Modern Physics/The Law of Gravitation

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Law of Gravitation

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

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.

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