Gravitation
In the following section, the basic Newtonian concept of gravitation will
be explained.
Newton's Investigation on Planetary Motion
About 1666, at the earlyt age of 24, Newton investigated the motion of
a planet moving in a circle round the sun S as centre, Figure 1.1(i). The
force acting on the planet of mass is mrw2 , where r
is the radius of the circle and w is the angular speed of the motion.
Since w= 2pi/T, where T is the period of the motion,
force on planet = mr(2pi/T)2 = 4pi2 mr/T2
This is equal to the force of attraction of the sun on the planet. Assuming
an inverse- square law for the distance r, then, if k is a constnat,
force on planet = km/r2
km/r2 = 4pi2mr/T2
T2 = 4pi2r3/k
T2
r3
Motion of Moon round Earth
Newton now tested the inverse-square law by applying it to the case
of the moon's motion round the earth, Figure 1.1(ii). the moon has a period
of revolution, T, about the earth of approxinately 27.3 days, and force
on it = mRw2, where R is radius of the moon's orbit and m is its mass.
force = mR(2pi/T)2 = 4pi2mR/T2
If the moon was at the earth's surface, the force of attraction on it
due to the earth would be mg, where g is the acceleration due to gravity,
Figure 1.1(ii). Assuming that the force of attraction varies as the inverse
square of the distance between the earth and the moon, then by ratio of
the two forces,
4pi2mR/T2 : mg = 1/R2
: 1/rE2
where rE is the radius of the earth. Cancelling m and simplifying,
4pi2mP/T2g = rE2/R2
g = 4pi2R3/rE2T2
..................(1)
Newton substitituted the then known values of R, rE and T,
but found that the answer for g was nowhere near the observed value of
9.8 ms-2. It was only years later that newer and more accurate estimates
of the earth's radius (rE =6.4 x 106m )were
made.
Newton's Law of Gravitation, G
Newton saw that a universal law could be stated for the gravitational attraction
between any two particles of matter. He suggested that the forcen of the
attraction between two given particles is inversely proportional to the
square of their distance apart.
From this law it follows that the force of attraction, F, between two
particles of masses m and M respectively, at a distance r apart, is given
by
F = G mM/r2 .............(2)
where G is a universal constant known as the gravitational constant.
This expression for F is Newton's law of gravitation. It is a universal
law because it appears to be true for masses all over the world.
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