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
$F=0$