(Solved): The cantilever probe of the atomic force microscope has a triangular shape, as shown. As the tip sc ...

The cantilever probe of the atomic force microscope has a triangular shape, as shown. As the tip scans surface of a biomaterial, force $F$ is applied at the tip of the probe, as shown. Calculate the deflection of the tip for a given $F$, knowing that the bending rigidity of the probe changes linearly along its longitudinal axis as $\mathrm{B}(x)={\mathrm{B}}_0(1?x/L)$, where ${\mathrm{B}}_0$ is the bending rigidity at $x=0$ and $L$ is the length of the probe. Compare this deflection with the deflection of a cantilever beam of a uniform cross-section and constant bending rigidity ${\mathrm{B}}_0$. What should be the length the uniform beam in order to have the same end deflection as the triangular beam? Would these two beams have the same end slopes if they have the same end deflections? Explain your answer. Neglect gravity effects.