Solving Force Integration Problem in UFM

[Caveman11]

Hi all,

I am having a problem which is something that I can't think what the reason is. I may well be making a very elementary mistake.

Here it is:

In UFM in order to write the force on the cantilever as a function of the tip surface separation then we must take into account the sample oscillations.

The sample oscillations are given by: \begin{equation}Acos(\omega t)\end{equation} where A is the oscillation amplitude.

By subtracting this from the tip sample z displacement we obtain the time dependant separation.

\begin{equation}
z -Acos(\omega t)
\end{equation}

Then integrating over one whole oscillation period we obtain

\begin{equation}
F=\int_{0}^{T}F(z -Acos(\omega t)) dt
\end{equation}

However this is where my problem is. In all literature I have read there is a factor of \begin{equation}\frac{1}{2\pi}\end{equation} infront of the integral.

But I can't think where is has come from?


Any response you have is greatly appreciated.

Thanks

 

[BigJDubs]

The factor of $\frac{1}{2\pi}$ is due to the fact that the integral must be scaled to a full oscillation period, which is given by $2\pi$. Since the integral is over one oscillation period, the factor $\frac{1}{2\pi}$ must be included.