Finding Quantized Speeds, Radii & Energies with Bohr's Rule

[nymphidius]

Homework Statement

Use Bohr's rule, L=n$\hbar$, to find the quantized speeds, radii, and energies for circular orbits of a particle of mass m bound to the origin by the Hooke's-law force, F(r) = -Kr.

Homework Equations

mass=m

ƩF=ma

F(r)=-kr

L=n$\hbar$

p=mvr

mvr=n$\hbar$

r=n$\hbar$/mv (a)

The Attempt at a Solution

mv²/r=-kr

mv²=-kr² (b)

subbing (a) into (b)

mv²=-kn²$\hbar²$/m²v²

solving for v...

(v²)²=-kn²$\hbar²$/m³

v=(-kn²$\hbar²$/m³)^¼

and this is where I get stuck because of the negative sign in the 4th root of v. If I make ma=kr it works out, BUT it seems cheap just to change the law from -kr to kr.

 

[BigJDubs]

Am I missing something? I am also stuck on the energy part. I figured it had something to do with 1/2mv²+kr=-E but I don't know how to incorporate the n\hbar part of Bohr's rule.Thank you for any help!