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

Options

A. `n^(2)`
B. n
C. `sqrt(n)`
D. None of these

Answer

C

Explanation

`-(1)/(2) xx P.E = K.E`
` = - (1)/(2) (-(1)/(2) mkr^(2)) = (1)/(2)mv^(2) , mvr = (nh)/(2pi)`
`v^(2) = (n^(2) h^(2))/(4pi^(2) m^(2) r^(2)) , r^(4) = (n^(2) h^(2))/(2pi^(2) m^(2)k)`
`or r prop sqrt(n)`