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A small particle of mass m moves in such...

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 Bohr model of quantization of angular
momentum and circular orbits, find the radius of nth allowed orbit.

Text Solution

Verified by Experts

The force at a distance `r is f = - dU// dr = - 2 ur`.
Suppose`r` be the redius of nth orbit . Then the necessary centripetal force is provided by the above force, then.
`(mv^(2))/( r) = 2 ur` (i)
Further the equatization of angular momentum gives
`mvr = (nh)/(2 pi)` (ii)
Solving Eqs. (i) and (ii) for `r` we get `r = ((n^(2) h^(2))/(8 um pi^(2))) ^1//4` (iii)
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