A small particle of mass m moves in such a way that its potential energy U=1/2 m ω² r² where ω is constant and r is the distance of the particle from origin. Assuming Bohr's quantization of momentum and circular orbit, the radius of nᵗʰ orbit will be proportional to,

Solution

Physics sir · 3 steps

The force grows with r here, not as 1/r², so find v from the force first and only then quantise.

1. Force from U. The force is F = -dU/dr = -mω² r. It points towards the origin, and its size mω² r supplies the centripetal force.
2. Speed on the orbit. Set mv²/r = mω² r, which gives v = ω r. Unlike the hydrogen atom, the speed grows with the radius.

   mvr = nh/2π ⇒ mω r² = nh/2π ⇒ r² = nh/(2π mω)

3. So. Since m, ω and h are constants, r² ∝ n, so r ∝ √n. The answer is (a).

The idea to keep. For any Bohr-type problem, get v in terms of r from the force balance, then put it into mvr = nh/2π. The scaling of r depends on the force law, not on a memorised formula.

Answer from the NTA final answer key. Solution by PrepWiser.