A small particle of mass m move in such a way the potential energy `U = (1)/(2) m^(2) omega^(2) r^(2)` where omega 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 √n

Explanation

The force at a distance r is
`F = (du)/(dr) = - m omega^(2)r`….(i)
Suppose the particle moves along a circle of radius r .The net force on it should be `mu^(2)//r` along readius, Comparing with (i),
`(m upsilon^(2))/®= m omega^(2) r`
`or, upsilon = omega r`...(ii)
The quantization of angular momentum given
`m upsilon e = ( pi h)/(2pi)`
or, `upsilon = (pi h)/(2 pi m r)` ...(iii)
From (ii) and (iii),
`r = ((pi h)/(2 pi m r))^(1//2)`
thus, the radius of the nth orbit is propotional to in