Part II:  Pressure Fluctuation Machine

For this part, we observed the properties of two “gases” of motorized molecules in a mechanical interaction.  We used the previous contained area, but we removed the Force Sensors and added a Motion Sensor to record the position of the moveable piston.  For each system, we observed the motion for 30 minutes.  The most probable position of the wall was then calculated for each system.  This experiment was modeled after a 2 previous studies done by Jeffrey Prentis in 1999 and 2000.

3 Motorized Molecules Interacting with 3 Motorized Molecules:

In the first system we observed, three motorized molecules interacted with a system of three motorized molecules.  By using the equations of motion we found in the earlier experiment, a plot was made of the expected force exerted on the wall at varying positions.  The theoretical average position of the wall was then calculated.

Fig. 19:  Approximate force each gas exerted on the wall at different positions.

Eq. 1:  Equations of motion for the forces in this system. <FL> is the equation of motion for the 3-ball system on the left and <FR> is the equation of motion for the 3-ball system on the right.

    According to the forces each system exerted on the wall, an equilibrium point should have been reached at 35.0 cm.  At this position, the force from the “gas” in the left system is equal to the force from the “gas” in the right system.  Because at this point the system is at equilibrium, the probability of encountering the piston should be high.

    After recording data for 30 minutes, we collected the following results:

Fig 20:  Movement of Piston during a period of 30 minutes.

The system was initiated when the piston was in the center of the enclosed area.  The piston oscillated around its central point varying by about 10 cm to either side.  Once we had compiled the data, we used the following equation to find the average position of the wall:

                    (Prentis 247)

Our values were as follows:

Because the observed systems had an equal number of molecules and the same total energy, we expected the average position of the wall to be in the middle of the enclosed area.  The experimental value we found was within 15% error of the expected value.  This difference can be attributed to sources of error we encountered in the experiment. 

    We also analyzed our data by creating a histogram of position vs. probability.  Using the equations for probability discussed in the theory, we found the following:

Fig. 21:  Probability of the location of the wall.

Eq. 2:  Theoretical probabilities for the system.

Our experimental data differed from the expected values.  The width of our histogram was much smaller than the theoretical.  This error is due to the experimental setup.  Due to external forces, the moveable wall was not able to move freely to any available state.  The friction between the piston and the walls resulted in a loss of energy each time the position of the piston changed.  The irregular failure of the balls was also a problem.  Two of our balls periodically gave up and had to be pushed back into motion. 

To add validation to our results, we compared our theoretical expectations and results to data received from a computer simulation of the same system.  The simulation was written by Dr. Wolfgang Christian of Davidson College. 

 Squiggle Balls Simulation: 

By placing three balls in each side, we found the following results.

Fig. 22:  Simulated probability of wall.

Our equation of probability models the motion of the piston to within a small error. This small error is caused by the moveable wall acting as a ball instead of a moving object.  Also, energy is conserved in this system and the moveable wall was not subject to a force of friction.  Because these experimental errors are eliminated, there is a greater probability the piston will be found at the extremes of the enclosure.  

4 Motorized Molecules Interacting with 2 Motorized Molecules:

The next system we observed was one of four motorized molecules interacting with two motorized molecules.  We choose these numbers based on the availability of functional balls.  By using the equations of motion we found in the earlier experiment, a plot was made of the expected force exerted on the wall at varying positions. From these equations, the theoretical average position of the wall was then calculated.

Fig. 23:  Approximate force each gas exerted on the wall at different positions.

Eq. 3:  Equations of motion for the forces in this system. <FL> is the equation of motion for the 4-ball system on the left and <FR> is the equation of motion for the 2-ball system on the right.

According to the force equations, an equilibrium point should have been reached at 47.61 cm.  This is our expected value for the average position.

    The following data was taken of a 30 minute run:

Fig 24:  Movement of Piston during a period of 30 minutes.

The system was once again initiated in the center of the enclosed area.  We used the following equation to find the average position of the wall:

                    (Prentis 247)

Our values were as follows:

Our experimental value was within 10% error of the theoretical value.  This error could be due to faulty squiggle balls.  Two of our balls periodically became stuck and needed to be pushed.  Our sail was also leaning backwards.  This may have effected the measurement of the motion sensor.

Our data was also analyzed by looking at a histogram of the position vs. the probability.  Using the equations for probability discussed in the theory, we found the following:

Fig. 25:  Probability of the location of the wall.

Eq. 4:  Theoretical probabilities for the system.

Our experimental data differed from the expected values.  The width of our histogram is much smaller than the theoretical.  This error is experimental.  Due to external forces, the moveable wall was not able to move freely to any available state.  The friction between the piston and the walls resulted in a loss of energy each time the position of the piston changed.  The irregular failure of the balls was also a problem.  Two of our balls periodically gave up and had to be pushed back into motion.

To add validation to our results, we compared our theoretical expectations and results to data received from a computer simulation of the same system.  The simulation was written by Dr. Wolfgang Christian of Davidson College. 

 Squiggle Balls Simulation

By placing four balls on one side and two on the other, we found the following results.

Fig. 26:  Simulated probability of wall.

Our equation of probability models the motion of the piston to within a small error. This small error is caused because the moveable wall acts as a ball instead of a moving object.  Also, energy is conserved in this system and the moveable wall was not subject to a force of friction.  Because these experimental errors are eliminated, there is a greater probability the piston will be found at the extremes of the enclosure.