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):
5 uN
To find the maximum force that can be applied with the cantilever, we need to calculate the maximum stress that the beam can handle without exceeding the maximum strain limit.
First, let's calculate the maximum stress (σmax) using Hooke's Law:
σmax = E * εmax
Where:
σmax is the maximum stress
E is the Young's modulus
εmax is the maximum strain
Given:
E = 300 GPa = 300 * 10^9 Pa
εmax = 0.1% = 0.001 (expressed as a decimal)
Substituting the values into the equation, we get:
σmax = (300 * 10^9 Pa) * 0.001
= 300 * 10^6 Pa
Now, let's calculate the maximum force (Fmax) the cantilever can handle:
Fmax = σmax * A
Where:
Fmax is the maximum force
σmax is the maximum stress
A is the cross-sectional area of the cantilever
The cross-sectional area (A) of the cantilever can be calculated by multiplying the length (l) and the width (b) of the rectangular cross-section:
A = l * b
Given:
l = 200 μm = 200 * 10^(-6) m
b = 20 μm = 20 * 10^(-6) m
Substituting the values into the equation, we get:
A = (200 * 10^(-6) m) * (20 * 10^(-6) m)
= 4000 * 10^(-12) m^2
= 4 * 10^(-9) m^2
Substituting the values into the equation for Fmax, we get:
Fmax = (300 * 10^6 Pa) * (4 * 10^(-9) m^2)
= 1200 * 10^(-3) N
= 1200 μN
Therefore, the maximum force that can be applied with the cantilever is 1200μN.