Question

A particle of mass m is moving in a horizontal circle of radius r under centripetal force equal to $-\frac{K}{r^2}$, where K is a constant. The total energy of the particle is:

• A. $-\frac{K}{2r^2}$
• B. $\frac{K}{2r}$
• C. $-\frac{K}{2r}$
• D. $-\frac{K}{r^2}$

Answer 1

Answer: (C)

$-\frac{K}{2r}$

Centripetal force $=\frac{m{v}^2}{r}=\frac{k}{r^2}$ ...(given)

Kinetic Energy $=\frac{1}{2}m{v}^2=\frac{1}{2}\frac{k}{r}$

Potential Energy $=-{\int }_{\infty }^{r}Fdr$

the lower limit has been taken as$\infty$ because potential energy is zero at infinty

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

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

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

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

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