Two rings with uniform distribution of mass having masses M and 2M respectively. having same radius are placed coaxially as shown in figure. Graviational potential at point P is $- \frac{GM}{R} [\frac{1}{\sqrt{x}} + \frac{y}{\sqrt{5}}]$ Find the value of xy.

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Detailed Solution

V on the axis of a circular ring of mass’M’ radius ‘R’ at a distance ‘x’ from centre is

$V = \frac{- GM}{\sqrt{R^{2} + x^{2}}};$ $∴ V_{1} = \frac{- GM}{\sqrt{R^{2} + R^{2}}} = \frac{- GM}{\sqrt{2} R}$

$V_{2} = \frac{- G 2 M}{\sqrt{R^{2} + 4 R^{2}}} = \frac{- 2 GM}{\sqrt{5} R}$

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

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