gravitational force

Force between two masses, also between mass and field, considered in butiran.

When we talk about force between two masses at distance some terms come up, such gravity, gravitational force, and gravitation. Gravity is a real force but not in the traditional sense, because at fundamental level there are not any action-at-distance forces, since it just a mass creating gravitational field and then the other mass moving and interacting with this field ¹. It is better to use the term gravitational force instead of gravity about the force between to masses that are not in contact ². Newton’s law of universal gravitation usually present the force in scalar form ³, which is not easy to use for interaction more than to masses. The vector form ⁴ will be used here since it is more general and easier to use in calculating force superposition. There is also gravitional force on a mass in a gravitational field as weight of a mass is calculated near earth surface ⁵.

Mass $m_i$ will experience gravitational force

$$\tag{1} \vec{F} _{G,i} = m_i \vec{g}$$

in gravitational field $\vec{g}$, where source of gravitational field is atually another mass or mass distribution.

Mass $m_i$ will experience gravitational force

$$\tag{2} \vec{F} _{G,ij} = -G \frac{m_i m_j}{r _{ij}^2} \hat{r} _{ij}$$

due to existence of other mass $m_j$, where position of mass $m_i$ is $\vec{r}_i$ and position of mass $m_j$ is $\vec{r}_j$.

Relative position of mass $m_i$ from mass $m_j$ is

$$\tag{3} \vec{r} _{ij} = \vec{r}_i - \vec{r}_j,$$

distance between two masses is

$$\tag{4} r_{ij} = |\vec{r} _{ij}| = \sqrt{\vec{r} _{ij} \cdot \vec{r} _{ij}},$$

and

$$\tag{5} \hat{r} _{ij} = \frac{\vec{r} _{ij}}{r _{ij}}$$

is the unit vector.

From Eqns (1) and (2) it can be obtained

$$\tag{6} \vec{g}_j(\vec{r}) = -G \frac{m_j}{|\vec{r} - \vec{r}_j|^3} (\vec{r} - \vec{r}_j)$$

as gravitational field produced by mass $m_j$ experienced by another mass in the surrounding of mass $m_j$.

Challenge 1. Show the steps to obtained Eqn (6) from Eqns (1) and (2). If necessary use also Eqns (3) and (5).