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 the gravitational potential at point $P$ is

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

Answer 1

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

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

As all the points on the periphery of either ring are at the 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 of 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 a 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 $V=-\frac{GM}{\sqrt{R^2+x^2}}$