Physical Pendulum

Imagine it's a nice day in the middle of the summer, and you are playing on a tire swing with your friend. Your friend gives you a big push on the swing, and you hold on tight to the rope attached to the tire as you swing up into the air. As you are swinging back down, the tire swing also begins to spin! Your grip tightens on the rope to keep you from flying off the swing. Not only are tire swings a fun activity for kids, but they are also a great example of a physical pendulum. In this article, we will define a physical pendulum, describe its motion, and consider the differences between simple and physical pendulums.

Fig. 1 - A tire swing acts as a physical pendulum.

Definition of a Physical Pendulum

In the article, "Simple Pendulum", we define a simple pendulum as an ideal pendulum that has all of the mass concentrated at one point. We can only use the idea of a simple pendulum when an object hanging from a string can be modeled as a point mass, and the mass of the string is negligible. In real life, however, most real pendulums cannot be modeled accurately as point masses on massless strings. We refer to such a real pendulum as a physical pendulum.

At the beginning of this article, we mentioned a tire swing as an example of a physical pendulum. The way a tire swing spins during its motion is evidence of how it cannot be modeled as a simple pendulum. Modeling the tire swing would be rather complicated considering how can rotate not only about the pivot point, but also the axis parallel to the rope. A more simple example of a physical pendulum is a straight rod with a pivot on one end, as shown below. We cannot ignore the mass of the rod as we did with the string in a simple pendulum, so we must consider the location of the center of mass and the moment of inertia when describing its motion.

Fig. 2 - A uniform rod with a pivot at one end is a physical pendulum.

Equation for a Physical Pendulum

Using Newton's second law, we can find an equation of motion for a physical pendulum. Consider a nonuniform object of mass \(m\) that hangs from a pivot so that it can rotate freely about that point. The distance from the pivot point to the center of gravity of the object is \(d.\) When the object experiences an angular displacement of \(\theta,\) the restoring torque on the object is given by:

\[\tau=-mgd\sin\theta,\]

where \(g\) is the acceleration due to gravity, \(g=9.8\,\mathrm{\frac{m}{s^2}}.\) There is a minus sign in this equation because the restoring torque points in the clockwise direction when the displacement is counterclockwise.

Since the restoring torque is proportional to \(\sin\theta\) instead of \(\theta,\) the motion is not simply harmonic. However, for small amplitudes of motion, we can approximate that \(\sin\theta\approx\theta.\) In this approximation, the restoring force is:

\[\tau=-mgd\sin\theta\approx-mgd\theta,\]

and the motion is approximately simply harmonic.

Now, we substitute the restoring force into Newton's second law in the rotational form to get an equation of motion for a physical pendulum:

\[\begin{align*}\sum\tau&=I\alpha\\[8pt]-mgd\theta&=I\frac{\mathrm{d}^2\theta}{\mathrm{d}t^2}\\[8pt]\frac{\mathrm{d}^2\theta}{\mathrm{d}t^2}&=-\frac{mgd}{I}\theta.\end{align*}\]

In this equation, \(I\) is the moment of inertia and \(\alpha\) is the angular acceleration. We recognize the term on the right hand side of the equation, \(\frac{mgd}{I},\) as the square of the angular frequency, \(\omega,\) so that \(\omega=\sqrt{\frac{mgd}{I}}.\) In terms of the angular frequency, the equation of motion is given by:

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

Period of a Physical Pendulum

The period of a physical pendulum can be derived from the angular frequency found in the section above. The period in terms of the angular frequency is given by:

\[\begin{align*}T&=\frac{1}{f}\\[8pt]&=\frac{2\pi}{\omega}\\[8pt]&=\frac{2\pi}{\sqrt{\frac{mgd}{I}}}\\[8pt]&=2\pi\sqrt{\frac{I}{mgd}}.\end{align*}\]

Moment of Inertia of a Physical Pendulum

The moment of inertia of an object with a complex shape can be calculated by first finding its period of motion. First, we determine the center of gravity of the object. Then, the object is allowed to pivot about a certain point that is a distance \(d\) away from the center of gravity. We finally measure the period of the oscillation. Solving for the moment of inertia in the equation found in the previous section gives us:

\[\begin{align*}T&=2\pi\sqrt{\frac{I}{mgd}}\\[8pt]\frac{T}{2\pi}&=\sqrt{\frac{I}{mgd}}\\[8pt]\left(\frac{T}{2\pi}\right)^2&=\frac{I}{mgd}\\[8pt]I&=mgd\left(\frac{T}{2\pi}\right)^2.\end{align*}\]

We then substitute in the measured values for the mass, distance from the center of gravity, and period of oscillation to calculate the moment of inertia.

Simple Pendulum vs. Physical Pendulum

A simple pendulum is a special case of physical pendulums that can be modeled as a point mass on a massless string. In the case of a simple pendulum, the period is only dependent on the length of the string \(l\) and the acceleration due to gravity:

\[T=2\pi\sqrt{\frac{l}{g}}.\]

Though the calculations for a simple pendulum are less complex than those for a physical pendulum, most real-life pendulums cannot be modeled as a simple pendulum and must be treated as a physical pendulum.

Physical Pendulum Examples

Let's do a couple of examples with physical pendulums for practice!