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9.2 The Measurement of G 277

9.2 THE MEASUREMENT OF G

The gravitational constant G is rather difficult to measure with precision.The trouble
is that gravitational forces between masses of laboratory size are extremely small, and
thus a very delicate apparatus is needed to detect these forces. Measurements of G are
usually done with a Cavendish torsion balance (see Fig. 9.5).Two equal, small spher-
ical masses m and m are attached to a lightweight horizontal beam which is suspended
at its middle by a thin vertical fiber. When the beam is left undisturbed, it will settle
into an equilibrium position such that the fiber is completely untwisted. If two equal,
large masses M and M are brought near the small masses, the gravitational attraction
between each small mass and the neighboring large mass tends to rotate the beam
clockwise (as seen from above).The twist of the fiber opposes this rotation, and the net
result is that the beam settles into a new equilibrium position in which the force on the
beam generated by the gravitational attraction between the masses is exactly balanced
by the force exerted by the twisted fiber. The gravitational constant can then be cal-
culated from the measured values of the angular displacement between the two equi-
librium positions, the values of the masses, their distances, and the characteristics of
the fiber.
Note that the mass of the Earth can be calculated from Eq. (9.6) using the known
values of G, R , and g:
E
2 6 2 2
R g (6.38 10 m) 9.81 m/s
M E
E 11 2 2
G 6.67 10 N m /kg HENRY CAVENDISH (1731–1810)
English experimental physicist and chemist.
24
5.98 10 kg His torsion balance for the absolute measure-
This calculation would seem to be a rather roundabout way to arrive at the mass of ment of the gravitational force was based on
an earlier design used by Coulomb for the
the Earth, but there is no direct route, since we cannot place the Earth on a balance.
measurement of the electric force.
Because the calculation requires a prior measurement of the value of G, the Cavendish
experiment has often been described figuratively as “weighing the Earth.”

m M FIGURE 9.5 Model of large torsion
m'
balance used by Cavendish.The small
M'
masses m, m hang from the ends of a
horizontal beam which is suspended at
its middle by a thin vertical fiber.