Procedure

Initial Setup

Mass Positioning

To begin we must have at least an idea of where to position the small mass. The large mass must also be placed such that it is in a similar position to the small mass at the opposite end such that the air resistance is similar from velocity. Ideally the masses should also have an identical cross sectional area for the same reason but the uncertainty introduced in the experiment due to this is hopefully small enough to not affect results.

The small mass and large mass positions can be placed using trial and error or by using the calculation outlined in the Theory section.

By positioning the large mass such that it touches a knife edge pivot, the small mass can be set somewhere around 2-4cm from the opposite knife edge. This seems to be a good positioning for the large mass.

A decision of how the position of the small mass is going to be measured must be made. The value of the measurement is relevant only to the other small mass position measurements but it's precision will affect the fit precision and ultimately the final result. By measuring from the bottom of the small mass, in the small mass up (SMU) orientation, to top flat portion of the closest knife edge, a digital caliper can be used as a precise measuring device. It is important to note, however, that the small mass can slide and pivot slightly even when tightened very well so it is very easy to over estimate the measurements precision.

Wall Mounted Bracket

The wall bracket, which supports the pendulum as it swings, must be mounted firmly on a solid structure. It must be level in both the x and y planes (taking z as up and x as the direction of desired oscillation). If the bracket is not level, the pendulum can "walk" very minutely which can prolong y oscillations which introduce error into the measurements. These oscillations can be practically undetectable to the naked eye.

Period Measurement

A photo-gate is a good way to measure periods of the pendulum as it swings. Considerations for the photo-gate include a rigid base for it to stand on, what part of the pendulum is breaking the beam, and what software is interpreting the photo-gate signal.

A solid base is necessary so that movement around the apparatus and natural vibrations of the lab do not cause unwanted motion of the gate.

Something other than the bar, which is approximately 2.5 cm thick, should break the photo-gate beam. A thin steel "needle" was fashioned to be attached to each end of the pendulum and can be seen in the image. This allows for much small oscillations meaning that the small angle approximation correction can be much smaller. Another point to consider is the distance from the pivot to the photo-gate. The angle of oscillation should be measured so that the data can be corrected for the small angle approximation. The distance the pendulum should be displaced can be calculated by:

$x=y\tan \theta$

Where $y$ is the distance from the pivot to the photo-gate, $\theta$ is the desired angle of oscillation, and $x$ is the horizontal distance the pendulum should be displaced at the height of the photo-gate.

The software easily available at CNU for interpreting the photo-gates is DataStudio. DataStudio has a photo-gate pendulum setup but use caution as the software interpolates the period based on your reported pendulum thickness. This setting can be used as a rough estimate of the period but using the state setup and taking time from either the leading or trailing edge of the data spikes is preferred.

Starting Oscillations

Doe to the rigid and massing nature of the Kater's pendulum, the system is very susceptible to oscillations in the y direction. These oscillations can cause large inaccuracies in the measurement of the period so great care must be taken to ensure they have dampened completely before taking data.

The two ways to combat this problem are to attempt to minimize these oscillations when starting the pendulum swinging and to wait a long enough period of time that they y oscillations dampen completely.

To start the pendulum with as little y oscillation as possible requires that no contact be made with the pendulum low enough the cause oscillations. A release by hand simply can not be done consistently without oscillation. Even attempting to use a rigid instrument as a release introduces y oscillations.

A method that produces relatively consistent results is as follows:

1. Align the pendulum's knife edge in the x direction on the bracket. It is helpful to mark a position on the bracket so that the alignment can be the same for each run. This mark should make sure that the pendulum does not come into contact with the bracket when it swings.
2. Align the pendulum's knife edge in the y direction by visually inspecting that the pendulum is centered in the slot of the bracket.
3. Check that the pendulums swing is parallel to the wall. This can be done by swinging the pendulum and seeing that the bottom of the pendulum is swinging parallel to a meter stick set parallel to the wall.
4. Stop all oscillations as best as possible.
5. Gently begin the pendulum swinging by putting slight pressure on the weight at the top of the pendulum. It may take a number of small applications of force to reach the desired swing.
6. Begin recording data using the DataStudio photo-gate and pendulum setup. Do this for at least 2 minutes and watch to see if the period changes over the 2 minutes. If the period "walks" it is an indicator that y oscillations may be still dampening. Take another 2 minutes of data and watch again. Once the period appears steady, switch to the DataStudio photo-gate state measurement and begin taking data. The figure shows what can happen to measurements if the oscillations in the y-axis are not allowed to dampen fully.
7. The moment you begin taking data, measure the swing by seeing the maximum and minimum points of the needle on the meter stick parallel to the wall. Half this distance is the $x$ value from the calculation in the Period Measurement section. Remember to adjust for the thickness of the needle if you used to outside edges to take the measurement. While this measurement will only be accurate to ~2-3mm it is plenty of precision to make an accurate small angle approximation correction to the final data.

Taking Data

The method of starting oscillations described in the section above should allow data to be taken in a given orientation. As described in the Theory section, a number of period measurements is required to even be in the ball park of the accuracy required. A large number of measurements should be taken in each orientation and each orientation should be measured a number of times.

As outlined above, recording the oscillation angle as data taking begins allows the small angle correction to be made. Similarly, retaking this measurement at the end will indicate the amount of degradation which has occurred during the run. This information can be used to interpolate the angle of oscillation midway through the run or to calculate a small angle correction for each data point.

Interpreting the Data

Keep in mind that the times recorded by the state setup of DataStudio are times of the leading and trailing edge of a voltage pulse which indicates when the "needle" broke and the stopped breaking the photo-gate beam. Be careful to ensure that the period used is extracted correctly from the raw data.

By taking data with various small mass positions which bracket the estimated equal period position in both the SMU and SMD orientations will allow two plots to be made. One plot is of the change of the SMU orientation's period as the mass moves, and the other will be of the SMD orientation. By plotting these data points on the same graph and plotting a best fit line for each data set, a crossing point can be found. This crossing point is the value where the periods should be precisely the same and should be used to calculate $g$.