The beat period of the pendulum is the time it takes to swing from one vertical position to the next without regard to the direction of swing. A full period would be considered to be the time between swings to the vertical position from the same direction of swing. There are therefore two beats in every complete period,
Figure 1 shows the elements of a simple pendulum which consists of:
The equation for the period for a simple pendulum is P = 2 π (L/g)1/2
External factors such as the strength of the local gravity and the maximum angle of the swing α will affect the period as will the weight of the string relative to the weight of the mass and the size of the mass relative to the length of the string. These effects are small and can be expected to introduce errors in length of less than one part per thousand.
The effect of a change in local gravity will directly change the length of a pendulum of a fixed period by about one half percent from a minimum at he equator to a maximum at the poles. This change would be less than one part per thousand at the latitude of ancient civilizations which were all close to 30 degrees north latitude.
Larger angles of swing increase the period as shown in the following equation. P = 4K (L/g)1/2
Where K is a function of the maximum angle of swing. It can be calculated only by finding the solution to an elliptical integral which is available in published mathematical tables. Increasing the swing angle from very small values to 30 degrees on each side would increase the period by about 1.7 percent. Limiting the swing to about to about 1/10 the length of the string on each side (0.1 radian) would increase the period by only about 0.625 parts per thousand. Reducing the swing to one half this value would reduce the change to about 0.156 parts per thousand.
Fortunately the following formula provides a reasonably accurate increase in the period without the use of an elliptical integral. In this formula the maximum angle of swing should be inputed in Radians
The Following is a sample calculation with a maximum swing angle of 1/10 radian = 5.72958 degrees
T /T0 = 1 + 0.000625 + 0.000000358 + 0.00000000002346 = 1.000625358
The beat period at a maximum swing angle of 1/10 radian will increase by 0.625 parts per thousand
The Length reduction required to maintain the original period would be 1.25 parts per thousand
Simply increasing mass of a real string will reduce the period below that of a simple pendulum as follows.
(P/P0)2 = ( M/m + 1/2)/ (M/m + 1) Where M/m is ratio of pendulum mass to string mass
However it will also Increase the inertia of the ball and the string tending to increase the period as follows
(P/P0)2 = ( 1 + 2/5 (R/L)2 + 1/3 (m/M) Where R/L is the ratio of the radius of the ball to the length of the String
Realistic ratios of the ball to string mass can completely cancel out the reduction in period caused by moderate angles of swing. For example the 0.625 parts per thousand increase in period induced by a swing angle of one tenth radian can be completely canceled by a ball to string mass of 122.31 when using a brass ball. The ratio is somewhat smaller at 121.46 when using less dense iron ball. When using a more dense Platinum Ball the mass/string ratio increases slightly to126.23
In an experiment intended to duplicate an ancient pendulum we used waxed flax string similar to what would have been available in ancient Egypt and Sumeria and a steel balls similar to the copper and bronze which would have been available.
The Sumerian pendulum we proposed had a length of 994 mm and a beat (1/2 period) of one second. It was allowed to swing 1/10 radian from the vertical. As a simple Pendulum, its predicted period would be to be 0.625 milliseconds too long.
Our Experimental pendulum used a 25 m diameter 67 gram steel ball and a 994 mm long 0.60 gm string.
We experienced a precise 1.0000 second period just as our mathematical formulas predicted.