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JEE Main 202227 JuneEvening Shift
Exam
JEE Main
Year
2022
Date
27 June
Shift
Evening Shift
Subject
Physics
Chapter
Gravitation

JEE Main 2022 · 27 June · Evening Shift · Physics · Gravitation

Four spheres each of mass m form a square of side d (as shown in...

Four spheres each of mass mm form aa square of side dd (as shown in figure). A fifth sphere of mass MM is situated at the centre of square. The total gravitational potential energy of the system is

Four spheres each of mass m form a square of side d (as shown in figure). A fifth sphere of mass M is situated at the

Options

  1. A−Gmd- \dfrac{Gm}{d}\left[ \left(4 + \sqrt{2}\right)$m +4\sqrt{2}$M\right]
  2. B−Gmd- \dfrac{Gm}{d}\left[ \left(4 + \sqrt{2}\right)$M +4\sqrt{2}$m\right]
  3. C−Gmd[3m2+42M]- \dfrac{Gm}{d}\left[3{m}^{2} + 4\sqrt{2}M\right]
  4. D−Gmd[6m2+42M]- \dfrac{Gm}{d}\left[6{m}^{2} + 4\sqrt{2}M\right]

Correct answer

A. −Gmd- \dfrac{Gm}{d}\left[ \left(4 + \sqrt{2}\right)$m +4\sqrt{2}$M\right]

Step-by-step solution

Four spheres each of mass m form a square of side d (as shown in figure). A fifth sphere of mass M is situated at the

The total gravitational potential energy of the system is

Usystem=(U12+U23+U34+U41)+(U13+U24)+(U15+U25+U35+U45){U}_{system} = \left({U}_{12} + {U}_{23} + {U}_{34} + {U}_{41}\right) + \left({U}_{13} + {U}_{24}\right) + \left({U}_{15} + {U}_{25} + {U}_{35} + {U}_{45}\right)

=−4Gmmd−2Gmmd2−42GMmd= - \dfrac{4Gmm}{d} - \dfrac{2Gmm}{d\sqrt{2}} - \dfrac{4\sqrt{2}GMm}{d}

=−(4+2)Gm2d−42GMmd= - \dfrac{\left(4 + \sqrt{2}\right)G{m}^{2}}{d} - \dfrac{4\sqrt{2}GMm}{d}

=−Gmd[(4+2)m+42M]= - \dfrac{Gm}{d}\left[\left(4 + \sqrt{2}\right)m + 4\sqrt{2}M\right]

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