Gravity and Gravitation

# Gravitational Field

## Gravitational Field

Gravitational force is an action at a distance force i.e. this force exists between two masses even when they are not in contact with each other. Gravitational field is the space surrounding a material body in which its gravitational force of attraction can be measured.

## Gravitational Field Intensity

​Gravitational field intensity due to a material body at a point in the field is defined as the force experienced by unit mass placed at a point, provided that unit mass itself does not produce any change in the field of the material body.

​Consider a material body of mass $M$ located at a point $O$. Suppose a body of mass $m$ is placed at a point $A$ which is $r$ distance apart from the centre of the material body. We have to determine the gravitational field intensity at the point $A$.

​Now, Gravitational force on mass $m$ due to the material body is, $F=\frac{GMm}{r^2}$ If $E$ is the gravitational field intensity then, $E=\frac{F}{m}$ $∴E=\frac{GM}{r^2}$

This gives the gravitational filed intensity at a point which is at distance $r$.

If the material body of mass $M$ is earth and the body of mass $m$ lies very close to the surface of the earth, then, $r = R$ (Radius of the earth) and, $E=\frac{GM}{R^2}$ also, we have, $g=\frac{GM}{R^2}$ Therefore, $E=g$

Thus, the gravitational field intensity near to the surface of the earth is approximately equal to its acceleration due to gravity.

## Gravitational Potential Energy

​Gravitational Potential Energy of a planet or body at a point is the amount of work done in bringing the body from infinity to that point inside the gravitational field of that planet.

​Consider earth to be a uniform sphere of radius $R$ and mass $M$. Suppose a body is placed at a point $A$ which is $r$ distance apart from the centre of the earth. Then, we have to determine the gravitational potential energy of the body.

​By definition, gravitational potential energy of the body at the point $A$ is,
$W =$ Work done in bringing the body from $∞$ to $A$.

Suppose the body is at a point $P$ which is $x$ distance apart from the centre of the earth. Then, the gravitational force of the earth on the body is given by, $F=\frac{GMm}{x^2}$

Small work done to move the body through infinitesimally small distance $dx$ is given by, $dW=F\;dx$ $dW={GMm}{x^2}\;dx$

Integrating within limits from $∞$ to $r$, $\int_0^W\;dW=\int_∞^r \frac{GMm}{x^2}\;dx$ $W=GMm\int_∞^r x^{-2}\;dx$ $W=GMm\left[ \frac{x⁻¹}{-1} \right]_∞^r$ $W=-GMm\left[ \frac{1}{r}-\frac{1}{∞} \right]$ $W=-\frac{GMm}{r}$

It gives the potential energy of body of mass $m$ which is located at a distance $r$ from the centre of the earth. It is denoted by $U$.

$∴U=-\frac{GMm}{r}$ The negative sign shows that the work is done to bring the body from infinity to a distance $r$.

## Gravitational Potential

​Gravitational Potential of a planet or body at a point is the amount of work done in bringing a body of unit mass from infinity to the point. It is denoted by $V$. We can obtain gravitational potential at a distance $r$ from earth’s centre by, $V=\frac{U}{m}$ $V=-\frac{GM}{R}$