituted into the equations of motion for simple harmonic motion, except you should note that x in this case, would refer to the angular displacement rather than the linear displacement in the x-direction. This is sometimes indicated by using the symbol theta (θ) in place of the x in this case.
In many cases in physics, complications like friction are neglected to make the calculations simpler in situations when they would likely be negligible anyway. There are expressions you can use if you need to calculate a case where friction becomes important, but the key point to remember is that with friction accounted for, oscillations become “damped,” meaning they decrease in amplitude with each oscillation. However, the period and frequency of the oscillation remain unchanged even in the presence of friction.
Forced Oscillations and Resonance
Resonance is basically the opposite of a damped oscillation. All objects have a natural frequency, which they “like” to oscillate at, and if the oscillation is forced or driven at this frequency (by a periodic force), the amplitude of the motion will increase. The frequency at which resonance occurs is called the resonant frequency, and in general, all objects have their own resonant frequency, which depends on their physical characteristics.
As with damping, calculating motion under these circumstances gets more complicated, but it is possible if you’re tackling a problem that requires it. However, understanding the key aspects of how the object behaves in these situations is enough for most purposes, especially if this is the first time you’re learning about the physics of oscillations!