Question

Find ${\int }_{x=\infty }^{x=2R}\frac{GMm}{x^2}dx$ $(G,M$ and $m$ are constants)

• A. $\frac{GMm}{R}$
• B. $-\frac{GMm}{2R}$
• C. $-\frac{GMm}{R}$
• D. $-\frac{GMm}{R^2}$

Answer 1

The correct option is B $-\frac{GMm}{2R}$

${\int }_{x=\infty }^{x=2R}\frac{GMm}{x^2}dx=GMm{\int }_{\infty }^{2R}\frac{1}{x^2}dx$
$=GMm{\int }_{\infty }^{2R}{x}^{-2}dx=GMm{\left[\frac{x^{-1}}{-1}\right]}_{\infty }^{2R}$
$=-GMm{\left[\frac{1}{x}\right]}_{\infty }^{2R}=-GMm[\frac{1}{2R}-\frac{1}{\infty }]$
$=-\frac{GMm}{2R}(\because \frac{1}{\infty }=0)$