We have designed an apparatus, which we have called a Partial Derivative Machine (PDM), that serves as a mechanical analogue of a thermodynamic system. Using this device, students have a tangible way to wrestle with issues related to partial derivatives and thermodynamics, such as which variables are held fixed, how many variables are independent, and how energy can be added to a system.

The Partial Derivative Machine is an apparatus consisting of a central spring system that can be stretched via four strings extending outward from the center. This central system is on a large piece of particle board which features a pulley on two adjacent corners (Corners C and D), and a knob on all four corners. By tightening the knobs at A and B, one can hold the system in place while adding weights to the hanging strings, allowing one to manipulate the state of the system.

In order to more easily measure the stretching of the system, a measuring tape is placed on the board parallel to each string and flags are added to the strings (Example labeled E). By labeling the axis from corner B to corner D as the “X-axis” and the axis from corner A to corner C as the “Y-axis” the instructor is able to define four quantities for this experiment:

• $x$, the distance between the flags on the X strings
• $y$, the distance between the flags on the Y strings
• $F_x$, the tension in the X oriented strings
• $F_y$, the tension in the Y oriented strings

There are two conditions under which the system can be manipulated. The first method involves tightening the knob on corner C or D to pin a third string, thereby fixing $x$ or $y$, and then increasing the mass on the freely hanging string. For example, pinning the knob at C would fix $y$, then adding weight to the X string would increase $x$, $F_x$, and $F_y$.

Alternatively, one can leave both the X and Y strings free and add weights to one or both. In doing so, placing weight on a string causes the system to stretch in one direction while compressing in the other direction as the system balances the forces. For example, adding weight to the X string would cause $x$ and $F_x$ to increase while $F_y$ stays constant and $y$ decreases.

It is important to note that as weights are added it is not uncommon for the system to shift from its centered position. In order to keep $F_x$ and $F_y$ orthogonal as the system is shifted, students are told to temporarily loosen all four knobs to recenter the system between measurements. If done correctly, this action is only a translation of the system. Thus it does not result in stretching or compressing the system and does not impact any of the measurements students are instructed to make.

Also, the utilization of the 'black box' will allow students to interact with the device first without seeing the internal mechanisms at work.

Corinne's Paradigms Map