46 Kinematics and Dynamics of a Point Massto study the motion of bodies under Earth's gravity. Even though many
modern historians question whether indeed, around 1590, he was dropping
different objects from the leaning tower of Pisa, in 1604 Galileo did con-
duct related experiments using an inclined plane; in 1608, he formulated
mathematically the basic laws of accelerated motion under the gravitational
force. The conjecture of Galilei that the acceleration due to gravity is es-
sentially the same for all kinds of matter has been verified experimentally.
Between 1905 and 1908, the Hungarian physicist VASAROSNAMENYI BARO
EOTVOS LORAND (1848-1919), also known as ROLAND EOTVOS, measured
a variation of about 5 x 10~^9 in the Earth's pull on wood and platinum;
somehow, the result was published only in 1922. In the 1950s, the Ameri-
can physicist ROBERT HENRY DiCKE (1916-1997) measured a difference of
(1.3 ± 1.0) x 10-11 for the Sun's attraction of aluminum and gold objects.
Following the historical developments, we derived relation (2.1.11) from
Newton's Second Law of Motion and Kepler's Third Law of Planetary Mo-
tion. Problem 2.1, page 413, presents a deeper insight into the problem. In
particular, more detailed analysis shows that, in our derivation of (2.1.11),
Kepler's Third Law can be replaced by his First Law, along with the as-
sumption that the gravitational force is attracting and central, that is,
acts along the line connecting the Sun and the planet. Moreover, the reader
who completes Problem 2.1 will see that all three Kepler's laws follow from
(2.1.1) and (2.1.11). This illustrates the power of mathematical models in
reasoning about physical laws.2.1.2 Parallel Translation of Frames
Recall that Newton's laws of motion hold only in inertial frames; see page 4
for the definition of frame. Ignoring the possible logical issues, we therefore
say that an inertial frame is a frame in which a point mass maintains
constant velocity in the absence of external forces. The same frame can
be (approximately) inertial in some situations and not inertial in others.
For example, a frame fixed to the surface of the Earth is inertial if the
objective is to study the motion of a billiard ball on a pool table. The same
frame is no longer inertial if the objective is to study the trajectory of an
intercontinental ballistic missile: the inertial frame for this problem should
not rotate with the Earth; the primary inertial frame fixed to the stars is
a possible choice, see page 43. In this and the following two sections, we
investigate how relation (2.1.1) changes if the frame is not inertial. The