A particle of mass m is moving in a circular orbit under the influence of the central force F(r)=-kr, corresponding to the potential energy V(r)=kr2/2, where k is a positive force constant and r is the radial distance from the origin. According to the Bohr's quantization rule, the angular momentum of the particle is given by L=n , where =h/(2 ), h is the Planck's constant, and n a positive integer. If v and E are the speed and total energy of the particle, respectively, then which of the following expression(s) is(are) correct?
- r2=n 1mk
- v2=n km3
- Lmr2= km
- E= n 2 km
Answer
(1) r2=n 1mk, (2) v2=n km3, (3) Lmr2= km
Solution
Centripetal: mv2r=kr ⇒ v=r km, i.e. = vr= km.
L=mvr=r2 mk=n ⇒ r2=n 1mk → (A) correct.
v2=r2 km=n km mk=n km3 → (B) correct.
Lmr2= vr= km → (C) correct.
E= 12mv2+ 12kr2=kr2=n km → (D) wrong (factor 12).
Answer: (A), (B), (C)