Problem $6\mathrm{.}1$ Atomic Force Microscope $(20$ points$)$

Recall we introduce the atomic force microscope $($AFM$)$ in Biomaterials course. The AFM can be used to evaluate the mechanical properties of cells and ligand$-$receptor interactions, by measuring the deflection of a very small cantilever beam when a conical tip with an extremely sharp point makes contact with a material.

Assume: the beam is $\backslash (100\backslash$mu $\backslash$mathrm$\{$~m$\}\backslash )$ long from the rigidly secured base to the tip, $\backslash (10\backslash$mu $\backslash$mathrm$\{$~m$\}\backslash )$ wide, and $\backslash (1\backslash$mu $\backslash$mathrm$\{$~m$\}\backslash )$ thick. The cantilever beam is usually constructed from silicon $($Si$)$ or silicon nitride $($Si$3$N$4)$ with an elastic modulus, E$,$ that is $200$ GPa in this problem. Neglect the mass of the beam and the tip. Prior to making contact, you model the cantilever beam as being flat.

As a first step to calibrate and work with the AFM, you decide to find the contact force, F $,$ given a measured vertical deflection of the tip of d meters?