(III) Three bodies of identical mass M form the vertices of...

(III) Three bodies of identical mass $M$ form the vertices of
an equilateral triangle of side $\ell$ and rotate in circular orbits
about the center of the triangle. They are held in place by
their mutual gravitation. What is the speed of each?

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Key Concepts

Newton’s Law of Universal Gravitation

This law states that every point mass attracts every other point mass by a force that is proportional to the product of their masses and inversely proportional to the square of the distance between their centers. It is fundamental when calculating the gravitational force that acts between bodies in any orbital configuration.

Centripetal Force in Circular Motion

In any circular motion, an object must experience an inward force that acts as the centripetal force, keeping it in its curved path. This concept is key when setting the required centripetal force equal to the net gravitational force acting on a body, thereby determining its orbital speed.

Geometry of Equilateral Triangles

The geometric properties of equilateral triangles, such as the constant side lengths and the specific distance from a vertex to the center (centroid), are essential. These properties allow for the determination of the effective radius of each body’s circular orbit when three bodies form such a triangle.

Balance of Forces in Orbital Motion

The idea that the inward gravitational forces can exactly provide the centripetal force necessary for circular orbits is a central concept in orbital mechanics. In symmetric systems, this balance simplifies the computation of orbital speeds and ensures stable, mutual orbits.

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