Question

Explain how Newton verified his law of gravitation.

Answer 1

The gravitational force experienced by the apple due to Earth.

F = `-("GM"_"E""M"_"A")/"R"^2`

M_A – Mass of the apple

M_E – Mass of the Earth

R – Radius of the Earth

Equating the above equation with Newton’s second law, we get

M_Aa_A = `-("GM"_"E""M"_"A")/"R"^2`

Simplifying the above equation we get,

a_A = `-("GM"_"E")/"R"^2`

a_A is the acceleration of apple that is equal to ‘g’.

Similarly the force experienced by Moon due to Earth is given by

F = `-("GM"_"E""M"_"m")/"R"_"m"^2`

R_m – distance of the Moon from the Earth.

M_m – Mass of the Moon.

The acceleration experienced by the Moon is given by,

a_m = `-("GM"_"E")/("R"_"m"^2)`

The ratio between the apple’s acceleration to Moon’s acceleration is given by

`"a"_"A"/"a"_"m" = "R"_"m"^2/"R"^2`

From the Hipparchrus measurement, the distance to the Moon is 60 times that of Earth's radius. R_m = 60R.

`"a"_"A"/"a"_"m" = (60  "R")^2/"R"^2` = 3600

The apple’s acceleration is 3600 times the acceleration of the Moon.

The same result was obtained by Newton using his gravitational formula.

The apple’s acceleration is measured easily and it is 9.8 m s⁻². The moon orbits the Earth once in 27.3 days and by using the centripetal acceleration formula,

`"a"_"A"/"a"_"m" = 9.8/0.00272` = 3600

which is exactly what he got through his law of gravitation.