Problem:
Four particles of equal mass are moving around a circle of radius r due to their mutual gravitational attraction. Find the angular velocity of each particle.

Solution:
To find the angular velocity of each particle, we need to consider the gravitational attraction between the particles and the centripetal force acting on each particle.

Step 1: Setting up the problem
- We have four particles of equal mass moving around a circle of radius r.
- Let's label the particles as A, B, C, and D.
- The gravitational attraction between any two particles is given by the formula F = G * (m^2) / r^2, where G is the gravitational constant, m is the mass of each particle, and r is the distance between the particles.
- The centripetal force acting on each particle is given by the formula F = m * (v^2) / r, where m is the mass of each particle, v is the velocity of each particle, and r is the radius of the circle.

Step 2: Finding the gravitational force
- The gravitational force between particle A and particle B is equal to the gravitational force between particle B and particle A.
- Similarly, the gravitational force between particle A and particle C is equal to the gravitational force between particle C and particle A, and so on.
- Therefore, the total gravitational force acting on particle A is the sum of the gravitational forces between A and B, A and C, and A and D.
- Similarly, the total gravitational force acting on particle B is the sum of the gravitational forces between B and A, B and C, and B and D.
- We can apply the same logic to find the total gravitational forces acting on particles C and D.

Step 3: Finding the centripetal force
- Each particle is moving in a circular path with radius r.
- The centripetal force acting on each particle is provided by the gravitational force between the particles.
- Therefore, the centripetal force acting on each particle is equal to the total gravitational force acting on that particle.
- We can set up equations for each particle by equating the gravitational force to the centripetal force and solving for the velocity of each particle.

Step 4: Finding the angular velocity
- The angular velocity of each particle is given by the formula ω = v / r, where ω is the angular velocity, v is the velocity of each particle, and r is the radius of the circle.
- We can substitute the values of the velocities we found in step 3 into this formula to find the angular velocity of each particle.

In conclusion, to find the angular velocity of each particle moving around a circle of radius r due to their mutual gravitational attraction, we need to find the gravitational forces between the particles, set up equations for the centripetal forces, solve for the velocities, and finally calculate the angular velocities using the formula ω = v / r.