JEE Main & Advanced Physics Gravitation / गुरुत्वाकर्षण Newton's law of Gravitation

Newton's law of gravitation states that every body in this universe attracts every other body with a force, which is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres. The direction of the force is along the line joining the particles.

Thus the magnitude of the gravitational force F that two particles of masses \[{{m}_{1}}\] and \[{{m}_{2}}\] are separated by a distance r exert on each other is given by \[F\propto \frac{{{m}_{1}}\,{{m}_{2}}}{{{r}^{2}}}\]

or                 \[F=G\frac{{{m}_{1}}{{m}_{2}}}{{{r}^{2}}}\]  

Vector form : According to Newton's law of gravitation

\[{{\overset{\to }{\mathop{F}}\,}_{12}}=\frac{-G{{m}_{1}}{{m}_{2}}}{{{r}^{2}}}\]\[{{\hat{r}}_{21}}\]\[=\frac{-\,G{{m}_{1}}{{m}_{2}}}{{{r}^{3}}}{{\overset{\to }{\mathop{r}}\,}_{21}}=\frac{-G{{m}_{1}}{{m}_{2}}}{|{{\overset{\to }{\mathop{r}}\,}_{21}}{{|}^{3}}}{{\overset{\to }{\mathop{r}}\,}_{21}}\]

Here negative sign indicates that the direction of \[{{\overset{\to }{\mathop{F}}\,}_{12}}\] is opposite to that of \[{{\hat{r}}_{21}}\].

Similarly \[{{\overset{\to }{\mathop{F}}\,}_{21}}=\frac{-G{{m}_{1}}{{m}_{2}}}{{{r}^{2}}}\]\[{{\hat{r}}_{12}}\]\[=\frac{-\,G{{m}_{1}}{{m}_{2}}}{{{r}^{3}}}{{\overset{\to }{\mathop{r}}\,}_{12}}=\frac{-\,G{{m}_{1}}{{m}_{2}}}{|{{\overset{\to }{\mathop{r}}\,}_{12}}{{|}^{3}}}{{\overset{\to }{\mathop{r}}\,}_{12}}\]

 \[=\frac{G{{m}_{1}}{{m}_{2}}}{{{r}^{2}}}\]\[{{\hat{r}}_{21}}\]  [\[\because \,{{\hat{r}}_{12}}=-{{\hat{r}}_{21}}]\]

\[\therefore \] It is clear that \[{{\overset{\to }{\mathop{F}}\,}_{12}}\]= - \[{{\overset{\to }{\mathop{F}}\,}_{21}}\]. Which is Newton's third law of motion.

Here G is constant of proportionality which is called 'Universal gravitational constant'.

If \[{{m}_{1}}={{m}_{2}}\] and \[r=1\] then \[G=F\]

i.e.  universal gravitational constant is equal to the force of attraction between two bodies each of unit mass whose centres are placed unit distance apart.

(i) The value of G in the laboratory was first determined by Cavendish using the torsional balance.

(ii) The value of G is \[\text{6}\text{.67}\times \text{1}{{\text{0}}^{-11}}\,N-{{m}^{2}}\,\,k{{g}^{-2}}\] in S.I. and \[\text{6}\text{.67}\times \text{1}{{\text{0}}^{-8}}\,dyne-c{{m}^{2}}-{{g}^{2}}\] in C.G.S. system.

(iii) Dimensional formula \[[{{M}^{-1}}{{L}^{3}}{{T}^{-2}}]\].

(iv) The value of G does not depend upon the nature and size of the bodies.

(v) It also does not depend upon the nature of the medium between the two bodies.

(vi) As G is very small, hence gravitational forces are very small, unless one (or both) of the mass is huge.