How To Calculate The Period Of Motion In Physics

The natural world is full of examples of periodic motion, from the orbits of planets around the sun to our own heartbeats.

All of these oscillations involve the completion of a cycle, whether it's the return of an orbiting body to its starting point, the return of a vibrating spring to its equilibrium point or the expansion and contraction of a heartbeat. The time it takes for an oscillating system to complete a cycle is known as its period.

The period of a system is a measure of time, and in physics, it's usually denoted by the capital letter T. Period is measured in time units appropriate for that system, but seconds are the most common. The second is a unit of time originally based on the rotation of the Earth on its axis and on its orbit around the sun, although the modern definition is based on vibrations of the cesium-133 atom rather than on any astronomical phenomenon.

The periods of some systems are intuitive, such as the rotation of the Earth, which is a day, or (by definition) 86,400 seconds. You can calculate the periods of some other systems, such as an oscillating spring, by using characteristics of the system, such as mass and its spring constant.

When it comes to vibrations of light, things get a little more complicated, because photons move transversely through space while they vibrate, so wavelength is a more useful quantity than period.

Period is the Reciprocal of Frequency

The period is the time it takes for an oscillating system to complete a cycle, whereas the **frequency (****f****)** is the number of cycles the system can complete in a given time period. For example, the Earth rotates once each day, so the period is 1 day, and the frequency is also 1 cycle per day. If you set the time standard to years, the period is 1/365 years while the frequency is 365 cycles per year. Period and frequency are reciprocal quantities with an inverse relationship:

\(T = \frac{1}{f}\)

In calculations involving atomic and electromagnetic phenomena, frequency in physics is usually measured in cycles per second, also known as Hertz (Hz), s^-1 or 1/sec. When considering rotating bodies in the macroscopic world, revolutions per minute (rpm) is also a common unit. Period can be measured in seconds, minutes or whatever time period is appropriate.

Period of a Simple Harmonic Oscillator

The most basic type of periodic motion is that of a simple harmonic oscillator, which is defined as one which always experiences an acceleration proportional to its distance from the equilibrium position and directed toward the equilibrium position; this results in simple harmonic motion. In the absence of frictional forces, both a pendulum and a mass attached to a spring can be simple harmonic oscillators.

It's possible to compare the oscillations of a mass on a spring or a pendulum to the motion of a body orbiting with uniform motion in a circular trajectory with radius r. If the angular velocity of the body moving in a circle is ω, its angular displacement (θ) from its starting point at any time t is θ = ωt, and the x and y components of its position are x = r cos(ωt) and y = r sin(ωt).

Many oscillators move only in one dimension, and if they move horizontally, they are moving in the x direction. If the amplitude, which is the farthest it moves from its equilibrium position, is A, then the position at any time t is x = A cos(ωt). Here ω is known as the angular frequency (measured in radians), and it's related to the frequency of oscillation (f) by the equation ω = 2πf. Because f = 1/T, you can write the period T of oscillation like this:

\(T = \frac{2π}{\omega}\)

Simple harmonic motion is typically represented graphically with a sinusoidal function like a sine or cosine function. This encodes a lot of information in what is called the wave function. In more advanced physics the equation of motion and harmonic motion for an object will likely take the form of partial differential equations (with many complex derivatives), but sine and cosine functions are often still solutions to these equations.

In many of these physical systems, the number of oscillations in a certain period of time, the amplitude of the motion, and the objects in question can help to describe the kinetic energy and potential energy of a system. This can apply to springs, electromagnetic radiation, sound waves, and so much more!

Springs and Pendulums: Period Equations

According to Hooke's Law, a mass on a spring is subject to a restoring force F = −kx, where the constant k is a characteristic of the spring known as the spring constant and x is the displacement. The minus sign indicates the force is always directed opposite the direction of displacement. According to Newton's second law, this force is also equal to the mass of the body (m) times its acceleration (a), so ma = −kx.

For an object oscillating with angular frequency ω, its acceleration is equal to −Aω² cos ωt or, simplified, −_ω__²x_. Now you can write m( −_ω²x_) = −kx, eliminate x and get ω = √(k/m). The period of oscillation for a mass on a spring is then:

\(T = 2 \pi \sqrt{\frac{m}{k}}\)

You can apply similar considerations to a simple pendulum, which is one on which all the mass is centered on the end of a string. If the length of the pendulum string is L, the period equation in physics for a small angle pendulum (i.e. one in which the maximum angular displacement from the equilibrium position is small), which turns out to be independent of the mass m, is

\(T = 2\pi \sqrt{\frac{L}{g}}\)

where g is the acceleration due to gravity. It is also important to note that one complete oscillation from a pendulum occurs when the mass returns to its initial position. The maximum displacement (on the other side) represents half of a complete oscillation.

Pendulums and springs can also have other forces and influences acting on them. Damping (which occurs from friction and resistance in the real world) reduces the frequency of oscillation, and it can also be intentionally applied to a system.

The Period and Wavelength of a Wave

Like a simple oscillator, a wave has an equilibrium point and a maximum amplitude on either side of the equilibrium point. However, because the wave is traveling through a medium or through space, the oscillation is stretched out along the direction of motion. A wavelength is defined as the transverse distance between any two identical points in the oscillation cycle, usually the points of maximum amplitude on one side of the equilibrium position.

The period of a wave is the time it takes for one complete wavelength to pass a reference point, whereas the frequency of a wave is the number of wavelengths that pass the reference point in a given time period. When the time period is one second, frequency can be expressed in cycles per second (Hertz) and period is expressed in seconds.

The period of the wave depends on how fast it's moving and on its wavelength (λ). The wave moves a distance of one wavelength in a time of one period, so the wave speed formula is v = λ/T, where v is the velocity. Reorganizing to express period in terms of the other quantities, you get:

\(T = \frac{\lambda}{v}\)

For example, if the waves on a lake are separated by 10 feet and are moving 5 feet per second, the period of each wave is 10/5 = 2 seconds.

Using the Wave Speed Formula

All electromagnetic radiation, of which visible light is one type, travels with a constant speed, denoted by the letter c, through a vacuum. You can write the wave speed formula using this value, and doing as physicists usually do, exchanging the period of the wave for its frequency. The formula becomes:

\(c = \frac{\lambda}{T} = f × \lambda\)

Since c is a constant, this equation allows you to calculate the wavelength of the light if you know its frequency and vice versa. Frequency is always expressed in Hertz, and because light has an extremely small wavelength, physicists measure it in angstroms (Å), where one angstrom is 10^-10 meters.