Vector Form of Newton’s Law of Gravitation

The vector form of Newton’s law of gravitation signifies that the gravitational forces acting between the two particles form action-reaction pair.

Derivation of Newton's Law of Gravitation

From the above figure, it can be seen that the two particles of masses are placed at a distance.

\begin{array}{l} \vec{�_{21}} = (\vec{�_{2}} - \vec{�_{1}}) \end{array}

The direction of the vector is from M1 to M2

Therefore, the force applied on M2 by M1 is

\begin{array}{l} \vec{�_{21}} = - \frac{� �_{1} �_{2}}{�_{21}^{2}} \hat{�_{21}} \end{array}

The negative sign indicates the attractive nature of the force.

Similarly, force on M1 and M2

\begin{array}{l} \vec{�_{12}} = - \frac{� �_{1} �_{2}}{�_{12}^{2}} \hat{�_{12}} \end{array}

Since,

\begin{array}{l} \hat{�_{12}} = - \hat{�_{21}} \end{array}

\begin{array}{l} \vec{�_{12}} = - \frac{� �_{1} �_{2}}{(- �_{21})^{2}} [\hat{- �_{21}}] \end{array}

\begin{array}{l} \vec{�_{12}} = \frac{� �_{1} �_{2}}{(�_{21})^{2}} [\hat{�_{21}}] \end{array}

\begin{array}{l} = - \vec{�_{21}} \end{array}

Hence, the applied forces are equal and opposite. Also, the gravitational force follows Newton’s third law.

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