Question

A particle of mass m is moving around the origin with a constant force F pulling it towards the origin. If Bohr model is used to describe its motion, the radius of the n^th orbit and the particle’s speed v in the orbit depend on n as ______.

Options

• r ∝ n^1/3; v ∝ n^1/3

• r ∝ n^1/3; v ∝ n^2/3

• r ∝ n^2/3; v ∝ n^1/3

• r ∝ n^4/3; v ∝ n^−1/3

Answer 1

A particle of mass m is moving around the origin with a constant force F pulling it towards the origin. If Bohr model is used to describe its motion, the radius of the n^th orbit and the particle’s speed v in the orbit depend on n as r ∝ n^2/3; v ∝ n^1/3.

Explanation:

F = `(mv^2)/r  ("constant given")   ...(i)`

From Bohr’s quantisation condition,

`mvr = (nh)/(2pi)   ...(ii)`

`r = (nh)/(2pimv)`

Putting the value of r in (i), we get

F = `(mv^2)/(nh) xx 2pimv`

= `(2pim^2v^3)/(nh)`

v³ ∝ n

v ∝ n^1/3

From (ii)

r ∝ `n/v ∝ n/n^(1//3)`

= n^2/3