Harmonic motion between sinusoid formulas and graphs

Harmonic motion between sinusoid formulas and graphs
Harmonic motion between sinusoid formulas and graphs

Harmonic motion is the latest member of the kinematic physics family; let’s say it is the cousin of circular motion. Do you know why a pendulum swings back and forth at such a constant rate? Or why do springs always return to their rest position after being stretched or compressed? Probably not and that’s okay, that’s what we’re here for.

In any case, the answers lie in the physical phenomena described by harmonic motion and its formulas. Whether you like it or not, harmonic motion is everywhere around us: from the movement of sea waves to the beating of the heart , to classwork and TOLCs. But what exactly does it mean and how can we describe it mathematically ? In the next scrolls, we will dive into the world of harmonic motion and its formulas.

Don’t worry if maths isn’t your strong point, you’re in good company with me, but I’ll do my best to make everything as understandable as possible .So let’s get started and clear things up a bit!

What is harmonic motion and what do the formulas mean

Let’s start with the basics and give a definition. Harmonic motion describes an oscillatory movement that follows a very specific rhythm .

An object in harmonic motion will oscillate back and forth around an equilibrium point in a very particular way, following a trajectory described by a sine or cosine function .

Sine and cosine functions between Cartesian axes.  The functions describe the trajectories of harmonic motion and its formulas.

This means that if you plot the position of the object over time, you will get a nice sinusoidal graph. Fantastic, right? No, but someone has to do it (and unfortunately, that someone is you). For a graph, however, we need formulas and equations that allow us to represent this type of movement.

So let’s look at harmonic motion and its formulas.
Specifically, we consider three of them, which allow us to calculate the positionspeed, and acceleration of an object in harmonic motion at any time.

  1. Hourly law of harmonic motion

The formula describes the position of an object in harmonic motion at a given time. It is given by: x(t) = A cos(ωt + φ) .

  1. Speed ​​in harmonic motion.

The speed of an object in harmonic motion varies over time. The formula for velocity is: v(t) = -ωA sin(ωt + φ) .

  1. Acceleration in harmonic motion

The acceleration in harmonic motion is always directed towards the equilibrium point. The formula for acceleration is given by: a(t) = -ω²A cos(ωt + φ) .

The hourly law

Let’s start again from the hourly law of harmonic motion and understand what the variables contained in its formula express. The latter is expressed by the function: x(t) = A cos(ωt + φ) .


  • x(t) is the position of the object at time t ;
  • A is the amplitude of motion (the maximum distance from equilibrium );
  • ω is the angular velocity ( 2π divided by the period of motion);
  • t is the time ;
  • φ is the initial phase (the position of the object at time t = 0 ).

Let’s make two small clarifications.

Frequency is the number of oscillations that an object performs in one second and is related to the angular velocity ω through the formula ω = 2πf .

It is measured in Hertz ( Hz ) and is the inverse of the period (we can see it immediately below, don’t panic 😉).
So, a higher frequency means more oscillations in a given amount of time.

Speed ​​in harmonic motion

As regards the other two formulas seen above (speed and acceleration) we can say that the speed is:

  • maximum when the object passes through the equilibrium point ;
  • minimum (i.e., zero) when the object reaches the extreme points of its motion.

The time required for a complete oscillation is given by the period .
The latter is given by the duration of a single motion cycle and is measured in seconds ( s ).

In the context of harmonic motion, period is the inverse of frequency ( T = 1/f ) and can be related to angular velocity via the formula ω = 2π/T .
When the period value is high it means that the moving object with harmonic motion takes longer to complete a cycle.

Hand with white watch on wrist while typing numbers on an orange calculator to carry out exercises on harmonic motion and its formulas.  In the background there are notebooks and pens with mathematical equations and numbers.

Acceleration in harmonic motion

In contrast to velocity, acceleration is :

  • maximum at the extreme points of the motion;
  • minimum (i.e. zero) when the object passes through the equilibrium point .

This allows us to understand why the object slows down or speeds up based on the distance from the equilibrium point . To understand this difference between speed and acceleration, we can take help of the analysis of the motion of the pendulum .

In fact, a pendulum swinging back and forth is an example of harmonic motion .
This is because the speed of the pendulum changes during the swing . In fact, it is maximum when it passes through the equilibrium point and minimum (zero) at the extreme points of the oscillation.
The acceleration , however, is always directed towards the equilibrium point , that is, the pendulum is always ” pushed ” towards the center .

Understanding formulas: examples and exercises

To better understand all the formulas and concepts we have just seen, let’s delve deeper into the movement of the pendulum.

  • The amplitude of a pendulum, A , would be the maximum distance the pendulum moves away from the center during the swing.
  • The initial phase, φ, is the initial position of the pendulum.
  • The pendulum’s period, T , would be the time it takes to make one complete swing back and forth.
  • The frequency, f , would be the inverse of the period and represent the number of oscillations that the pendulum makes in one second.
  • Finally, the angular velocity, ω, is 2π/T or 2πf and represents the speed at which the pendulum swings.

Let’s see these concepts on harmonic motion and its formulas better with a multiple-choice exercise.

A pendulum has an oscillation period of 1s. If the length of the pendulum is quadrupled, what value will the period take on? The answer alternatives are:

  • A. 0.1 s;
  • B. 0.25 s;
  • C. 4s;
  • Q. 2s.

To solve this problem, we need to use the period formula of a simple pendulum, which is:
T = 2π √(L/g) .

In this case, we know that the initial period of the pendulum is 1s.

We want to know what the new period will be if the length of the pendulum is quadrupled.

If we call T1 the initial period and L1 the initial length, and T2 the new period and L2 the new length, then we have:

  • T1 = 2π √(L1/g) ;
  • T2 = 2π √(L2/g) .

But since L2 = 4L1 (because the length of the pendulum is quadrupled), we can write:

  • T2 = 2π √(4L1/g) .

You can simplify this expression by taking the square root of 4, getting:

  • T2 = 2 * 2π √(L1/g) .

This tells us that the new period is twice the initial period.
So, if the initial period was 1s , the new period will be 2s

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