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Adding detail to the model of the pendulum.

Solution

We consider first the usual Simple Model of a Pendulum

And then a slightly more detailed model

With two variations.

The simple model has the virtue that it is extremely simple to calculate themoment of inertia, I, of the pendulum about the axis of rotation. We assume a massless stick of length L and a point mass m at the end. The moment of inertia is simply

\begin{large} \[ I = mL^{2} \]\end{large}

If we pull the pendulum away from its vertical equilibrium position by an angle θ, then the restoring force F_res is given by

\begin{large} \[ F_{res} = m g sin(\theta) \]\end{large}

And the natural frequency, as noted in the vocabulary entry on pendulum, is given by

\begin{large} \[ \omega = \sqrt{\frac{g}{L}} \]\end{large}

Now consider the second diagram above. Now, instead pf a point particle, the mass of the pendulum ihas real extent. It is a disc of radius r. We can calculate the real moment of inertia by using the parallel axis theorem. According to that, we can calculate the moment of inertia about a point other than the center of mass (about an axis parallel to one running through the center of mass) by simply adding md² to it, where d is the distance between the center of mass and the axis of rotation. For a uniform disc, rotating about its center, the moment of inertia is

\begin{large} \[ I = \frac{1}{2}mr^{2} \]\end{large}

In this case d is the length of the pendulum L, so the moment of inertia is

\begin{large} \[ I = \frac{1}{2}mr^{2} + mL^{2} \]\end{large}

and the angular frequency is given by

\begin{large} \[ \omega = \sqrt{\frac{mLg}{I}} = \sqrt{\frac{mLg}{\frac{1}{2}mr^{2} + mL^{2}}} \]\end{large}

After some algebra, this reduces to:

\begin{large} \[ \omega = \sqrt{\frac{g}{L + \frac{r^{2}}{2L}}} \]\end{large}

So taking into account the physical size of the mass reduces the frequency. Why is this? The answer is that now we must consider not only the moment associated with moving the center of mass from side to side, we must also consider the moment of inertia associated with rotating that mass as well. In the simple model we simply ignored this, since you can't rotate a point particle. A real physical pendulum would behave the same way if it were mounted to the end of the stick of length L by a perfectly frictionless axle, leaving it free to rotate. Since nothing exerts a torque on the mass that requires it to rotate (only to move its center of mass), , the mass does not have to rotate, and the frequency will be the same as for a point mass. Notice that, in the limit as r approaches zero, the physical pendulum result tends towards the simple pendulum result. For a radius significantly smaller than L, the physical pendulum will behave like a simple pendulum.

What this means is that, if you have two pendulums of identical dimensions, but with one having the mass rigidly mounted and the other mounted on a freely rotatring bearing, the latter will have a higher rate of oscillation. That's a surprising result.

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