Final answer:
Two particles of equal mass m orbiting each other due to gravity will have a speed given by the square root of (Gm/(2r)), where G is the gravitational constant, m is the mass, and r is the radius of the orbit.
Step-by-step explanation:
The question involves the concept of particles orbiting each other due to gravitational attraction, which is a topic in physics, specifically within the field of classical mechanics. When two particles of equal mass m orbit around a common center of mass due to their mutual gravitational attraction, we can find the speed of each particle using the formula for centripetal force arising from gravitational attraction.
For an object orbiting in a circle of radius r, the gravitational force provides the necessary centripetal force for circular motion. The gravitational force between two masses m and M separated by a distance r is given by Newton's universal law of gravitation:
Fg = G(mM/r2),
where G is the gravitational constant. Since the particles have equal masses and orbit around their common center of mass, the distance between them is 2r, so each particle experiences a centripetal force due to gravity:
Fc = mv2/r,
Equating the gravitational force and the centripetal force, we get:
G(m2/(2r)2) = mv2/r or Gm/r = 2v2
Solving for v, the speed of each particle, yields:
v = √(Gm/(2r))