page_title: Cavendish experiment
section_title: Derivation of G and the Earth's mass
hierarchical_section_title: Cavendish experiment / Derivation of G and the Earth's mass
context_page_description: The Cavendish experiment, performed in 1797–1798 by English scientist Henry Cavendish, was the first experiment to measure the force of gravity between masses in the laboratory and the first to yield accurate values for the gravitational constant. Because of the unit conventions then in use, the gravitational constant does not appear explicitly in Cavendish's work. Instead, the result was originally expressed as the specific gravity of the Earth, or equivalently the mass of the Earth. His experiment gave the first accurate values for these geophysical constants.
  The experiment was devised sometime before 1783 by geologist John Michell, who constructed a torsion balance apparatus for it. However, Michell died in 1793 without completing the work. After his death the apparatus passed to Francis John Hyde Wollaston and then to Cavendish, who rebuilt the apparatus but kept close to Michell's original plan. Cavendish then carried out a series of measurements with the equipment and reported his results in the Philosophical Transactions of the Royal Society in 1798.
context_section_description: The following is not the method Cavendish used, but describes how modern physicists would calculate the results from his experiment.  From Hooke's law, the torque on the torsion wire is proportional to the deflection angle θ of the balance. The torque is κθ where κ is the torsion coefficient of the wire. However, a torque in opposite direction is also generated by the gravitational pull of the masses. It can be written as a product of the attractive forces between the balls and the distance to the suspension wire.  Since there are two pairs of balls, each experiencing force F at a distance  from the axis of the balance, the torque is LF. At equilibrium (when the balance has been stabilized at an angle θ), the total amount of torque must be zero, as these two sources of torque cancel out. Thus, we can equate their intensities given by the formulas above, which gives the following:
  For F, Newton's law of universal gravitation is used to express the attractive force between the large and small balls:
  Substituting F into the first equation above gives
  To find the torsion coefficient (κ) of the wire, Cavendish measured the natural resonant oscillation period T of the torsion balance:
  Assuming the mass of the torsion beam itself is negligible, the moment of inertia of the balance is just due to the small balls:
  ,
  and so:
  Solving this for κ, substituting into (1), and rearranging for G, the result is:
  Once G has been found, the attraction of an object at the Earth's surface to the Earth itself can be used to calculate the Earth's mass and density:
caption: Diagram of torsion balance