What Are the Permitted Radii Using the Bohr Model for a Fifth Force Potential?

[mcel]

bohr model: permitted radii?

Homework Statement

A new (fifth) force has been proposed that binds an object to a central body through a potential
energy function given by:
U(r) = -Dr^{\frac{-3}{2}}
2 r > 0 and D > 0

(a) What is the (central) force F(r) associated with this potential energy function?
The object (with mass m) is in an orbit around a central body. The central body is electrically
neutral so we can ignore the Coulomb force. Further, we can ignore the gravitational force
between the object and the central body as it is insignificant compared to the “fifth force.”

(b) What is the total energy of this body in orbit? Is it bound to the central body? Explain
your reasoning.

(c) Using the Bohr model that says that angular momentum is quantized, L = mvr = n\hbar, determine the permitted values of the object’s radius r_{n}.

(d) The total energy of the object associated with r_{n} is also quantized. The general form of this energy expression is En = an^{b} where a and b are constants. Determine these constants.

Homework Equations

E=U+K
F=ma (I think..)

The Attempt at a Solution

a:
Got this one:
Differentiate the first formula to dU(r) = {\frac{3}{2}}Dr^{\frac{-5}{2}}

b:
this one too:
E=U+K
with K=\frac{1}{2}mv^{2} and U=-Dr^{\frac{-3}{2}}

c:
I tried {\frac{3}{2}}Dr^{\frac{-5}{2}}=ma and a=\frac{v^{2}}{r_{n}} with gives r_{n}=\frac{9}{4}(\frac{D}{nv\hbar})^{2} but I don't know if it is correct.

d. I don't know how tackle this.

 

[turin]

A new (fifth) force has been proposed that binds an object to a central body through a potential
energy function given by:
U(r) = -Dr^{\frac{-3}{2}}
2 r > 0 and D > 0
(b) What is the total energy of this body in orbit? Is it bound to the central body? Explain
your reasoning.

b:
E=U+K
with K=\frac{1}{2}mv^{2} and U=-Dr^{\frac{-3}{2}}

What is v? Did you answer the rest of the question?

(c) Using the Bohr model that says that angular momentum is quantized, L = mvr = n\hbar, determine the permitted values of the object’s radius r_{n}.

c:
I tried {\frac{3}{2}}Dr^{\frac{-5}{2}}=ma and a=\frac{v^{2}}{r_{n}} with gives r_{n}=\frac{9}{4}(\frac{D}{nv\hbar})^{2} but I don't know if it is correct.

Again, what is v? In this case, you definitely shouldn't have a v-dependent r, unless v is also quantized. Is it? Ultimately, it is probably best to remove v from all considerations, since this problem is getting at the QM, and v is much more CM than QM.

 

[mcel]

I got it. Thanks for the help :)