Question

The particles $A$ and $B$ of mass $m$ each are separated by a distance $r$. Another particle $C$ of mass $M$ is placed at the midpoint of $A$ and $B$. Find the work done in taking $C$ to a point equidistant $r$ from $A$ and $B$ without acceleration ( $G=$ Gravitational constant and only gravitational interaction between $A, B$ and $C$ is considered)

• A. $\frac{ GMm }{r}$
• B. $\frac{2 G M m}{r}$
• C. $\frac{3 GMm }{r}$
• D. $\frac{4 G M m}{r}$

Answer 1

Answer: (B)

Since particle $C$ is moved without any acceleration,
$\Rightarrow \Delta K . E =0$
$\Rightarrow $ Work done by external agent $+W_{\text {gravitation }}=0 $
$\Rightarrow $ Work done by external agent $=-W g $
$=-(-\Delta U) $
$=\Delta U $
$=U_{t}-U_{\text {in }}$
$U_f = -\frac{GMm}{r/2} - \frac{GMm}{r} = -\frac{2GMm}{r}$
$U_i = -\frac{GMm}{r/2} -\frac{GMm}{r/2} = -\frac{4GMm}{r}$
$\Rightarrow $ Work done $=\frac{2 G M m}{r}$