**THE GREAT WEIGHTED WHEEL**

A Great Weighted Wheel similar to the one pictured above was built at the Tower of London during the 17th. century by Edward Somerset, The Second Marquis of Worcester. It is one of the many early attempts to achieve perpetual motion.

This great wooden wheel was designed to achieve a permanent 'out-of-balance' condition at the spindle. The wheel was weighted with 40 cannon balls, each weighing about 50 pounds (approximately 23 kg), the weights being suspended by pairs of one-foot cords attached to the outer and inner rims. The outer rim is 14 feet in diameter and there is a one-foot gap between the rims. Looking at the picture above it seems that the weights on the right are constrained by their cords to hang further away from the central hub than the weights on the left. As the wheel rotates the various cords will tighten and slacken and the weights are expected to continually shift across so as to perpetuate this clockwise turning moment about the hub.

A description of the Great Weighted
Wheel (and 99 of his other remarkable inventions) may be found in a
curious book written by Somerset and called 'The Century of Inventions'.

Somerset's book is included as part of a volume
written by Henry Dircks entitled "
The Life, Times and Scientific Labors of [Edward
Somerset] the Second Marquis of Worcester, to which is added, a reprint of his
Century of Inventions, 1663." London, 1865. 558 pp. A copy of this book and details of the wheel
(Invention Number 56) may be found (towards the end of Section One) at

http://www.history.rochester.edu/steam/dircks

The torque and potential energy of the wheel have recently been analysed in an article included in the August 2000 issue of 'Mathematics Today' published by The Institute of Mathematics and Its Applications. The following links point to supporting documents for that article, in which it is shown mathematically, by means of an equlibrium analysis in the complex plane, that perpetual motion does not take place.

Click here to see a portrait of the Marquis of Worcester or here for a famous and colourful portrait of Charles I.

- Here is an
applet that allows the wheel to be rotated from the
reference position specified in the article. The number of spokes in each
sector, the total potential energy and the total torque are printed on the
screen as the rotation occurs. The equilibrium positions may most easily
be found by observing the positions where the torque changes sign. The
stable equilibrium occurs at 0.48
^{0}anticlockwise from the reference position (i.e. between 4 and 5 tenths of a degree on the scroll bar). The unstable position is at 4.99^{0}anticlockwise from the reference position (between 49 and 50 tenths of a degree on the scroll bar).

- Another
applet written by a colleague, Mr. A.A.R. Townsend, may be found on his
site at http://www.comapps.com

- Here is a a Mathcad 2000 document (128 kB) (or as a 36 kB zip file) showing the details of the calculation of the equilibrium positions and here is the same document saved as a Mathcad 7 file (or 36kB zip file). The calculation is done twice by slightly different methods.
- The Mathcad document is also available as a pdf file (68 kB)

**
ERRATA:** There is a printing error in the Mathematics
Today article on page 110 in the left-hand column, 12^{th} line from
the bottom of the page. Click here for the correct equation as a short
Word doc file (15 kB)

Download the full text of the paper as a zipped Word
doc file (672 kB)

TJF 15/August/00 (email terryf@mathapps.net)