Explanation:

In this scenario, we have two particles each with mass m separated by a distance d. Both particles are moving in a uniform circle under the action of their mutual gravitational force of attraction. We need to determine the speed of each particle.

Key points to consider:

1. Gravitational Force: The force of gravitational attraction between two objects is given by Newton's Law of Universal Gravitation:
- F = G * (m1 * m2) / r^2
- Where F is the force of attraction, G is the gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between their centers.

2. Circular Motion: For an object to move in a circle, there must be a centripetal force acting towards the center of the circle. In this case, the gravitational force provides the necessary centripetal force.

3. Centripetal Force: The centripetal force required for an object of mass m to move in a circle of radius r with speed v is given by:
- Fc = m * v^2 / r

Derivation of speed:

Considering the gravitational force acting as the centripetal force, we can equate the two equations:
- G * (m1 * m2) / r^2 = m * v^2 / r

Simplifying the equation, we can solve for the speed v:
- v^2 = G * (m1 * m2) / r

Taking the square root of both sides, we get:
- v = sqrt(G * (m1 * m2) / r)

Final answer:

The speed of each particle moving in a uniform circle under the action of their mutual gravitational force of attraction is given by the equation:
- v = sqrt(G * (m1 * m2) / r)

Conclusion:

By calculating the square root of the product of the gravitational constant and the product of the masses of the particles, divided by the distance between them, we can determine the speed at which each particle is moving in the uniform circle.

9m /a