Suspending the ioLab remote from a length of copper wire, with both ends securely attached, creates a torsional pendulum. Setting off small axial oscillations and monitoring the gyroscope readings should make it possible to measure the frequency of oscillations quite precisely. It will depend on the effective torsional constant of the wire, \( k \), and on the moment of inertia \( I_0 \) of the object undergoing the torsional oscillations, the ioLab itself, as \( \omega = \sqrt{k/I_0} \). You can vary both the length of the wire, and the moment of inertia by adding a pair of additional weights (Al or steel washers from the kit of Lab#6, loonies/toonies) in pairs, held to the outside of the ioLab body by rubber bands. The additional moment of inertia due to two equal point masses \( m \) at a distance \( r \) from the rotation axis is \( I = 2mr^2 \) and you can refine this by doing an integration over the shape of the washer, as discussed in class.

Some notes on the experiment:

Repeat for three different wire lengths (keep all lengths ≥20cm, to help reduce the off-axis oscillations), with no washers, Al washers (mass 2.11g, thickness 1mm) only, steel washers (mass 6.76g, thickness 1mm) only, and Al and steel washers together.

Acquire 10-15 s of data, keeping the ioLab oscillations small enough (see the video above) for damping due to air resistance to not be a factor.

Analyze the data to obtain the torsional constant of this copper wire, and the moment of inertia of the ioLab remote. Report the latter as its effective radius of gyration and compare that to the known dimensions of the ioLab remote.

The torque required to twist a uniform cylinder of length \( L \), radius \( R \), and shear modulus \( S \), by angle \( \theta \) is
\( \tau = S J \theta/L \) where \( J = \frac{1}{2} \pi R^4 \). The product \( S J \) is called torsional rigidity. Look up known shear modulus of copper and see if your data is consistent with it, within the experimental error.

You have been provided with 28 AWG gauge wire (nominal diameter 0.321 mm). If you prefer to measure the wire thickness directly, wind a few turns on a cylindrical object, placing the wires tightly together, and measure to total length with a ruler, however this does not take into account the thickness of lacquer insulation, so you may need to use fine sandpaper to gently scrape off the insulation first without losing any copper thickness. The lacquer itself is quite flexible, so it should have minimal effect on the effective torsional rigidity.

One of the images above shows how to dampen any off-axis vibrations and get a very smooth spin of the ioLab, by starting the torsional oscillations, then gently pressing a pen against the wire to dampen the unwanted swings, and slowly backing it off.

To develop your extrema script beforehand, feel free to use this sample data set.

Optional: determine the location of the accelerometer on the board of the ioLab. When spinning about the y-axis, the x-component of acceleration is the centripetal acceleration as measured at the location of the accelerometer. Since \( a_c = \omega^2 r \), one can determine \( r \) as the fit parameter. You can then open up the ioLab, identify the accelerometer chip, and use a ruler to measure its distance from the midline of the printed circuit board. Include images in your lab report. (Credit for this idea goes to Mats Selen, the inventor of the ioLab device.)