In the case of equal speeds, the centripetal accelerations of the two point masses will be different due to the difference in their respective radii.

The first step to solving this problem is to determine the speed, v, of the particles. Using the formula for circular motion v = 2πR/ T, where R is the radius and T is the period of rotation, we can express the speed in terms of R and T.

When comparing the two centripetal accelerations, we can plug in the value of v into the formula for centripetal acceleration, a = v^2/R. In this case, the accelerations will be proportional to the ratio of the radii, as v is the same for both particles. Since it is given that R₁ = 2R₂, we can conclude that the centripetal acceleration of the first particle will be twice that of the second particle.

For the second case, where the periods of rotation are equal, we can use the same formula for circular motion, v = 2πR/T, to express the speeds in terms of the radii and periods. When looking at the formula for centripetal acceleration, a = v^2/R, we can see that the accelerations will be directly proportional to the square of the radii, as v and T are both the same for both particles. Therefore, the first particle with radius R₁ will have a centripetal acceleration four times that of the second particle with radius R₂.