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Understanding gravitational force is key when working with celestial bodies, like stars. Gravitational force is a natural phenomenon by which all things with mass are brought toward one another. In this context, the gravitational force between two stars of mass \( M \) can be calculated using Newton's Law of Universal Gravitation: \( F = \frac{G M^2}{L^2} \).

Here, **\( G \)** is the gravitational constant, \( 6.674 \times 10^{-11} \text{ m}^3 \text{ kg}^{-1} \text{ s}^{-2} \). This constant allows us to compute the attractive force based on the mass of the objects and the distance between them.

Key points to remember about gravitational force:

• It is always attractive.
• It depends on the product of the masses involved.
• It decreases with the square of the distance between the centers of the two masses.

In an equilateral triangle of stars, each star exerts a gravitational pull on the others, contributing to the net force acting towards the center of the formation.

Centripetal force is crucial in understanding rotational motion and the behavior of objects moving along a curved path. When any object moves in a circle, it constantly changes direction, which requires a force directed towards the center of the circle, known as centripetal force.

For a star rotating around the center of an equilateral triangle, the centripetal force required to maintain its circular path can be expressed as \( F_c = \frac{M v^2}{R} \), where:

• \( M \) is the mass of the star.
• \( v \) is the velocity of the star.
• \( R \) is the radius of the circular path, specifically \( \frac{L}{\sqrt{3}} \), derived from the geometry of the equilateral triangle.

The gravitational forces between the stars provide the necessary centripetal force for this rotational motion. Understanding how these forces balance allows us to solve for the speed of the stars.

An equilateral triangle is a geometric shape where all three sides are of equal length, and all three interior angles are \( 60^\circ \). This symmetry plays a significant role in the problem presented, as it dictates the way the forces interact among the stars.

In the context of three stars forming an equilateral triangle, the centroid acts as the point of rotation. The centroid is the spot where all the "mass" of the triangle can be considered to be concentrated, and it is equidistant from all the vertices.

Important properties of an equilateral triangle:

• All sides, by definition, are the same length, which simplifies calculations of forces.
• Its symmetry means any vertex is always \( 60^\circ \) apart from another.
• The distance from the centroid to any vertex (\( R = \frac{L}{\sqrt{3}} \)) is crucial in determining rotational dynamics.

In our scenario, using these geometrical properties allows us to translate the gravitational interactions into useful centripetal forces, leading us to calculate the velocity of the stars in the rotating system.