Question

A ring shaped object of radius $R$ contains mass $M$ distributed non-uniformly. Which of the following graphs correctly represents the variation of gravitational potential at a point on the axis of ring?
(Assume that ring is in YZ plane with origin as centre)

• A.
• B.
• C.
• D.

Answer 1

The correct option is D


Any point $\text{P}$ on the axis of ring is equidistant from all the point masses present on the ring.

Thus, non- uniform distribution will not affect the gravitational potential at any axial point.

So, the gravitational field on any axial point at distance $x$ from the centre of the ring is,
$$V=-\frac{GM}{\sqrt{x^2+R^2}}$$

As the value of $x\uparrow$ on both sides w.r.t. centre of ring, the potential will decreases.

For $x=0,$ the point $\text{P}$ lies at centre of ring, the potential at this point is

$\Rightarrow V_c=-\frac{GM}{{\sqrt{R}}^2}=-\frac{GM}{R}$

Thus, potential at the centre of ring is maximum and finite and decreases on both sides of axis continuously.

$\therefore$ Correct graphical representation is option $\left(d\right)$.

Why this question ? To make the student understand that the potential at any axial point due to a ring having non-uniformly distributed mass can be calculated simply as done in this problem.