Question

INFINITE POINT UNIT MASSES ARE PLACED AT 1 CM, 2 CM , 4CM,8 CM 16CM......... from origin on x axis. Find gravitational field at origin. NO UNWANTED ANSWERS PLZ​

Answer 1

The magnitude of the net gravitational field at origin will be,

$\sf{\longrightarrow E=\dfrac{G(1)}{(1\times10^{-2})^2}+\dfrac{G(1)}{(2\times10^{-2})^2}+\dfrac{G(1)}{(4\times10^{-2})^2}+\dfrac{G(1)}{(8\times10^{-2})^2}+\dots}$

$\sf{\longrightarrow E=\dfrac{G}{10^{-4}}\left[\dfrac{1}{1^2}+\dfrac{1}{2^2}+\dfrac{1}{4^2}+\dfrac{1}{8^2}+\dots\right]}$

Or,

$\sf{\longrightarrow E=6.67\times10^{-7}\left[1+\dfrac{1}{4}+\dfrac{1}{4^2}+\dfrac{1}{4^3}+\dots\right]}$

The series is an infinite GP of first term $\sf{a=1}$ and common ratio $\sf{r=\dfrac{1}{4}<1.}$ Then,

$\sf{\longrightarrow E=6.67\times10^{-7}\times\dfrac{1}{1-\dfrac{1}{4}}}$

$\sf{\longrightarrow E=6.67\times10^{-7}\times\dfrac{4}{3}}$

$\sf{\longrightarrow\underline{\underline{E=8.89\times10^{-7}\ N\,kg^{-1}}}}$