Question

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

• A. $-\frac{GM}{R}[\frac{1}{\sqrt{2}}+\frac{2}{\sqrt{5}}]$
• B. $-\frac{GM}{R}[1+\frac{2}{2}]$
• C. zero
• D. cannot be determined form the given information

Answer 1

Answer: (A)

$-\frac{GM}{R}[\frac{1}{\sqrt{2}}+\frac{2}{\sqrt{5}}]$

Gravitational potential due to ring (M,R) at an axial point which is x unit away from the center, $V=-\frac{GM}{\sqrt{x^2+R^2}}$

Thus gravitational potential at P, $V_P=(-\frac{GM}{\sqrt{R^2+R^2}})+(-\frac{G\left(2M\right)}{\sqrt{(2R{)}^2+R^2}})$

$V_P=-GM[\left(\frac{1}{\sqrt{2R^2}}\right)+\left(\frac{2}{\sqrt{5R^2}}\right)]$

$V_P=-\frac{GM}{R}[\frac{1}{\sqrt{2}}+\frac{2}{\sqrt{5}}]$