13.2 Gravitational Force Between Point Masses

The Point Mass Approximation

• For a point outside a uniform sphere, the mass of the sphere can be considered to be completely concentrated as a point mass at its centre.

Conditions

• The approximation is only valid if the sphere has uniform density
• Distance $r$ is greater than the radius of the sphere.

Newton’s Law of Gravitation

Newton's Law of Gravitation

• The gravitational force of attraction between two point masses is directly proportional to the product of the masses
• And inversely proportional to the square of their separation.

Representation of M, m, r, F

Derivation: Newton's Law of Gravitation

$$\begin{array}{rl}F& \propto \frac{Mm}{r^2}\\ & \\ F& =\frac{GMm}{r^2}\end{array}$$

Where $G$ is the Gravitational Constant, $G=6.67\times 10^{-11}\mathrm{N}{\mathrm{m}}^2\mathrm{k}{\mathrm{g}}^{-2}$

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