Galileo first got the concept of a pendulum whilst watching a swinging lamp in a Pisa cathedral. He then timed its period (how long each swing is) by measuring it against his pulse. Initially it was thought that the period of swing depended solely on the length of the pendulum rod (isochronous), and not on the arc it swang through. However a Dutchman called Christiaan Huygens discovered that as the arc increases a slight increase in the rate occurs as well.

For a pendulum to be isochronous it would need to swing in a cycloidal path, see here for an external site explaining a cycloid.


You can see the difference between cycloid and circular above.

Attempts to get a pendulum swinging in a cycloid arc have been achieved, but with limited success, so a circular arc is usually adopted. The resultant small error that this introduces is therefore called circular error
These errors can be dismissed though - so long as the arc never changes. If the arc causes a loss of 5 seconds a day, you simply adjust the clock to run 5 seconds faster to offset the error. If the arc varies however, you start to get problems, as the allowance you have made for 5 seconds no longer applies for the new arc.

To find the size of the change in seconds per day that an altered arc would cause you need to know 2 things- (1) the angle size of the original semi arc* and (2) the angle of the semi arc* that it has changed to.

You then work out the circular error for each and subtract one from another.

Use the formula Error in day = (secs in a day/16) multiplied by the arc in radians (squared)

This equates to



So a 2 degree arc that changes to a 2.5 degree arc would work out as follows.

2 degrees = 0.03490658503989 radians
5400 x (0.03490658503989 x 0.03490658503989) = 6.579736
circular error in secs

2.5 degrees = 0.04363323129986 radians
5400 x (0.04363323129986 x 0.04363323129986) = 10.280837
circular error in secs

10.280837 - 6.579736 = 3.701101

So the 2 degree arc that changes to a 2.5 degree arc will cause a loss of 3.7 seconds daily.

 

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