# Universal law of Gravitation

## Universal law of gravitation

• Everybody in the universe attracts every other body with a force which is directly proportional to the product of their masses and inversly proportional to the square of distance between them.
• Mathematically Newton's gravitation law is if F is the force acting between two bodies of masses M1 and M2 and the distance between them is R then magnitude of force is given as

$F=\frac{Gm_{1}m_{2}}{r^{2}}$
In vector notation, Force on $m_2$ due to $m_1$ is given
$F=\frac{Gm_{1}m_{2}(-\hat{\boldsymbol{\mathbf{r}}})}{r^{2}}$
$F=-\frac{Gm_{1}m_{2}\hat{\boldsymbol{\mathbf{r}}}}{r^{2}}$ or
$F=-\frac{Gm_{1}m_{2}{\boldsymbol{\mathbf{r}}}}{r^{3}}$
where G- universal gravitational constant
$\hat{\boldsymbol{\mathbf{r}}}$ is unit vector from m1 to m2 and ${\boldsymbol{\mathbf{r}}} = \boldsymbol{\mathbf{r_{2}}} - \boldsymbol{\mathbf{r_{1}}}$
Gravitational force constant is
SI - G= 6.67 x 10-11 nm2 kg-2
CGS - G= 6.67 x 10-8 dyn cm2 g-2
Dimensional formula of C1 is [m-1L3T-2]
• Since force is attractive, direction of force is along $-\mathbf{r}$
• We can represent force on $m_2$ by $m_1$ as $\mathbf{F_{21}}$ and $m_1$ by $m_2$ as $\mathbf{F_{12}}$,then
$\mathbf{F_{21}} = - \mathbf{F_{12}}$
• If we have a collection of point masses,the force on any one of them is the vector sum of the gravitational forces exerted by the other point masses

## How to calculate the Force of Gravitation

• Write down the force vector due to each of point masses
• Choose a convenient Coordinate system and choose the unit vectors
• Derive each of the above forces in these unit vectors
• Now force of gravity can be calculated easily
• Incase of system of continous masses, we can use calculas and symtery to calculate the force of attraction

## Important Conclusions

• The force of attraction between a hollow spherical shell of uniform density and a point mass situated outside is just as if the entire mass of the shell is concentrated at the centre of the shell

$F= \frac {GMm}{r^2}$
• The force of attraction due to a hollow spherical shell of uniform density, on a point mass situated inside it is zero
$F=0$