A small particle of mass `m` moves in such a way that the potential energy `U=ar^(2)` where `a` is a constant and `r` is the distance of the particle from the origin. Assuming Bohr's model of quantization of angular momentum and circular orbits, find the radius of `n^(th)` allowed orbit.

Explanation

To solve the problem, we need to find the radius of the nth allowed orbit for a particle moving under the influence of a potential energy given by \( U = ar^2 \). We will use the principles of classical mechanics and Bohr's quantization condition for angular momentum.

### Step-by-Step Solution:

1. **Identify the Force from Potential Energy**:
The force \( F \) acting on the particle can be derived from the potential energy \( U \). The force is given by:
\[
F = -\frac{dU}{dr}
...