# Gravitational constant of the Earth

August 31, 2015

#### The Controversy over Newton's Gravitational Constant

In 1686 Isaac Newton realized that the motion of the planets and the moon as well as that of a falling apple could be explained by his Law of Universal Gravitation, which states that any two objects attract each other with a force equal to the product of their masses divided by the square of their separation times a constant of proportionality. Newton estimated this constant of proportionality, called , perhaps from the gravitational acceleration of the falling apple and an inspired guess for the average density of the Earth. However, more than 100 years elapsed before was first measured in the laboratory; in 1798 Cavendish and co-workers obtained a value accurate to about 1%. When asked why he was measuring , Cavendish replied that he was "weighing the Earth"; once is known the mass of the Earth can be obtained from the 9.8m/s2 gravitational acceleration on the Earth surface and the Sun's mass can be obtained from the size and period of the Earth orbit around the sun. Early in this century Albert Einstein developed his theory of gravity called General Relativity in which the gravitational attraction is explained as a result of the curvature of space-time. This curvature is proportional to .

Naturally, the value of the fundamental constant has interested physicists for over 300 years and, except for the the speed of light, it has the longest history of measurements. Almost all measurements of have used variations of the torsion balance technique pioneered by Cavendish. The usual torsion balance consists of a 'dumbbell' (two masses connected by a horizontal rod) suspended by a very thin fiber. When two heavy attracting bodies are placed on opposite sides of the dumbbell, the dumbbell twists by a very small amount. The attracting bodies are then moved to the other side of the dumbbell and the dumbbell twists in the opposite direction. The magnitude of these twists is used to find . In a variation of the technique, the dumbbell is set into an oscillatory motion and the frequency of the oscillation is measured. The gravitational interaction between the dumbbell and the attracting bodies causes the oscillation frequency to change slightly when the attractors are moved to a different position and this frequency change determines . This frequency shift method was used in the most precise measurement of to date (reported in 1982) by Gabe Luther and William Towler from the National Bureau of Standards and the University of Virginia. It was published in 1982. Based on their measurement, the Committee on Data For Science and Technology, which gathers and critically analyzes data on the fundamental constants, assigned an uncertainty of 0.0128% to . Although this seems quite precise, the fractional uncertainty in is thousands of times larger than those of other important fundamental constants, such as Planck's constant or the charge on the electron. As a result, the mass of the Earth is known far less precisely than, for instance, its diameter. In fact, if the Earth's diameter were known as poorly as its mass, it would be uncertain by one mile. This should be compared to the 3 cm uncertainty in the distance between the Earth and Moon, which is determined using laser ranging and the well-known speed of light!

Recently the value of has been called into question by new measurements from respected research teams in Germany, New Zealand, and Russia. The new values disagree wildly. For example, a team from the German Institute of Standards led by W. Michaelis obtained a value for that is 0.6% larger than the accepted value; a group from the University of Wuppertal in Germany led by Hinrich Meyer found a value that is 0.06% lower, and Mark Fitzgerald and collaborators at Measurement Standards Laboratory of New Zealand measured a value that is 0.1% lower. The Russian group found a curious space and time variation of of up to 0.7% The collection of these new results suggests that the uncertainty in could be much larger than originally thought. This controversy has spurred several efforts to make a more reliable measurement of .

Source: www.npl.washington.edu
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##### What is the universal gravitational constant, the mass of the earth, and the radius of the earth?

The universal gravitational constant, denoted "G," is equal to 6.672e-11. The mass is 5.9736 e24 kg, and the radius is 6378.1 km at the equator and 6356.8 km at the poles.