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Force or potential energy of masses at the corners of a square

This type of question, asked 4 times: 2021 Paper 20, 27 Jun 2022, 31 Jan 2024, 22 Jan 2026 S1. All 4 questions below, oldest first.

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  1. Question 1

    2021 Paper 20

    21
    A body of mass (2M) splits into four masses {m,M−m,m,M−m}\mathbf{\{ m,M - m,m,M - m\}}, which are rearranged to form a square as shown in the figure. The ratio of Mm\frac{M}{m} for which, the gravitational potential energy of the system becomes maximum is x:1x:1. The value of xx is.
  2. Question 2

    27 Jun 2022

    96
    Four spheres each of mass mm form a 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:

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  3. Question 3

    31 Jan 2024

    7
    Four identical particles of mass mm are kept at the four corners of a square. If the gravitational force exerted on one of the masses by the other masses is (22+132)Gm2 L2\left( \frac{2\sqrt{2}+ 1}{32} \right)\frac{Gm^{2}}{{\text{ }L}^{2}}, the length of the sides of the square is

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  4. Question 4

    22 Jan 2026 S1

    10
    Net gravitational force at the centre of a square is found to be F1F_{1} when four particles having mass M,2M,3MM,2M,3M and 4M4M are placed at the four corners of the square as shown in figure and it is F2F_{2} when the positions of 3M3M and 4M4M are interchanged. The ratio F1F2\frac{F_{1}}{F_{2}} is α5\frac{\alpha}{\sqrt{5}}. The value of α\alpha is ____\_\_\_\_ .

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