Can we measure $10^{-12}\ \mathrm{N}$ force? I would be interested to measure a very small force, say in the order of $10^{-12}\ \mathrm{N}$? Is this possible? What equipment is needed?
My setup
Assume that I have a relatively heavy machine say between 5-10 kg that I want to measure if it produces this thrust, which according to calculations should be of this feeble magnitude. But (according to the predictions) this should be periodic, with a frequency of about 200Hz and it should last for about a quarter of the time period. I should also mention that this apparatus is expected to vibrate (a little), since inside a disk is supposed to be rotated at about 12k rpm.
My research
I have read about torsion balance as a possible method. I am also thinking about some piezo-electric crystals. Would be feasible?
What piezoelectric cells would be recommended? I read that the Atomic Force Microscopy devices are also implemented using piezoelectric materials.
 A: The question is, a $10^{-12}\rm\,N$ force applied to what. A force of $10^{-12}\rm\,N$ applied to a hydrogen atom, with mass $10^{-27}\rm\,kg$, would produce an acceleration $F/m = 10^{+15}\rm\,m/s^2$.
A torsion pendulum is absolutely a way to allow very feeble forces to cause observable, macroscopic motion.
My favorite underrated classic paper is Beth’s 1936 experiment which transferred angular momentum from a beam of circularly polarized light to a torsion pendulum. There was a parity-violation experiment in the 1960s that used a torsion pendulum as a detector for circular polarization in photons emitted from a parity-violating weak interaction process. And the 2001-ish proposal that gravity might be non-Newtonian at short distances has been mostly ruled out by torsion-pendulum measurements of gravitational attraction between coin-sized test masses.
In all of those cases, you accumulate the very small force by doing the experiment many times, repeating at a frequency near the resonant frequency of the pendulum.
For more direct measurements of very tiny forces, you might read about the operation of an atomic force microscope.
A: Not to answer your question completely but on the AFM point- we can model the AFM cantilever as a spring in contact mode. The spring constant can be ~0.2 N/m or lower.
Now you can, without much work get z-direction sensitivities of 0.1 nm without too much trouble, taking into account experimental noise.
Very roughly speaking this gives a resolution of 0.02 nN or 20 pN. So, not far off for measuring forces on a tiny cantilever. However, for a mass that size I'm not sure it's possible.
