A small particle of mass m moves in such a way that the
potential energy `U = ar^2`, where a is constant and r is the distance of the
particle from the origin. Assuming Bhor model of quantization of angular
momentum and circular orbits, find the rodius of nth allowed orbit.

Explanation

To find the radius of the nth allowed orbit for a small particle of mass \( m \) moving under the potential energy \( U = ar^2 \), we will follow the steps based on the Bohr model of quantization of angular momentum and circular orbits.

### Step 1: Understand the Forces Acting on the Particle
In a circular orbit, the centripetal force is provided by the gradient of the potential energy. The centrifugal force can be expressed as:

\[ F_{\text{centrifugal}} = \frac{mv_n^2}{r_n} \]

where \( v_n \) is the velocity of the particle in the nth orbit and \( r_n \) is the radius of that orbit.

### Step 2: Calculate the Force from Potential Energy
The force acting on the particle can also be derived from the potential energy \( U \):

\[ F = -\frac{dU}{dr} = -\frac{d(ar^2)}{dr} = -2ar \]

Thus, the magnitude of the force is:

\[ F = 2ar \]

### Step 3: Set the Forces Equal
For the particle in circular motion, we equate the centripetal force to the force derived from potential energy:

\[ \frac{mv_n^2}{r_n} = 2ar_n \]

Rearranging gives:

\[ mv_n^2 = 2ar_n^2 \quad \text{(Equation 1)} \]

### Step 4: Apply the Quantization of Angular Momentum
According to Bohr's model, the angular momentum \( L \) of the particle is quantized:

\[ L = m v_n r_n = \frac{n h}{2\pi} \]

From this, we can express the velocity \( v_n \):

\[ v_n = \frac{n h}{2\pi m r_n} \]

### Step 5: Substitute \( v_n \) into Equation 1
Substituting the expression for \( v_n \) into Equation 1:

\[ m \left(\frac{n h}{2\pi m r_n}\right)^2 = 2ar_n^2 \]

This simplifies to:

\[ \frac{n^2 h^2}{4\pi^2 m r_n^2} = 2ar_n^2 \]

### Step 6: Rearranging the Equation
Multiplying both sides by \( r_n^2 \) gives:

\[ n^2 h^2 = 8\pi^2 m a r_n^4 \]

### Step 7: Solve for \( r_n^4 \)
Rearranging for \( r_n^4 \):

\[ r_n^4 = \frac{n^2 h^2}{8\pi^2 m a} \]

### Step 8: Solve for \( r_n \)
Taking the fourth root gives us the radius of the nth allowed orbit:

\[ r_n = \left(\frac{n^2 h^2}{8\pi^2 m a}\right)^{1/4} \]

### Final Result
Thus, the radius of the nth allowed orbit is:

\[ r_n = \left(\frac{n^2 h^2}{8\pi^2 m a}\right)^{1/4} \]

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