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Theoretical Question 3 - Theoretical Question 3 Page 1/3...

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Theoretical Question 3 Page 1/3 Theoretical Question 3 Atomic Probe Microscope Atomic probe microscopes (APMs) are powerful tools in the field of nano-science. The motion of a cantilever in APM can be detected by a photo-detector monitoring the reflected laser beam, as shown in Fig. 3.1. The cantilever can move only in the vertical direction and its displacement z as a function of time t can be described by the equation F kz dt dz b dt z d m = + + 2 2 , (3.1) where m is the cantilever mass, 2 0 ω m k = is the spring constant of the cantilever, b is a small damping coefficient satisfying 0 ) / ( 0 > >> m b ω , and finally F is an external driving force of the piezoelectric tube. Figure 3.1 A schematic diagram for a scanning probe microscope (SPM). The inset in the lower right corner represents a simplified mechanical model to describe the coupling of the piezotube with the cantilever. [Part A] (a) [1.5 points] When t F F ω sin 0 = , ) ( t z satisfying Eq. (3.1) can be written as ) sin( ) ( φ ω = t A t z , where 0 > A and π φ 0 . Find the expression of the V
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