A particle is moving along a horizontal circle ‘r’ under a centripetal force –c/r² where’c’ is a constant. Then. the total energy of the particle is

Given,

centripetal force is $\frac{-c}{r^2}$

we know that kinetic energy is $=\frac{1}{2}m{v}^2=\frac{1}{2}\frac{c}{r}$

potential energy = $-{\int }_{\infty }^{r}Fdr$

Because it is known that potential energy is 0 at infinity, the lower limit has been taken as

$=-{\int }_{\infty }^r\frac{c}{r^2}dr$

$=-c{\int }_{\infty }^r{r}^{-2}dr$

$=-c{{\left|\frac{r^{-1}}{-1}\right|}^r}_{\infty }$

$=\frac{-c}{r}$

$Total energy=\frac{c}{2r}-\frac{c}{r}=-\frac{c}{2r}$

Hence the correct answer is $-\frac{c}{2r}.$