You are designing a sensor to detect biological pathogens in drinking water. Your sensor is based on a cantilever which is micromachined from silicon, with a length of 300?m, a width of 100 ?m, and a thickness of 1 ?m. (Assume these dimensions are exact.) Because silicon is a crystal, the relevant value of Young's modulus depends on the direction relative to the crystal axes; assume it is exactly 165 GPa for this problem. Assume the density of silicon is exactly 2,330 kg/m3.
(a) What is the oscillation frequency f for your cantilever? (Give a result with eight signi?cant ?gures, assuming all the numbers used as inputs are exact.)
(b)You now coat your cantilever with a layer of receptor molecules, which can bind the pathogen. Each receptor molecule has a molecular weight of 278.1 u, where 1 u = 1.66053886 × 10?27 kg. The receptor molecules coat all the exposed surfaces of your cantilever, each occupying an average area of exactly (5 nm)^2. Assume that this distributed mass counts toward the effective mass in the same way that the mass of the silicon itself counts. What is the new oscillation frequency?
(c) Each of the pathogens you must detect has a mass of 20,000 u. Assuming that 1% of the receptor molecules bind a pathogen, what is the new oscillation frequency? (Again, assume that the distributed mass of the bound pathogens counts the same way that the mass of the silicon itself counts.)
(d) What is the minimum time for which you must make a frequency measurement to detect the difference between the frequencies in parts (b) and (c)?