Modern Physics/The Law of Gravitation

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Law of Gravitation[edit]

Of Newton's accomplishments, the discovery of the universal law of gravitation ranks as one of his greatest. Imagine two masses, M1 and M2, separated by a distance r. The gravitational force has the magnitude

 F_g = G \frac{M_1 M_2}{r^2}

where G is the gravitational constant:

G \approx 6.67 \times 10^{-11} \frac{m^3}{kg \cdot s^2}

The force is always attractive, and acts along the line joining the centre of the two masses.

Vector Notation[edit]

Let's say that we have two masses, M and m, separated by a distance r, and a distance vector R. The relationship between R and r is given by:

|\vec{\mathbf{R}}| = r

We will also change our force into a force vector, acting in the direction of R:

\vec{F}_g = G \frac{M_1 M_2}{r^2} \cdot \frac{\vec{\mathbf{R}}}{r}

And this gives us our final vector equation:

\vec{F}_g = G \frac{M_1 M_2 \vec{\mathbf{R}}}{r^3}

Notice that since the ratio between R and r is normalized, the addition of these terms does not alter the equation, only the direction in which the force is acting.