Gravity

The force that causes objects to fall toward the surface of the Earth, maintains the Moon in orbit about the Earth, and maintains the planets in orbit around the Sun, is called gravity, and was first correctly described by Isaac Newton in 1687. According to Newton, any two point mass objects (or spherically symmetric objects of finite extent) exert a force of attraction on one another. This force points along the line of centers joining the objects, is directly proportional to the product of the objects' masses, and inversely proportional to the square of the distance between them.

Consider a system consisting of two point mass objects. Let object $1$ have mass $m_1$ and displacement ${\bf r}_1$. Let object $2$ have mass $m_2$ and displacement ${\bf r}_2$. The gravitational force exerted on object 2 by object 1 is written

$${\bf f}_{21} =- G\,m_1\,m_2\,\frac{{\bf r}_2-{\bf r}_1}{\vert{\bf r}_2-{\bf r}_1\vert^3}.$$ (1.238)

The constant of proportionality, $G$, is called the universal gravitational constant, and takes the value

$$G = 6.67430\times 10^{-11}\,{\rm m^3\,kg^{-1}\,s^{-2}}.$$ (1.239)

This constant was first `measured' by Henry Cavendish in 1798 (to be more exact, the result $G=6.74\times10^{-11} \,{\rm m^3\,kg^{-1}\,s^{-2}}$ can be inferred from Cavendish's results). An equal and opposite force to (1.238) acts on object 1.

Suppose that we have a system of $N$ point mass objects. Let the $i$th object have mass $m_i$ and displacement ${\bf r}_i$. Now, it is an experimentally verified fact that gravity is a superposable force. In other words, the gravitational force exerted on object $i$ by object $j$ is unaffected by the presence of any other objects in the universe. Hence, the net gravitational force experienced by object $i$ is

$${\bf f}_i = -G\,m_i\sum_{j=1,N}^{j\neq i}m_j\,\frac{{\bf r}_i-{\bf r}_j}{\vert{\bf r}_i-{\bf r}_j\vert^3}.$$ (1.240)

Note that object $i$ is missing from the sum on the right-hand side of the previous equation because this object cannot exert a gravitational force on itself.

Suppose, finally, that a point object of mass $m$ is located at the origin of our coordinate system. It follows from Equation (1.238) that the gravitational acceleration, due to the gravitational attraction of mass $m$, experienced by another point object whose displacement is ${\bf r}$ is

$${\bf g}= -G\,m\,\frac{{\bf r}}{r^3}.$$ (1.241)