Did you know that besides being fun to play on, playground swings can provide a good lesson in physics? That is because the back-and-forth motion a swing makes is a perfect example of the motion of a pendulum. For example, you might notice that you swing faster standing up on your swing than your buddy who is sitting on their swing! You can read why this is so and more in this article about pendulums.
- Definition of a Pendulum
- Parts of a Pendulum
- Types of Pendulums
- Physical Pendulums
- Simple Pendulums
- Torsional Pendulums
- Pendulum Formulas
- Physical Pendulums
- Simple Pendulums
- Torsional Pendulums
- Period of Pendulums
- Mechanical Energy in Pendulums
Definition of a Pendulum
The word "pendulum" comes from the Latin word pendulus, which means "dangling" or "suspended".
A pendulum is a mass suspended from a pivot so that it can swing back and forth freely.
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 \(\mathrm{kg}\)).
If you hang a clothespin on a drying rack, you are essentially creating a physical pendulum! The pivot point is where the clothespin touches the rack, the clothespin itself is completely rigid, and it can swing back and forth freely, so this is indeed a physical pendulum.
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.
An example of a pendulum that is close to being a simple pendulum is the Foucault pendulum. This is a big pendulum found in lots of science museums that showcase the rotation of the Earth around its axis. How this exactly works is outside the scope of this article but definitely interesting to read about!
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 of a string is the proportionality constant between how much torque the string exerts and how twisted it is: in some sense, it describes the stiffness of the string.
The torsion constant describes the amount of torque per radian, so its units are \(\mathrm{N\,m}\).
A tire swing is a good example of a torsional pendulum: when you turn the tire, the twist in the rope makes the tire start turning back once you let go of it, and the tire will then "overshoot" its equilibrium position and turn even further the other way, and so on until all the energy in the system has dissipated and the tyre returns to its rest position.
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).
Why do we need to assume small swing angles, you ask? As you might know or will see in more specialized articles, the actual differential equation is
\[\frac{\mathrm{d}^2\theta}{\mathrm{d}t^2}=-\frac{mgd}{I}\sin\theta,\]
where \(\theta\) is the angle the string makes with the vertical. The sine comes from figuring out the triangles in the free-body diagram of this pendulum. For small angles \(\theta_\text{small}\), we have \(\sin\theta_\text{small}\approx\theta_\text{small}\), so then the differential equation becomes, to a good approximation,
\[\frac{\mathrm{d}^2\sin\theta}{\mathrm{d}t^2}=-\frac{mgd}{I}\sin\theta.\]
Multiplying both sides by the length \(L\) of the string and using \(L\sin\theta=x\), we end up with our differential equation:
\[\frac{\mathrm{d}^2x}{\mathrm{d}t^2}=-\frac{mgd}{I}x.\]
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
\[T_\text{phys}=2\pi\sqrt{\frac{I}{mgd}}.\]
Remember that a simple harmonic motion always looks like \(x=A\sin(\omega t)\), so from any formula about \(x\) we can immediately read off the angular frequency \(\omega\) by looking at what's inside the sine function. In turn, from the angular frequency \(\omega\), we can easily calculate the period \(T\) using the formula relating them: \(T=\frac{2\pi}{\omega}\).
Actually, the general form of the differential equation of simple harmonic motion is
\[\frac{\mathrm{d}^2x}{\mathrm{d}t^2}=-\omega^2x,\]
so this is another way to read off the angular frequency of the motion resulting from the differential equation. In the case of the physical pendulum, we see that
\[\omega^2=\frac{mgd}{I}\]
so we can also conclude via this way that the angular frequency is
\[\omega_\text{phys}=\sqrt{\frac{mgd}{I}}.\]
Let's study an example to see how we can use these equations in practice. Suppose our clothespin that is hanging freely on the drying rack has a moment of inertia of \(3.0\times 10^{-5}\,\mathrm{kg\,m^2}\), a mass of \(9.0\,\mathrm{g}\) and its center of mass is \(3.6\,\mathrm{cm}\) away from the pivot point. Then the period of this clothespin is
\begin{align*}T_\text{clothespin}&=2\pi\sqrt{\frac{I}{mgd}}\\&=2\pi\sqrt{\frac{3.0\times 10^{-5}\,\mathrm{kg\,m}^2}{9.0\times 10^{-3}\,\mathrm{kg}\times 9.8\mathrm{\tfrac{m}{s^2}}\times 3.6\times 10^{-2}\,\mathrm{m}}}\\&=0.61\,\mathrm{s}.\end{align*}
The angular frequency of the clothespin is therefore
\[\omega_\text{clothespin}=\frac{2\pi}{T_\text{clothespin}}=10\,\mathrm{\frac{rad}{s}}.\]
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:
\[\frac{\mathrm{d}^2\theta}{\mathrm{d}t^2}=-\frac{c}{I}\theta.\]
Note that this moment of inertia is now around another axis, namely around the axis that overlaps the string.
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
\[T=2\pi\sqrt{\frac{I}{c}}.\]
The angular frequency of the simple harmonic motion of the torsional pendulum should not be confused with the angular speed of the bob itself! The latter changes with time (and is equal to \(\tfrac{\mathrm{d}\theta}{\mathrm{d}t}\)), while the former is constant and merely tells us in what part of one full cycle we are. The angular frequency of the torsional pendulum is simply \(2\pi/T\), which is indeed constant.
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.
The kinetic energy of the torsional pendulum bob is \(K=\frac{1}{2}I\omega^2\), where \(\omega\) is now the angular speed of the bob itself and not the angular frequency of the motion of the pendulum! Thus,
\[\omega=\frac{\mathrm{d}\theta}{\mathrm{d}t}=\theta_\text{max}\sqrt{\frac{c}{I}}\cos\left(\sqrt{\frac{c}{I}}t\right).\]
At the equilibrium position, the angular speed is at its maximum, so
\[K_\text{eq}=\frac{1}{2}I\left(\theta_\text{max}\sqrt{\frac{c}{I}}\right)^2=\frac{1}{2}c\theta_\text{max}^2.\]
We see once again that all the energy of this pendulum is stored in the string: the moment of inertia of the bob has no effect on the total mechanical energy of the torsional pendulum!
Pendulum - Key takeaways
- A pendulum is a weight suspended from a pivot so that it can swing back and forth under the influence of gravity.
- A physical pendulum is an extended object that hangs from a pivot point that is displaced from the center of mass, about which the object is free to rotate.
- A simple pendulum is a special case of physical pendula that results when the hanging object can be modeled as a point mass a distance l from the pivot point.
- For small amplitudes of motion, the period of a physical pendulum is derived from the application of Newton’s second law in rotational form.
- The period of a pendulum is the time it takes the pendulum to make one full back-and-forth swing.
References
- Fig. 1 - A general pendulum, StudySmarter Originals.
- Fig. 2 - A clothespin as a physical pendulum, StudySmarter Originals.
- Fig. 3 - A simple pendulum, StudySmarter Originals.
- Fig. 4 - A torsional pendulum, StudySmarter Originals.
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