The Perpetual Motion Wheel

Introduction

This applet has been designed to show that a wheel of the design in Figure 1 cannot work as a perpetual motion machine. On inspection of this wheel, one's immediate reaction is that the arrangement of the weights for each nine-degree turn of the wheel gives rise to a turning moment that doesn't appear to alter. This turning moment appears to be always there and thus the wheel should turn by itself, given that the friction of the spindle bearings is made small enough. A consequence of this immediate reaction is that if the wheel was made big enough and the weights heavy enough, this wheel should be able to do some useful work, like turning a generator and generating electricity, etc. At this point, you have a suspicion that the something is not quite right. This is because you ask, "where is the energy coming from to turn a generator?" It could be incorrectly answered that it is coming from the Earth's gravitational field. Incorrect because the Earth's gravitational field is conservative and thus energy cannot be taken from it. So if it is not coming from the Earth's gravitational field, then, where is it coming from? Many inane questions probably will go through your mind, one such being "could this be the exceptional case showing that energy can be derived from the Earth's gravitational field?"

Figure 1 The Marquis of Worcester's Wheel (courtesy of University of Rochester)

The answer to the problem lies in what is happening between the nine-degree intervals as the wheel rotates clockwise from the position shown in Figure 1.

The construction of the wheel

Reference 1, describes the construction of the wheel as shown in Figure 1. It is a wheel of outer radius seven feet, having 40 spokes, seven feet each. It has an inner rim of radius six feet. Thus, the annulus between the outer circle and inner circle is given by inner radius divided by six. As there are 40 spokes, each spoke is inclined at 360/40 = 9 degrees to the next spoke. On each spoke there is a peg attached at the inner radius and one attached to the outer radius, as shown in Figure 2. The purpose of these pegs is for suspending the weights. In this applet, the pegs protrude out from the plane of the wheel so that the weights are freely suspended. There are 40 balls or weights, each one hanging in the centre of a cord two feet long. Each weight is fixed so that it does not move away from the centre of the cord. One end of this cord is attached to the outer radius peg at the top of the centre spoke C. The other end of the cord to the next right hand spoke inner radius ring, as shown in Figure 2. In a like manner each weight bearing cord is attached with every other spoke in succession. Doing this, it will be found, that, at A, the cord will have the position shown outside the inner circle; while at B, C, and D, it will also take the respective positions, as shown on the outside. The result in this case will be, that, all the weights on the side A, C, D, hang from the outer circle, while on the side B, C, D, all the weights are suspended from the lesser or inner circle.

Figure 2 Attachment of cords to inner and outer radius pegs

Analysis

A necessary condition for stability is that the potential energy of the system is a minimum. This applet shows that there is only one potential energy minimum for this system. This occurs at 5.15°. Also, the torque is zero at this angle. The wheel can be rotated by an external force applied to it and it will continue to rotate until friction between the spindle and the spindle bearings, cords on the pegs, swaying of the weights, etc. uses up the external energy initially supplied to the system. Once this energy has been expended, the wheel will come to rest at 5.15° from the initial position shown in Figure 1. An interesting point is that the wheel shown in Figure 1 has a small negative torque and a high potential energy. This would mean that if the wheel was not restrained and in a frictionless state, it would self-rotate anti-clockwise by 3.85° (9° -5.15°) before coming to rest.

The philosophy behind the analysis is to determine the total potential energy of the system by adding the heights h, multiplied by each of the weights mg, both above and below a horizontal line drawn through the centre of the wheel. Also, the turning moment of the system is determined by adding up the distances of each weight along this same horizontal line. Positive distances are to the right and negative distances are to the left of the centre of the wheel. Also, positive moments are clockwise and negative moments are anti-clockwise. These computations are done for each small amount the wheel rotates from the position shown in Figure 1. In the case of this applet, there are 900 increments for the nine-degree turn. Each angle increment being 0.01°. There is no need to analyse outside of the nine-degree region, because after any number of nine-degree rotations from the initial position the wheel resumes the same initial state as shown in Figure 1.

The Applet

There are two modes to this applet. These are the manual mode and the automatic mode. The manual mode allows you to change the angle of rotation and observe the potential energy and torque at your chosen angle. Also there is a graph of potential energy versus torque that allows you to determine the angle at which the torque is zero and the potential energy minimum for the system. You can only increase the angle from zero to nine degrees. If you try to decrease the angle from say 6 degrees to 5 degrees, the applet will reset itself to zero and you will have to click on the screen to review the 5 degree angle. This is due to the way the applet is programmed. A screen capture of the applet is shown in Figure 3.

Figure 3 The applet in manual mode

The automatic mode allows you to see plots of the potential energy and the torque against angle of rotation. Since there are 900 x 0.01° sets of calculations, the plot will take a couple of minutes to complete. This mode also shows the trajectories of the weights as the wheel rotates. Both modes are self-explanatory and there are sufficient screen instructions to permit you to work your way through the applet. The inner radius can be varied, however the wheel will remain scaled to that provided by Reference 1. The length of the string can be varied, but if it is chosen not to be the same as the annulus length in the applet, the results of torque and potential energy cannot be depended upon. This is because the equations used are applicable only for cord lengths that are scaled to Reference 1. In the applet, the string length is taken to be the annulus length. In Reference 1, the cord length is taken to be twice the annulus length. A screen capture of the applet is shown in Figure 4.

Figure 4 The applet in automatic mode

The use of Netscape Communicator 4 to run the applet is not recommended. This is because the applet runs too slow. It is recommended, due to its performance, to use Microsoft Explorer 4. You will need all the screen space available, so it is recommended that you turn off the tool bars. Should you scroll the screen whilst the applet is calculating, the results are not predictable. The screen area should be 800 x 600 pixels.

Acknowledgements

To my colleague, Dr. Terence J. Fairclough, who started this project and provided many constructive comments. His mathematical analysis of this project may be found in "The Great Weighted Wheel", Mathematics Today, Vol.36 No.4, pp107, August 2000.

References

  1. "The Century of Inventions", Edward Somerset, Marquis of Worcester, 1655;


    2. A Verbatim Reprint of the first edition, published in 1663, with an introduction and commentary by Henry Dirks, Esq., Civil Engineer, author of "Perpetuum Mobile, or history of the search after self-motive power;" Contribution to the history of "Electro-Metallurgy;" and "The Life Of Samuel Hartlib;" Also inventor of the "Dircksian Phantasmagoria," producing the optical illusions popularly called "The Ghost !"

  2. A search on the internet for "Bessler's wheel" will provide many references to an earlier wheel than that of the Marquis of Worcester. However, details of this wheel are unknown and so could not be used for analysis.
 


The source code (version 98/12/01) is available according to the GNU Public License.


Tony Townsend, tonyart@ieee.org