Newton's Laws of Motion and Gravitation 45recover Newton's argument by combining equation (1.3.29), page 36, with
his second law (2.1.1). Take an inertial frame with origin at the Sun and
assume that a planet of mass m executes a circular motion around the
origin with constant angular speed w. Then (2.1.1) and (1.3.29) result in
F = ma = -moj^2 r(t) = -mw^2 Rr{t),where r = r/\r\ is the unit radius vector pointing from the Sun to the
planet. On the other hand, u> — 2n/T, and therefore the magnitude
of F is ||F|| = m(4rr^2 /T^2 )R. Applying Kepler's Third Law, we obtain
||F|| = m{Air^2 /KsR?)R = Cm/R^2 , where C = 4n^2 /Ks. In other words,
the gravitational force exerted by the Sun on the planet of mass m at a
distance R is proportional to m and R~^2. By Newton's third law, there
must be an equal and opposite force exerted by m on the Sun. By the
same argument, we conclude that the magnitude of the force must also be
proportional to M and R~^2 , where M is the mass of the Sun. Therefore,
11*11 = ^, (2.1.11)
where G is a constant. Newton postulated that G is a universal
gravitational constant , that is, has the same value for any two masses,
and therefor (2.1.11) is a Universal Law of Gravitation. In 1798, the
English scientist HENRY CAVENDISH (1731-1810), in his quest to determine
the mass and density of Earth, verified the relation (2.1.11) experimentally
and determined a numerical value of G: G « 6.67 x 10-11 m^3 /(kg- sec^2 ).
Since then, the Universal Law of Gravitation has been tested and verified
on many occasions. An extremely small discrepancy has been discovered
in the orbit of Mercury that cannot be derived from (2.1.11), and is ex-
plained by Einstein's law of gravitation in the theory of general relativity;
see Problem 2.2 on page 414.
In our derivation of (2.1.11), we implicitly used the equivalence principle
that the two possible values of m, its inertial and gravitational masses, are
equal. A priori, this is not at all obvious. Indeed, the mass m in equa-
tion (2.1.1) of Newton's Second Law, the inertial mass, expresses an
object's resistance to external force: the larger the mass, the smaller the
acceleration. The mass m in (2.1.11), the gravitational mass, expresses
something completely different, namely, its gravitational attraction: the
larger the mass, the stronger the gravitational attraction it produces. The
equivalence principle is one of the foundations of Einstein's theory of gen-
eral relativity and can be traced back to Galilei, who was among the first