Question

Two identical spherical masses of mass $M$ are kept at some distance. Potential energy of the system when a mass $M$ is taken from the surface of one of the sphere to the other

• A. increases continuously.
• B. decreases continuously.
• C. first increases, then decreases.
• D. first decreases, then increases.

Answer 1

The correct option is C first increases, then decreases.

Suppose the identical spheres are at distance $d$.

Let at any time $t$, mass $M$ is at distance $r$ from one of the sphere.


Potential energy of the system at this time,
$U=U\left(\text{between two spheres}\right)+U\left(\text{between left sphere and mass M}\right)$
$+U\left(\text{between right sphere and mass M}\right)$

$\Rightarrow U=-\frac{G{M}^2}{d}-\frac{G{M}^2}{r}-\frac{G{M}^2}{(d-r)}$

$\Rightarrow U=-\frac{G{M}^2}{d}-G{M}^2(\frac{1}{r}+\frac{1}{d-r})$

$\Rightarrow U=-\frac{G{M}^2}{d}-G{M}^2\left(\frac{d}{r(d-r)}\right)$

Now let us take different cases.

$\left(i\right)$ When mass is at point $r=\frac{d}{4}$,
$U_1=-6.3\times \frac{G{M}^2}{d}$ ...........$\left(1\right)$

$\left(ii\right)$ When mass is at point $r=\frac{d}{2}$,
$U_2=-5.0\times \frac{G{M}^2}{d}$ ...........$\left(2\right)$

$\left(iii\right)$ When mass is at point $r=\frac{3d}{4}$,
$U_3=-6.3\times \frac{G{M}^2}{d}$ ...........$\left(3\right)$

Hence, option (c) is the correct alternative.

From $\left(1\right)$, $\left(2\right)$ and $\left(3\right)$, we can see that potential energy of the system first increases then decreases.