Simple Pendulum

The Motion of Simple Pendulums

For many different topics, it’s often useful to identify a simplified version of a more complex form of motion in order to both understand the system and solve problems. So, how can we analyze the motion of a simple pendulum? Relying on what we know from kinematics alone would be difficult. However, if we can show that simple pendulums exhibit simple harmonic motion, we can apply the same tools we’ve learned from simple harmonic motion.

Let’s start by considering the case that a simple pendulum exhibits simple harmonic motion. This means that the restoring force$F_r$must be proportional to the displacement$d$. Mathematically, we can write this relationship as:

$F_r\propto d$.

A simple pendulum exhibits rotational motion. In our case, this means the displacement is equal to the angle, or$\theta$times the radius. For our simple pendulum, the radius is just the length of the string$L$:

$F_r\propto L\theta$.

However, because the length of the string is constant, we’re only interested in the theta dependence of our pendulum. Therefore, we can drop the term for length from our proportionality, since we aren’t solving an equation just yet:

$F_r\propto \theta$.

Now, we should have enough information to draw a free-body diagram and determine if our restoring force shows this relationship with the angle. The only forces we have acting on our simple pendulum are the force of gravity acting downwards, and the tension in the string keeping our pendulum held in its rotational motion.

A simple pendulum free-body diagram, StudySmarter Originals

We can split up our force of gravity into its x and y-components to get our restoring force. Gravity supplies the restoring force because, again, the tension in the string maintains the rotational motion of our mass.

A pendulum free-body diagram with the individual components of the involved forces sketched out, StudySmarter Originals

Here, the gravity in the x-direction is the restoring force, the force that acts against its direction of motion. Note here that no matter where our mass is in the pendulum’s arc, the tension is perpendicular to the motion. This is why it’s gravity that provides the restoring force and not the tension. Next, we need to solve for$F_{g_x}$in terms of$\theta$:

$\begin{array}{rcl}\sin \left(\theta \right)& =& \frac{F_{g_x}}{F_g}\\ F_{g_x}& =& F_g\sin \left(\theta \right)\\ F_{g_x}& =& -mg\sin \left(\theta \right)\end{array}$.

Here,$m$is the mass and$g$is the acceleration due to gravity. Thus, our restoring force is proportional to$\sin \left(\theta \right)$, not$\theta$! However, we have a powerful mathematical tool at our hands to simplify this relationship for us.

In other words, this means we don’t need to evaluate the$\sin \left(\theta \right)$in a calculation as long as$\theta$is small. Therefore, for small angles, a simple pendulum indeed exhibits simple harmonic motion! Let’s walk through an example using the small-angle approximation to see this tool in action.

Let’s take a look at the most important formulas we need to know for working with simple pendulums.

Simple Pendulum Formulas

Now that we’ve shown the small-angle approximation holds for simple pendulums as long as the angle isn’t too large, we can relate simple pendulums to simple harmonic motion by exploring some of their properties. Let’s first consider our new restoring force assuming the small-angle approximation:

$\begin{array}{rcl}{F}_{g_x}& \approx & -mg\theta \\ F_{g_x}& \approx & -mg\left(\frac{x}{L}\right)\\ F_{g_x}& \approx & -\left(\frac{mg}{L}\right)x\end{array}$.

Recall that simple harmonic motion obeys Hooke’s law, the theory that states a linear proportionality between the displacement and a constant factor of proportionality (such as stiffness) with the force:

$F=-kx$.

Thus, comparing the equation for the restoring force to Hooke’s law, we can see that our force constant$k$is:

$k=\frac{mg}{L}$.

Now, we can apply all of our formulas for simple harmonic motion to a simple pendulum. Let’s start by considering the period, the time it takes to complete a cycle of motion. The period$T$is given by:

$T=2\pi \sqrt{\frac{m}{k}}$.

The period is measured in units of time, most commonly seconds,$\mathrm{s}$. Plugging in our value for the force constant$k$yields the period for a simple pendulum:

$\begin{array}{rcl}T& =& 2\pi \sqrt{\frac{m}{\left(\frac{mg}{L}\right)}}\\ T& =& 2\mathrm{\pi }\sqrt{\frac{\mathrm{L}}{\mathrm{g}}}\end{array}$.

The frequency, or the number of occurrences of some periodic event per unit time, is simply the inverse of the period:

$\begin{array}{rcl}ƒ& =& \frac{1}{T}\\ ƒ& =& \frac{1}{2\mathrm{\pi }}\sqrt{\frac{g}{L}}\end{array}$.

The frequency is measured in units of inverse time, most commonly inverse seconds,$\frac{1}{\mathrm{s}}$, also known as Hertz,$\mathrm{Hz}$. Now, we can use the formula for the period of a pendulum to solve for the angular frequency$\omega$, the frequency of a periodic event. Recall that the formula for the period can be given by:

$T=\frac{2\mathrm{\pi }}{\mathrm{\omega }}$.

Thus, we find the angular frequency to be:

$\begin{array}{rcl}\omega & =& \frac{2\pi }{T}\\ \omega & =& \frac{2\pi }{2\pi \sqrt{\frac{L}{g}}}\\ \omega & =& \sqrt{\frac{g}{L}}\end{array}$.

Angular frequency is usually measured in radians per second,$\frac{\mathrm{rad}}{\mathrm{s}}$. One interesting thing to note is that none of these formulas rely on the displacement angle or the mass of our pendulum. Within the bounds of the small-angle approximation, no matter how large of an amplitude we give our pendulums, the period and frequency remain the same, as long as the pendulum length remains constant!

To summarize, here are the formulas you should be ready to use to tackle simple pendulum problems:

• We can find the period of a simple pendulum using the formula$\begin{array}{rcl}T& =& 2\mathrm{\pi }\sqrt{\frac{\mathrm{L}}{\mathrm{g}}}\end{array}$.
• We find the frequency of a simple pendulum using the inverse of the previous formula,${ƒ}_m=\frac{1}{2\mathrm{\pi }}\sqrt{\frac{g}{L}}$(or more easily, the inverse of the answer you find for the period).
• We can find the angular frequency for a simple pendulum using the formula$\begin{array}{rcl}\omega & =& \sqrt{\frac{g}{L}}\end{array}$.

Simple Pendulum Applications

Applications of simple pendulums are more common in everyday life than you might think! We’ve already recognized that old clocks, particularly “grandfather clocks”, are a classic application of a pendulum in action. We also briefly considered that the back-and-forth motion on a swingset, a wrecking ball, or the throwing of a bowling ball can be approximated as simple pendulums. Each of these motions involves an approximately rigid attachment to a fixed point and a repetitive swing.

What are some other applications? Consider the classic amusement park ride of a swinging pirate ship:

The swinging boat ride exhibits the swinging motion of a simple pendulum, Public Domain

In this ride, the boat displays simple harmonic motion as it swings back and forth between two extreme heights about its rigid metal attachment to a central support beam. Metronomes, the tool musicians use to keep a precise rhythm and time as they play an instrument, also obey the behavior of a simple pendulum, with gravity acting on counterweights for the restoring force.

Examples of Simple Pendulums

Let’s work through an example solving for some variables of periodic motion given the length of a pendulum.

Let’s walk through another example of a simple pendulum, this time examining how this type of motion differs in different surface gravity.

Simple pendulums are just another form of simple harmonic motion, a periodic motion that can be found in many different everyday scenarios. Although we make approximations to analyze these systems, our simplified calculations are still valuable for understanding this type of motion.