Four particles of equal masses M move along a circle of radius R under the action of ual gravitational attraction maintaining a square shape. The speed of each particle is

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Four masses at the vertices of a square inscribed in a circle of radius R: side $s = R 2$ , diagonal $2 R$ .

Force on one mass: two neighbours along the sides plus the opposite mass, projected toward the centre:

$F = 2 \cdot \frac{G M ^{2}}{2 R ^{2}} cos 4 5^{\circ} + \frac{G M ^{2}}{4 R ^{2}} = \frac{G M ^{2}}{R ^{2}} (\frac{1}{2 2} + \frac{1}{4}) = \frac{G M ^{2}}{R ^{2}} \cdot \frac{2 2 + 1}{4}$

$\frac{M v ^{2}}{R} = F \Rightarrow v = \frac{GM}{R} (\frac{2 2 + 1}{4})$

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