$2.$ Atomic force microscopy $($AFM$)$ is a powerful technique that enables the imaging of almost any type of surface, including polymers, ceramics, composites, glass and biological samples. AFM is used to measure and localize many different forces, including adhesion strength, magnetic forces and mechanical properties. An AFM generates images by scanning a small cantilever over the surface of a sample. The sharp tip on the end of the cantilever contacts the surface, bending the cantilever and changing the amount of laser light reflected into the photodiode.

The tip deflection, $\backslash (\backslash$delta $\backslash ),$ of a cantilever beam is a function of tip load, $\backslash ($ W $\backslash ),$ beam length, $\backslash ($ l $\backslash ),$ second moment of area, $\backslash ($ I $\backslash ),$ and Young's modulus, $\backslash ($ E $\backslash ).$ Perform a dimensional analysis.

NOTE: The second moment of area, also known as area moment of inertia, is a geometrical property of an area which reflects how its points are distributed with respect to an arbitrary axis.

$($a$)$ List all the possible variables $($parameters$)$ with unit.

$($b$)$ How many non$-$dimensional groups exists? $($c$)$ Develop non$-$dimensional groups by performing Buckingham $\backslash (\backslash$pi $\backslash )-$theorem. Find the functional relationship between the non$-$dimensional groups. $($d$)$ Apply the exponent method to find the degree relationship between the tip deflection, $\backslash (\backslash$delta $\backslash )$ and other variables.

$($e$)$ By comparing the results from $($c$)$ and $($d$),$ find the degree of each non$-$dimensional group.