Quantizing Radii of a Mass in Circular Orbit

[quickclick330]

This is yet another sample exam problem which I don't quite know how to approach. Any help would be greatly appreciated.

5) Consider an object of mass m that moves in a circular orbit caused by a central
force given by F=-kr, where k is a constant. Suppose that the Bohr quantization
condition (L=mvr=nh) is applied to this motion. What are the allowed radii?
a. rn=(n4h4/m2k)1/4, n = 1,2,…
b. rn=(n2h2/mk)1/4, n = 1,2,…
c. rn=(n2h2/mk)1/3, n = 1,2,…
d. rn=(n2h2/mk)1/2, n = 1,2,…
e. Not enough information given to compute levels
f. Radii for this force law cannot be quantized

 

[Avodyne]

You first find v by requiring the force to produce the centrepital acceleration of circular motion. Then set L=mvr=nh and solve for r.

 

[ogb p]

I would approach this problem by first understanding the concepts involved. The Bohr quantization condition is a principle in quantum mechanics that states that the angular momentum of an electron in an atom is quantized and can only take on certain discrete values. In this problem, we are applying this concept to the motion of an object in a circular orbit, which is caused by a central force given by F=-kr.

To determine the allowed radii, we can use the equation L=mvr=nh, where L is the angular momentum, m is the mass, v is the velocity, r is the radius, n is a positive integer, and h is Planck's constant. We can rearrange this equation to solve for r, which gives us r=nh/2πm.

Now, we can plug in the given force law F=-kr into the equation for angular momentum, L=mvr, and solve for v. This gives us v=√(k/m)r. Substituting this into the equation for r, we get r=nh/2π√(k/m).

To determine the allowed radii, we need to find the values of n that satisfy this equation. Looking at the given options, we can see that option d, rn=(n2h2/mk)1/2, is the only one that matches this equation. Therefore, the allowed radii are given by the equation r=n2h2/mk, where n is a positive integer.

In conclusion, the correct answer to this problem is d, and the allowed radii for an object in circular orbit under a central force given by F=-kr are given by the equation r=n2h2/mk, where n is a positive integer.