Question

State and explain the universal law of gravitation. Give its vector form.

Answer 1

Hint: The universal law of gravitation gives us the force of attraction between two massive particles at a distance apart. Obtain the mathematical expression for the universal law of gravitation. Express it in vector form by giving the direction of force and the direction of position or the distance between the objects.

Complete step by step answer:
The universal law of gravitation gives the gravitational force of attraction between any two massive bodies at a distance apart from each other. Any bodies which have mass are attracted by other masses and the force of gravity is always acting on them.
If we consider two bodies of mass m and M, then the force of attraction between them is directly proportional to the product of the masses m and M. Again, if the two massive bodies are at a distance R apart from each other, then the force of attraction between the two bodies is inversely proportional to the square of the distance between them.
So, we can write that, the force of attraction between two bodies of mass m and M at a distance R apart is,
$F\propto \dfrac{mM}{{{R}^{2}}}$
We can equate the above equation by introducing a constant of proportionality G.
$F=G\dfrac{mM}{{{R}^{2}}}$
Where, G is the universal gravitational constant, the value of which is $G=6.67\times {{10}^{-11}}N{{m}^{2}}k{{g}^{-2}}$ . The value of G is always constant in any point or condition in the universe.
Consider object 1 and 2 of mass m and M respectively. The gravitational force of attraction in vector form is given by the mathematical expression,
${{F}_{21}}=-G\dfrac{{{m}_{1}}{{m}_{2}}}{{{\left| {{r}_{21}} \right|}^{2}}}{{\hat{r}}_{21}}$
Where, ${{F}_{21}}$ is the force on object 2 due to object 1.
G is the gravitational constant.
$\left| {{r}_{21}} \right|=\left| {{r}_{2}}-{{r}_{1}} \right|$ is the distance between the objects
${{\hat{r}}_{21}}=\dfrac{{{r}_{2}}-{{r}_{1}}}{\left| {{r}_{2}}-{{r}_{1}} \right|}$ is the unit vector from the object 1 to object 2

Note:
The gravitational force of attraction between two massive objects depends on the mass of the objects and the distance between them. As the mass of the objects increases, the force of attraction will also increase and as the distance between the objects increases, the force of attraction will decrease.