The value of g

A wide variety of engineering calculations in the cement industry make use of the acceleration due to gravity, g. The value used in these calculations is generally a "text-book" value, and may differ substantially from the actual value. Particularly in Great Britain, a vast number of actual accurately-determined values of g have been measured all over the country at a high geographical frequency, so it is possible to obtain a reliable value at any desired location.

Newton's Law of Gravitation as applied to the Earth is F = G m M / r², where F is the gravitational force acting on a body of mass m, G is the universal gravitational constant, M is the mass of the Earth, and r is the distance of the body from the centre of the earth. g is the factor in the equation F = m g, so g is given by:

Both G and M are empirical constants, and g has an inverse-square relationship to r, the distance from the centre of mass of the earth.

The measured value of g is affected by three major factors:

Latitude. This has two effects:

• because the earth is an ellipsoid, the distance of a point on the surface from the centre diminishes with latitude, raising g
• the spin of the earth produces a centrifugal effect opposing gravity, which is at a maximum at the equator and zero at the poles.

These two effects conspire to produce an increase of g with latitude. Their magnitudes are readily calculated by simple geometry.

Altitude. The latitude effect is calculated on the basis of the standard surface of the geoid, which is the spheroid at sea level. Points above sea level are progressively further from the centre of the earth, so g diminishes with altitude in a predictable manner.

Gravity Anomaly. In practice the value of g varies somewhat from the value predicted geometrically from latitude and altitude. Positive variation is caused by:

• the mass of the local above-sea-level topography
• above-average density of underlying rocks

The components of the Earth's structure have a variety of densities. The geometrical model of gravity conceives of the Earth as a series of onion-skin layers, each with uniform density (and this is nearly the case). An individual layer, because it is of uniform density, has its centre of mass coincident with that of the Earth. However, if a layer has a small patch of higher density rock, then the centre of mass is displaced towards the patch, reducing r, and so increasing g.

The geometric effects are calculated by a standard equation: the equation recommended by the UK National Physical Laboratory is that used currently (2017) by the International Union of Geodesy and Geophysics. The equation is:

where φ is the latitude and H is the altitude in metres.

The effect of latitude is around +0.000 85 m.s^-2 per degree in mainland Great Britain.

The effect of altitude is -0.000 308 8 m.s^-2 per 100 metres.

The value of the gravity anomaly in Great Britain varies from -0.00055 to +0.00137 m.s^-2, but most values are much nearer zero.

The effect of latitude is therefore dominant.

Values at cement plants

Values have been calculated using the open-source database provided by the British Geological Survey, which covers Great Britain and offshore waters. Anomaly values for Ireland are not open-source (Note 1), and values have been estimated from various third party graphical sources.

The values given (m.s^-2) are for plants with rotary kilns, at ground level at the hot ends of the kilns.

Note 1. In fact the Irish data, held by the Dublin Institute for Advanced Studies, is charged at €0.30 per point, so the BGS database, if charged in the same way, would cost €50,000. The market for this data probably has a high demand flexibility.