Analysis of the simple Pendulum

Now lets turn to the experiment of the day: the pendulum. Our goal will be to combine logic, dimensional analysis, and experiment to determine as did Galileo which factors affect the period of oscillation for a pendulum. We will not be able to really derive the expression, for that we need to know Newtons laws. Instead we will derive an empirical expression which has the correct dimension of time and is consistent with experiments.

First I will define the period of oscillation. It is the time it takes for the pendulum to complete one complete cycle of its repetitive motion. We will call this time T and we wish to derive an expression for it in terms of the physical properties of the simple pendulum. First let me hear which properties of the oscillating simple pendulum we should examine for their possible impact on the period.

• Length of pendulum, of dimension, [L]
• Mass of pendulum, m of dimension [M]
• Amplitude of Oscillation a of dimension [L]
• Acceleration of Gravity g of dimension

In square brackets I have indicated the dimension of the physical property. Now lets try to combine these properties to form an expression of dimension time.

The corresponding expression which keeps track of dimensions is

From this expression we can immediately conclude that q=0 because mass cannot appear on the right when it does not appear on the left. Moreover we can conclude that s=-0.5 since that is the only way we can end up with one power of time on the right. Finally to cancel out the lengths we must have

That is as far as we can get without experiment. We start off verifying the perhaps surprising conclusion that the period is independent of the mass of the pendulum. To accurately measure the period I let the pendulum swing 5 periods and divide the total time by five. With 50 g I get T= 4 s. I increase the mass by a factor of four while keeping everything else the same and indeed find the same period within error.

Now lets ask if the amplitude affects the period. This is something our crude dimensional analysis could not settle. We decrease the amplitude by a factor of four and re-measure the period to get the exact same result. Thus to within our experimental error we have r=0 which implies that p=0.5 and the final expression should be

Now to verify this we have to check that the period in fact depends on the length of the pendulum in the way suggested. Lets calculate what period we should expect according to our formula if we scale the length with a factor f:

The period scales with a factor because enters in the square root. Over here I have a pendulum with a length of 1 m. One quarter of the length of the long pendulum. The scale factor being a quarter I should expect this pendulum to have half the period of the long pendulum that is 2.0 s. Lets check that out. Indeed our empirical expression gives the correct result.

Using the experiments we can also determine the constant in the expression for T:

Thus in conclusion from our experiment and empirical dimensional analysis we suggest that the period of a simple pendulum is given by

When we get to chapter 13 we will be able to derive this expression from Newtons classical equations of motion and show that the constant in fact is .

 
• Walking and the Pendulum