Question

Two rings having masses $M$ and $2M$, respectively, having the same radius are placed coaxially as shown in the figure. If the mass distribution on both the rings is non-uniform, then gravitational potential at point $P$ is

• A. $-\frac{GM}{R}[\frac{1}{\sqrt{2}}+\frac{2}{\sqrt{5}}]$
• B. $-\frac{GM}{R}[1+\frac{1}{\sqrt{2}}]$
• C. $-\frac{GM}{R}[\frac{1}{\sqrt{2}}-\frac{2}{\sqrt{5}}]$
• D. Zero

Answer 1

The correct option is A $-\frac{GM}{R}[\frac{1}{\sqrt{2}}+\frac{2}{\sqrt{5}}]$

Given that,
Mass of left ring $=M$
Mass of the right ring $=2M$
Radius of the each ring $=R$

As all the points on the periphery of either ring are at same distance from point $P$. The potential at point $P$ due to the whole ring can be calculated as
$V=-\frac{GM}{\sqrt{R^2+x^2}}$
where $x$ is the axial distance from the centre of the ring and $G$ is the gravitational constant.

This expression is independent of the fact whether the distribution of mass is uniform or non-uniform.

Axial distance of point $P$ from left ring $=R$
Axial distance of point $P$ from right ring $=3R-R=2R$
The gravitational potential at $P$ due to both the rings is
$V_{\text{net}}=-\frac{GM}{\sqrt{R^2+R^2}}-\frac{G\times 2M}{\sqrt{R^2+\left(2R^2\right)}}$
$\Rightarrow V_{\text{net}}=-\frac{GM}{\sqrt{2}R}-\frac{G\times 2M}{\sqrt{5}R}$
$\Rightarrow V_{\text{net}}=-\frac{GM}{R}[\frac{1}{\sqrt{2}}+\frac{2}{\sqrt{5}}]$

Hence, option (a) is correct.

Why this question? Note: Even though the ring has non-uniform mass distribution, the potential expression remains the same. This is because potential is scalar quantity and all points on the ring are equidistant from the point on the axis. Key Concept: The gravitational potential due to a ring on its axis is given by $V=-\frac{GM}{\sqrt{R^2+x^2}}$