A small particle of mass m move in such a way the potential energy `(U = (1)/(2) m^(2) omega^(2) r^(2))` when 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 , show that radius of the nth allowed orbit is proportional to in

Explanation

To solve the problem, we need to show that the radius of the nth allowed orbit is proportional to \( n \). We start with the given potential energy and use the principles of circular motion and Bohr's quantization condition.

### Step-by-Step Solution:

1. **Given Potential Energy**:
The potential energy of the particle is given by:
\[ U = \frac{1}{2} m \omega^2 r^2 \]

2. **Calculate the Force**:
The force acting on the particle can be derived from the potential energy using:
\[ F = -\frac{dU}{dr} \]
We differentiate \( U \):
\[ F = -\frac{d}{dr} \left( \frac{1}{2} m \omega^2 r^2 \right) = -m \omega^2 r \]

3. **Centripetal Force**:
For a particle moving in a circular orbit, the centripetal force required to keep the particle in circular motion is given by:
\[ F_c = \frac{mv^2}{r} \]
Setting the magnitudes equal, we have:
\[ m \omega^2 r = \frac{mv^2}{r} \]

4. **Express Velocity**:
Rearranging the above equation gives:
\[ v^2 = \omega^2 r^2 \]
Taking the square root:
\[ v = \omega r \]

5. **Bohr's Quantization Condition**:
According to Bohr's model, the angular momentum \( L \) is quantized:
\[ L = mvr = n \frac{h}{2\pi} \]
Substituting \( v \) from the previous step:
\[ L = m(\omega r)r = m \omega r^2 \]
Setting this equal to the quantization condition:
\[ m \omega r^2 = n \frac{h}{2\pi} \]

6. **Solve for Radius \( r \)**:
Rearranging gives:
\[ r^2 = \frac{n h}{2\pi m \omega} \]
Taking the square root:
\[ r = \sqrt{\frac{n h}{2\pi m \omega}} \]

7. **Proportionality**:
Since \( \frac{h}{2\pi m \omega} \) is a constant, we can express the radius \( r \) in terms of \( n \):
\[ r \propto \sqrt{n} \]

### Conclusion:
Thus, we have shown that the radius of the nth allowed orbit is proportional to \( \sqrt{n} \):
\[ r_n \propto \sqrt{n} \]