PC1672 Advanced dynamics

3.1 Universal law of gravitation

Newton's universal law of gravitation states that all pieces of matter in the universe exert attractive forces on each other. The force exerted by a particle of mass on a particle of mass has the form

$\begin{displaymath}{\bf F}=-{GMm\over r^2}{\bf e}_r\end{displaymath}$

where

is the distance between the particles and ${\bf e}_r$ is a unit vector in the direction direction from

. (The force on

is equal in magnitude and opposite in direction.)

The gravitational force has the following important properties:

It is attractive.
It is universal.
It is weak. The gravitational constant is $G=6.67\times 10^{-11}\ \hbox{\rm N m$^2$\space kg$^{-2}$ }$ .
It is long-ranged, falling off according to an inverse-square law.
It is proportional to the mass of an object.
It is conservative.

Since this force is proportional to mass, it is useful to define the gravitational field ${\bf g}({\bf x})$ as the gravitational force on a unit mass at ${\bf x}$ . In a given gravitational field, all objects have the same acceleration whatever their mass.

The gravitational field at the surface of the Earth is $g=9.81\ \hbox{ms$^{-2}$ }$ . Knowing , we can use this to deduce that the mass of the Earth is $M_E=6\times 10^{24}\ \hbox{\rm kg}$ . This was first done by Cavendish in 1797, although Maskelyne and Playfair had already estimated the mass of the Earth in 1774 by measuring the deflection of a pendulum near Schiehallion, a conical mountain in Perthshire.

More on equivalence: A lift in free fall (in a uniform gravitational field) cannot be distinguished from a lift moving with constant velocity in outer space. All objects in the falling lift will be accelerating at the same rate and so, from the point of view of someone stuck inside it, they appear to be weightless. Einstein noted that for such an observer the frame defined by the lift is just like an inertial frame. Gravity is then just an inertial force observed by us in a (noninertial) frame at rest on the surface of the Earth (see above).

Textbook references

K&K 2.5 (8.4 for the principle of equivalence
F&C 6.1
B&O 1.2 (8.4 for the principle of equivalence)
M 2.7
M&T 5.1

Mike Birse
17th May 2000