What if we do replace the factor of \(\cfrac{1}{3}\) that results from considering only one of the three dimensions in space,
\(3*P=3*\cfrac{2}{3}\rho_n*\cfrac{4}{3}\pi{(2c)^3}*\cfrac{1}{2}m_av^2_{rms}\)
and instead consider a volume in 3D space.
We have,
\(\rho_n=\cfrac{dn_s}{df}=\cfrac{8\pi}{c^3}f^2\)
\(P=3*\cfrac{2}{3}*\cfrac{8\pi f^2}{c^3}*\cfrac{4}{3}\pi{(2c)^3}*\cfrac{1}{2}m_av^2_{rms}\)
\(\cfrac{8^3}{6}\pi^2*f^2*m_ac^2=1*\cfrac{4}{3}\pi*f^3*m_ac^2\)
\(f=64*\pi=201.06\,Hz\) wrong!
Are we done? Don't freak out with \(67.02\,Hz\), do \(201.06\,Hz\)...
It won't turn off???
