Theoretical Background:
When three particles move in a circle with the same speed but different angular velocities, they will meet at some point in time. The time interval after which they meet can be calculated using the concept of angular displacement and time.

Solution:
Let's assume that the time interval after which the particles meet is t seconds.

Angular Displacement:
The angular displacement of a particle is the angle covered by the particle in a circular path.
Angular displacement can be calculated using the formula:
θ = ωt
where θ is the angular displacement, ω is the angular velocity, and t is the time.

Relationship between Angular Displacement and Linear Displacement:
The linear displacement of a particle is the distance covered by the particle in a circular path.
The linear displacement can be calculated using the formula:
s = rθ
where s is the linear displacement, r is the radius of the circle, and θ is the angular displacement.

Meeting Point:
Since all three particles are moving in a circle of radius r, their linear displacements will be the same when they meet.
Therefore, we can equate their linear displacements:
s_p = s_q = s_r
rθ_p = rθ_q = rθ_r
θ_p = θ_q = θ_r

Calculating Time Interval:
We are given the angular velocities of particles P, Q, and R as 5π, 3π, and 2π respectively.
Using the formula θ = ωt, we can calculate the angular displacement of each particle at time t:
θ_p = 5πt
θ_q = 3πt
θ_r = 2πt

Since θ_p = θ_q = θ_r, we can equate the angular displacements and solve for t:
5πt = 3πt = 2πt
5t = 3t = 2t

By equating the angular displacements, we find that t = 0 is a valid solution. However, this is the initial time when the particles are at their starting positions. We are interested in the time interval after which they meet.

Therefore, we can set t as the time interval after which they meet:
5t = 3t = 2t
t = 0 is not a valid solution.
t = 2π/5

Hence, the time interval after which particles P, Q, and R meet is 2π/5 seconds.