Determining the value of G
To determine the value of the gravitational constant G, we have to measure the gravitational force between two bodies of known masses m1 and m2 at a known distance r. The force is extremely small for bodies that are small enough to be brought into the laboratory, but it can be measured with an instrument called a torsion balance, which Sir Henry Cavendish used in 1798 to determine G.
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Figure 13.4 shows a modern version of the Cavendish torsion balance. A light, rigid rod shaped like an inverted T is supported by a very thin, vertical quartz fiber. Two small spheres, each of mass m1, are mounted at the ends of the horizontal arms of the T. When we bring two large spheres, each of mass m2, to the positions shown, the attractive gravitational forces twist the T through a small angle. To measure this angle, we shine a beam of light on a mirror fastened to the T. The reflected beam strikes a scale, and as the T twists, the reflected beam moves along the scale.
After calibrating the Cavendish balance, we can measure gravitational forces and thus determine G. The presently accepted value is
To three significant figures, G = 6.67 * 10^{-11} N . m^{2}/kg^{2} . Because 1 N = 1 gm/s ^{2} ,
the units of G can also be expressed as m^{3}/(kg.s^{ 2} ). Gravitational forces combine vectorially. If each of two masses exerts a force on a third, the total force on the third mass is the vector sum of the individual forces of the first two. Example 13.3 makes use of this property, which is often called superposition of forces (see Section 4.1).