Pendulum

Definition of a Pendulum

The word "pendulum" comes from the Latin word pendulus, which means "dangling" or "suspended".

In this article, we will assume that none of the pendulums have any friction.

Parts of a Pendulum

You can probably guess what parts make up a pendulum just from looking at the definition, but let's quickly review them. We need a bob and an attachment point around which the bob can pivot or rotate. Depending on the type of pendulum, we might also need a string to attach the bob to the attachment point. Essentially, a pendulum looks something like the image below.

Fig. 1 - A general pendulum with a bob (blue), a string (green), and a pivot point (C).

Types of Pendulums

There are several types of pendulums, but we will focus on three main types: physical pendulums, simple pendulums, and torsional pendulums.

Physical Pendulums

A physical pendulum is the only type of pendulum that does not necessarily need a string. It is a pendulum that consists of a rigid object hanging off a pivot point, as illustrated below.

The quantities describing this pendulum are the moment of inertia \(I\) of the bob (with units \(\mathrm{kg\,m^2}\)), the distance \(d\) from the pivot to the center of mass of the bob (with units \(\mathrm{m}\)), and the total mass \(m\) of the bob (with units \(\mathrm{kg}\)).

Simple Pendulums

A simple pendulum is a pendulum that consists of a point mass hanging off a string that is attached to a pivot point. See the illustration below.

Fig. 3 - A simple pendulum with a bob (blue) that is a point mass of mass \(m\), a string (green), and a pivot point (C).

The quantities describing this pendulum are the length \(l\) of the string and the mass \(m\) of the bob. The string has negligible mass compared to the mass of the bob, so can be ignored.

Torsional Pendulums

A torsional pendulum is a pendulum that doesn't swing back and forth, but rather rotates back and forth. It consists of a rigid body hanging off a string that is attached to a pivot point as illustrated below. When the bob is turned one way, the twist in the string exerts a torque that pushes the bob toward the equilibrium position. The quantities describing this type of pendulum are the moment of inertia \(I\) of the bob and the torsion constant \(c\) of the string.

The torsion constant describes the amount of torque per radian, so its units are \(\mathrm{N\,m}\).

Pendulum Formulas

Of course, as physicists, we want to look at the behavior of these pendulums in a quantitative way, so we are looking to find some formulas that describe their motion. All these formulas will be derived in the respective specialized StudySmarter articles on these topics but they will simply be given here as an overview. All pendulums are described by differential equations, and these differential equations describe everything physically interesting (i.e. period, angular frequency, and amplitude) regarding the pendulum, as you will see in the examples below.

Physical Pendulums

For small swing angles \(\theta<10^\circ\), the differential equation governing the motion of a physical pendulum is given by

\[\frac{\mathrm{d}^2x}{\mathrm{d}t^2}=-\frac{mgd}{I}x,\]

where \(x\) is the distance from the center of mass of the bob to its equilibrium position (right below the pivot point).

There is no real systematic way to get to the solution of such a differential equation, and guessing solutions is hard without any experience, so you are never expected to do this. Therefore, we will just give the solution to the differential equation to you:

\[x=x_\text{max}\sin\left(\sqrt{\frac{mgd}{I}}t\right).\]

This means that the angular frequency \(\omega_\text{phys}\) of the physical pendulum is given by

\[\omega_\text{phys}=\sqrt{\frac{mgd}{I}},\]

and that the period \(T_\text{phys}\) of the physical pendulum is

\[T_\text{phys}=2\pi\sqrt{\frac{I}{mgd}}.\]

Simple Pendulums

A simple pendulum is just a special case of a physical pendulum in the sense that we can express the distance from the pivot to the center of mass of the bob as \(d=l\) and we can express the moment of inertia of the bob as \(I=ml^2\). This means that for small swing angles, we have

\[x=x_\text{max}\sin\left(\sqrt{\frac{g}{l}}t\right),\]

so the angular frequency \(\omega_\text{simple}\) of the simple pendulum is

\[\omega_\text{simple}=\sqrt{\frac{g}{l}}\]

and the period \(T_\text{simple}\) of the simple pendulum is

\[T_\text{simple}=2\pi\sqrt{\frac{l}{g}}.\]

Torsional Pendulums

A torsional pendulum is different in the sense that we measure the angular displacement \(\theta\) of the bob and not its translational displacement \(x\). For small enough rotation angles (which is everything under a maximum angular displacement of around \(270^\circ\)), we have the following differential equation:

\[\frac{\mathrm{d}^2\theta}{\mathrm{d}t^2}=-\frac{c}{I}\theta.\]

The solution to this differential equation is

\[\theta=\theta_\text{max}\sin\left(\sqrt{\frac{c}{I}}t\right).\]

This means that the angular frequency \(\omega_\text{torsion}\) of the simple harmonic motion of the torsional pendulum is

\[\omega_\text{torsion}=\sqrt{\frac{c}{I}}\]

and that the period \(T_\text{torsion}\) of the torsional pendulum is

\[T=2\pi\sqrt{\frac{I}{c}}.\]

Period of Pendulums

To recap, the period \(T_\text{phys}\) of a physical pendulum is

\[T_\text{phys}=2\pi\sqrt{\frac{I}{mgd}},\]

the period \(T_\text{simple}\) of a simple pendulum is

\[T_\text{simple}=2\pi\sqrt{\frac{l}{g}},\]

and the period \(T_\text{torsion}\) of a torsional pendulum is

\[T_\text{torsion}=2\pi\sqrt{\frac{I}{c}}.\]

Mechanical Energy in Pendulums

As pendulums are oscillating systems exhibiting simple harmonic motion, we can quite simply figure out the total mechanical energy of a pendulum. We do this by looking at a moment in time where the potential energy is zero: this always happens at the equilibrium position. This way, the kinetic energy \(K\) at that moment is equal to the total mechanical energy of the pendulum.