Work done in Gravitational Field

How to Calculate Work Done in Gravitational Field ?

Gravitational Field as a Conservative Field

Consider the gravitational field of a point particle of mass M located at the origin. A test particle of mass m is placed at the point P (x, y, z)

The position vector of P ,

$ \displaystyle \vec{OP} = \vec{r} = x\hat{i} + y\hat{j} + z\hat{k} $

$ \displaystyle \vec{OP} = r = \sqrt{x^2 + y^2 +z^2} $

The gravitational force acting on the particle at P,

$ \displaystyle \vec{F} = -\frac{G M m}{r^2}\hat{r} = -\frac{G M m}{r^3}\vec{r}$

The work done by the gravitational field for a small displacement ( $ \displaystyle \vec{dr} = dx\hat{i} + dy\hat{j} + dz\hat{k} $ ) is

$ \displaystyle dW = \vec{F}.\vec{dr} = -\frac{G M m}{r^3}\vec{r}.\vec{dr} $

$ \displaystyle dW = -\frac{G M m}{r^3} (xdx + ydy + zdz) $

$ \displaystyle dW = -\frac{G M m}{r^3} r dr $

$ \displaystyle dW = -\frac{G M m}{r^2} dr $

For a closed loop,

$ \displaystyle dW = \oint\vec{F}.\vec{dr} = -\oint\frac{G M m}{r^2} dr = [\frac{G M m}{r}]_{r_p}^{r_p} = 0 $

The gravitational field of a point particle is conservative. But what about an arbitrary gravitational field ?
we use the superposition principle.

Here, we use the superposition principle:

$ \displaystyle \oint\vec{F}.\vec{dr} = \oint ( \vec{F_1} + \vec{F_2} + \vec{F_3} + ……) .\vec{dr}$

Where $\vec{F_1}$ , $\vec{F_2}$ ,$\vec{F_3}$ . . . are the gravitational forces due to each point particle constituting an extended object, which exerts the net force $\vec{F}$

$ \oint\vec{F_1}.\vec{dr}+ \oint\vec{F_2}.\vec{dr} + \oint\vec{F_3}.\vec{dr} + …. = 0 + 0 + 0 + …= 0$

 Gravitational field is a conservative force field. This means that one can define potential energy for this field

Also Read :

∗ Newton’s Law of Gravitation & Gravitational Field Intensity
∗ Acceleration due to Gravity & Variation of g
∗ Gravitational Potential Energy
∗ Gravitational Potential
∗ Escape Velocity
∗ Kepler’s Laws of Planetary Motion
∗ Motion of Satellites

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