An atomic force microscope cantilever, of rectangular cross section, has a length, l, of 200μm, a width, b, of 20μm and a depth, h, of 1μm. It is made of silicon nitride, which has a Young's modulus, E, of 300GPa. What is the maximum force that can be applied with the cantilever, if the maximum strain in the beam is to be limited to 0.1%?
Pmax (in μN) = ?
To calculate the maximum force that can be applied with the cantilever, we need to determine the maximum stress that the cantilever can withstand before reaching the maximum strain limit.
First, let's calculate the moment of inertia (I) of the rectangular cross-section of the cantilever. The moment of inertia of a rectangular cross-section is given by the formula:
I = (b * h^3) / 12
Substituting the given values, we have:
I = (20μm * (1μm)^3) / 12
I = (20 * 1 * 1 * 1) / 12
I = 20 / 12
I = 1.67 μm^4
The maximum stress (σ_max) that the cantilever can withstand is given by Hooke's Law:
σ_max = E * ε_max
Where E is the Young's modulus and ε_max is the maximum strain limit. Substituting the given values, we get:
σ_max = 300GPa * (0.001)
σ_max = 300N/μm^2 * (0.001)
σ_max = 300μN/μm^2
Now, to calculate the maximum force (P_max), we use the equation for stress:
σ = (M * y) / I
Where M is the bending moment and y is the distance from the neutral axis to the point of interest (in this case, the top of the cantilever).
Considering a cantilever, the maximum bending moment (M_max) occurs at the base, and y is the distance from the neutral axis to the top of the cantilever (which is half of the height, h/2).
M_max = σ_max * I / y
M_max = (300 μN/μm^2) * (1.67 μm^4) / (0.5 μm)
M_max = 1005 μN μm
Finally, to calculate the maximum force (P_max), we use the equation for the bending moment:
M = P_max * l
Substituting the values we have:
1005 μN μm = P_max * 200 μm
P_max = 1005 μN μm / 200 μm
P_max = 5.025 μN
Therefore, the maximum force that can be applied with the cantilever is 5.025 μN.