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We know the gravitational constant is $G=6.67545 \times 10^{-11} m^3 kg^{-1} s^{-2}$. But how this value is determined at first?

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The gravitational constant has been experimentally measured. Wikipedia has a brief but quite nice article on the history of measurements of $G$. The first measurement was made in 1798 (yes, that long ago!) by Henry Cavendish, who measured the gravitational force between lead spheres. The force from small masses is exceedingly small, but with sensitive equipment it can be measured.

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  • $\begingroup$ It is also interesting to note that the first measurement of $G$ was made in 1798, well after Newton's publication of the law of universal gravitation in 1687. $\endgroup$ – NeutronStar May 23 '14 at 14:57
  • $\begingroup$ It is also interesting to note that the Cavendish experiment did not measure the value of G. Cavendish was trying to "weigh the Earth", and that is the result he obtained. That his experiment did implicitly measure G is an after the fact observation (a one hundred year after the fact observation). It's also interesting to note that the Cavendish experiment wasn't the first. Arguably, the Bouguer experiment (1739) was, except it didn't work. The Schiehallion experiment (1778) did get a positive result, but possibly that was sheer dumb luck. $\endgroup$ – David Hammen Jul 28 '14 at 21:12

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